id sid tid token lemma pos cana-1164 1 1 communications communication NOUN cana-1164 1 2 on on ADP cana-1164 1 3 applied apply VERB cana-1164 1 4 nonlinear nonlinear ADJ cana-1164 1 5 analysis analysis NOUN cana-1164 1 6 issn issn NOUN cana-1164 1 7 : : PUNCT cana-1164 1 8 1074 1074 NUM cana-1164 1 9 - - PUNCT cana-1164 1 10 133x 133x NUM cana-1164 1 11 vol vol NOUN cana-1164 1 12 31 31 NUM cana-1164 1 13 no no NOUN cana-1164 1 14 . . PUNCT cana-1164 2 1 6s 6s NUM cana-1164 2 2 ( ( PUNCT cana-1164 2 3 2024 2024 NUM cana-1164 2 4 ) ) PUNCT cana-1164 2 5 97 97 NUM cana-1164 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 2 7 some some DET cana-1164 2 8 properties property NOUN cana-1164 2 9 of of ADP cana-1164 2 10 the the DET cana-1164 2 11 spectrum spectrum NOUN cana-1164 2 12 of of ADP cana-1164 2 13 the the DET cana-1164 2 14 power power NOUN cana-1164 2 15 digraph digraph NOUN cana-1164 2 16 ๐šช(๐’ ๐šช(๐’ PROPN cana-1164 2 17 , , PUNCT cana-1164 2 18 ๐’Œ ๐’Œ ADJ cana-1164 2 19 ) ) PUNCT cana-1164 2 20 sanjay sanjay PROPN cana-1164 2 21 kumar kumar PROPN cana-1164 2 22 thakur thakur PROPN cana-1164 2 23 ๐Ÿโˆ— ๐Ÿโˆ— PROPN cana-1164 2 24 , , PUNCT cana-1164 2 25 gautam gautam PROPN cana-1164 2 26 chandra chandra PROPN cana-1164 2 27 ray ray PROPN cana-1164 2 28 ๐Ÿ ๐Ÿ NUM cana-1164 2 29 , , PUNCT cana-1164 2 30 pinkimani pinkimani NOUN cana-1164 2 31 goswami goswami NOUN cana-1164 2 32 ๐Ÿ‘ ๐Ÿ‘ PROPN cana-1164 2 33 1department 1department NUM cana-1164 2 34 of of ADP cana-1164 2 35 mathematics mathematic NOUN cana-1164 2 36 , , PUNCT cana-1164 2 37 science science NOUN cana-1164 2 38 college college NOUN cana-1164 2 39 , , PUNCT cana-1164 2 40 kokrajhar kokrajhar PROPN cana-1164 2 41 , , PUNCT cana-1164 2 42 india india PROPN cana-1164 2 43 , , PUNCT cana-1164 2 44 e e NOUN cana-1164 2 45 - - NOUN cana-1164 2 46 mail mail NOUN cana-1164 2 47 : : PUNCT cana-1164 2 48 sanj26sc@yahoo.com sanj26sc@yahoo.com PROPN cana-1164 3 1 2department 2department NUM cana-1164 3 2 of of ADP cana-1164 3 3 mathematics mathematic NOUN cana-1164 3 4 , , PUNCT cana-1164 3 5 cit cit PROPN cana-1164 3 6 , , PUNCT cana-1164 3 7 kokrajhar kokrajhar PROPN cana-1164 3 8 , , PUNCT cana-1164 3 9 india india PROPN cana-1164 3 10 , , PUNCT cana-1164 3 11 e e NOUN cana-1164 3 12 - - NOUN cana-1164 3 13 mail mail NOUN cana-1164 3 14 : : PUNCT cana-1164 3 15 gc.roy@cit.ac.in gc.roy@cit.ac.in PROPN cana-1164 3 16 3department 3department NUM cana-1164 3 17 of of ADP cana-1164 3 18 mathematics mathematic NOUN cana-1164 3 19 , , PUNCT cana-1164 3 20 university university NOUN cana-1164 3 21 of of ADP cana-1164 3 22 science science NOUN cana-1164 3 23 and and CCONJ cana-1164 3 24 technology technology NOUN cana-1164 3 25 , , PUNCT cana-1164 3 26 baridua baridua NOUN cana-1164 3 27 , , PUNCT cana-1164 3 28 e e NOUN cana-1164 3 29 - - NOUN cana-1164 3 30 mail mail NOUN cana-1164 3 31 : : PUNCT cana-1164 4 1 pinkimanigoswami@yahoo.com pinkimanigoswami@yahoo.com X cana-1164 4 2 โˆ—corresponding โˆ—corresponde VERB cana-1164 4 3 author author NOUN cana-1164 4 4 article article NOUN cana-1164 4 5 history history NOUN cana-1164 4 6 : : PUNCT cana-1164 4 7 received receive VERB cana-1164 4 8 : : PUNCT cana-1164 4 9 23 23 NUM cana-1164 4 10 - - SYM cana-1164 4 11 05 05 NUM cana-1164 4 12 - - PUNCT cana-1164 4 13 2024 2024 NUM cana-1164 4 14 revised revise VERB cana-1164 4 15 : : PUNCT cana-1164 4 16 10 10 NUM cana-1164 4 17 - - SYM cana-1164 4 18 07 07 NUM cana-1164 4 19 - - PUNCT cana-1164 4 20 2024 2024 NUM cana-1164 4 21 accepted accept VERB cana-1164 4 22 : : PUNCT cana-1164 4 23 22 22 NUM cana-1164 4 24 - - SYM cana-1164 4 25 07 07 NUM cana-1164 4 26 - - PUNCT cana-1164 4 27 2024 2024 NUM cana-1164 4 28 abstract abstract NOUN cana-1164 4 29 : : PUNCT cana-1164 4 30 for for ADP cana-1164 4 31 every every DET cana-1164 4 32 positive positive ADJ cana-1164 4 33 integer integer NOUN cana-1164 4 34 ๐‘› ๐‘› PROPN cana-1164 4 35 and and CCONJ cana-1164 4 36 ๐‘˜ ๐‘˜ X cana-1164 4 37 , , PUNCT cana-1164 4 38 a a DET cana-1164 4 39 power power NOUN cana-1164 4 40 digraph digraph NOUN cana-1164 4 41 modulo modulo PROPN cana-1164 4 42 ๐‘› ๐‘› PROPN cana-1164 4 43 , , PUNCT cana-1164 4 44 denoted denote VERB cana-1164 4 45 by by ADP cana-1164 4 46 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 4 47 , , PUNCT cana-1164 4 48 ๐‘˜ ๐‘˜ NOUN cana-1164 4 49 ) ) PUNCT cana-1164 4 50 is be AUX cana-1164 4 51 constructed construct VERB cana-1164 4 52 with with ADP cana-1164 4 53 the the DET cana-1164 4 54 vertex vertex NOUN cana-1164 4 55 set set VERB cana-1164 4 56 โ„ค๐‘› โ„ค๐‘› NOUN cana-1164 4 57 = = PRON cana-1164 4 58 { { PUNCT cana-1164 4 59 0,1,2,โ‹ฏ 0,1,2,โ‹ฏ PROPN cana-1164 4 60 , , PUNCT cana-1164 5 1 ๐‘› ๐‘› DET cana-1164 5 2 โˆ’ โˆ’ NOUN cana-1164 5 3 1 1 NUM cana-1164 5 4 } } PUNCT cana-1164 5 5 , , PUNCT cana-1164 5 6 and and CCONJ cana-1164 5 7 a a DET cana-1164 5 8 directed direct VERB cana-1164 5 9 edge edge NOUN cana-1164 5 10 from from ADP cana-1164 5 11 a a DET cana-1164 5 12 vertex vertex NOUN cana-1164 5 13 ๐‘ฅ ๐‘ฅ NOUN cana-1164 5 14 to to ADP cana-1164 5 15 a a DET cana-1164 5 16 vertex vertex NOUN cana-1164 5 17 ๐‘ฆ ๐‘ฆ NOUN cana-1164 5 18 exists exist VERB cana-1164 5 19 if if SCONJ cana-1164 5 20 and and CCONJ cana-1164 5 21 only only ADV cana-1164 5 22 if if SCONJ cana-1164 5 23 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ X cana-1164 5 24 โ‰ก โ‰ก PROPN cana-1164 5 25 ๐‘ฆ(๐‘š๐‘œ๐‘‘ ๐‘ฆ(๐‘š๐‘œ๐‘‘ PROPN cana-1164 5 26 ๐‘› ๐‘› PROPN cana-1164 5 27 ) ) PUNCT cana-1164 5 28 , , PUNCT cana-1164 5 29 where where SCONJ cana-1164 5 30 ๐‘ฅ ๐‘ฅ NOUN cana-1164 5 31 , , PUNCT cana-1164 5 32 ๐‘ฆ ๐‘ฆ NOUN cana-1164 5 33 โˆˆ โˆˆ PROPN cana-1164 5 34 โ„ค๐‘›. โ„ค๐‘›. PROPN cana-1164 5 35 in in ADP cana-1164 5 36 this this DET cana-1164 5 37 work work NOUN cana-1164 5 38 , , PUNCT cana-1164 5 39 we we PRON cana-1164 5 40 define define VERB cana-1164 5 41 the the DET cana-1164 5 42 out out NOUN cana-1164 5 43 - - PUNCT cana-1164 5 44 adjacency adjacency NOUN cana-1164 5 45 ( ( PUNCT cana-1164 5 46 ๐ดฮณ ๐ดฮณ PROPN cana-1164 5 47 + + PROPN cana-1164 5 48 ) ) PUNCT cana-1164 5 49 and and CCONJ cana-1164 5 50 the the DET cana-1164 5 51 in in ADP cana-1164 5 52 - - PUNCT cana-1164 5 53 adjacency adjacency NOUN cana-1164 5 54 ( ( PUNCT cana-1164 5 55 ๐ดฮณ ๐ดฮณ PROPN cana-1164 5 56 โˆ’ โˆ’ PROPN cana-1164 5 57 ) ) PUNCT cana-1164 5 58 matrices matrix NOUN cana-1164 5 59 of of ADP cana-1164 5 60 the the DET cana-1164 5 61 digraph digraph ADJ cana-1164 5 62 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 5 63 , , PUNCT cana-1164 5 64 ๐‘˜ ๐‘˜ NOUN cana-1164 5 65 ) ) PUNCT cana-1164 5 66 and and CCONJ cana-1164 5 67 some some DET cana-1164 5 68 results result NOUN cana-1164 5 69 on on ADP cana-1164 5 70 ๐ดฮณ ๐ดฮณ PROPN cana-1164 5 71 + + CCONJ cana-1164 5 72 and and CCONJ cana-1164 5 73 ๐ดฮณ ๐ดฮณ PROPN cana-1164 5 74 โˆ’ โˆ’ PROPN cana-1164 5 75 are be AUX cana-1164 5 76 discussed discuss VERB cana-1164 5 77 . . PUNCT cana-1164 6 1 it it PRON cana-1164 6 2 is be AUX cana-1164 6 3 proved prove VERB cana-1164 6 4 that that SCONJ cana-1164 6 5 the the DET cana-1164 6 6 matrices matrix NOUN cana-1164 6 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 6 8 + + CCONJ cana-1164 6 9 and and CCONJ cana-1164 6 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 6 11 โˆ’ โˆ’ PROPN cana-1164 6 12 are be AUX cana-1164 6 13 singular singular ADJ cana-1164 6 14 if if SCONJ cana-1164 6 15 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 6 16 ) ) PUNCT cana-1164 6 17 or or CCONJ cana-1164 6 18 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 6 19 , , PUNCT cana-1164 6 20 for for ADP cana-1164 6 21 some some DET cana-1164 6 22 prime prime ADJ cana-1164 6 23 ๐‘. ๐‘. NOUN cana-1164 6 24 some some DET cana-1164 6 25 spectral spectral ADJ cana-1164 6 26 properties property NOUN cana-1164 6 27 of of ADP cana-1164 6 28 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 6 29 , , PUNCT cana-1164 6 30 ๐‘˜ ๐‘˜ NOUN cana-1164 6 31 ) ) PUNCT cana-1164 6 32 are be AUX cana-1164 6 33 also also ADV cana-1164 6 34 presented present VERB cana-1164 6 35 . . PUNCT cana-1164 7 1 moreover moreover ADV cana-1164 7 2 , , PUNCT cana-1164 7 3 it it PRON cana-1164 7 4 is be AUX cana-1164 7 5 proved prove VERB cana-1164 7 6 that that SCONJ cana-1164 7 7 the the DET cana-1164 7 8 algebraic algebraic ADJ cana-1164 7 9 multiplicity multiplicity NOUN cana-1164 7 10 of of ADP cana-1164 7 11 1 1 NUM cana-1164 7 12 as as ADP cana-1164 7 13 an an DET cana-1164 7 14 eigenvalue eigenvalue NOUN cana-1164 7 15 of of ADP cana-1164 7 16 ๐ดฮณ ๐ดฮณ PROPN cana-1164 7 17 + + CCONJ cana-1164 7 18 is be AUX cana-1164 7 19 the the DET cana-1164 7 20 number number NOUN cana-1164 7 21 of of ADP cana-1164 7 22 components component NOUN cana-1164 7 23 of of ADP cana-1164 7 24 the the DET cana-1164 7 25 digraph digraph ADJ cana-1164 7 26 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 7 27 , , PUNCT cana-1164 7 28 ๐‘˜ ๐‘˜ NOUN cana-1164 7 29 ) ) PUNCT cana-1164 7 30 . . PUNCT cana-1164 8 1 keywords keyword NOUN cana-1164 8 2 : : PUNCT cana-1164 8 3 digraph digraph NOUN cana-1164 8 4 , , PUNCT cana-1164 8 5 adjacency adjacency NOUN cana-1164 8 6 matrices matrix NOUN cana-1164 8 7 of of ADP cana-1164 8 8 power power NOUN cana-1164 8 9 digraph digraph NOUN cana-1164 8 10 ( ( PUNCT cana-1164 8 11 mod mod NOUN cana-1164 8 12 ๐‘› ๐‘› PROPN cana-1164 8 13 ) ) PUNCT cana-1164 8 14 , , PUNCT cana-1164 8 15 eigenvalues eigenvalue VERB cana-1164 8 16 . . PUNCT cana-1164 9 1 ams am NOUN cana-1164 9 2 subject subject ADJ cana-1164 9 3 classification classification NOUN cana-1164 9 4 : : PUNCT cana-1164 9 5 11a07 11a07 NUM cana-1164 9 6 , , PUNCT cana-1164 9 7 05c50 05c50 NUM cana-1164 9 8 1 1 NUM cana-1164 9 9 . . PUNCT cana-1164 10 1 introduction introduction NOUN cana-1164 10 2 in in ADP cana-1164 10 3 recent recent ADJ cana-1164 10 4 years year NOUN cana-1164 10 5 , , PUNCT cana-1164 10 6 exploring explore VERB cana-1164 10 7 the the DET cana-1164 10 8 interconnections interconnection NOUN cana-1164 10 9 between between ADP cana-1164 10 10 graph graph NOUN cana-1164 10 11 theory theory NOUN cana-1164 10 12 , , PUNCT cana-1164 10 13 group group NOUN cana-1164 10 14 theory theory NOUN cana-1164 10 15 , , PUNCT cana-1164 10 16 and and CCONJ cana-1164 10 17 number number NOUN cana-1164 10 18 theory theory NOUN cana-1164 10 19 has have AUX cana-1164 10 20 emerged emerge VERB cana-1164 10 21 as as ADP cana-1164 10 22 an an DET cana-1164 10 23 attractive attractive ADJ cana-1164 10 24 and and CCONJ cana-1164 10 25 effective effective ADJ cana-1164 10 26 study study NOUN cana-1164 10 27 area area NOUN cana-1164 10 28 , , PUNCT cana-1164 10 29 for for ADP cana-1164 10 30 example example NOUN cana-1164 10 31 , , PUNCT cana-1164 10 32 [ [ X cana-1164 10 33 3 3 NUM cana-1164 10 34 , , PUNCT cana-1164 10 35 4 4 NUM cana-1164 10 36 , , PUNCT cana-1164 10 37 6 6 NUM cana-1164 10 38 , , PUNCT cana-1164 10 39 8 8 NUM cana-1164 10 40 , , PUNCT cana-1164 10 41 10 10 NUM cana-1164 10 42 , , PUNCT cana-1164 10 43 11 11 NUM cana-1164 10 44 , , PUNCT cana-1164 10 45 13 13 NUM cana-1164 10 46 , , PUNCT cana-1164 10 47 14 14 NUM cana-1164 10 48 , , PUNCT cana-1164 10 49 17 17 NUM cana-1164 10 50 , , PUNCT cana-1164 10 51 18 18 NUM cana-1164 10 52 , , PUNCT cana-1164 10 53 20 20 NUM cana-1164 10 54 ] ] PUNCT cana-1164 10 55 . . PUNCT cana-1164 11 1 in in ADP cana-1164 11 2 this this DET cana-1164 11 3 article article NOUN cana-1164 11 4 , , PUNCT cana-1164 11 5 for for ADP cana-1164 11 6 each each DET cana-1164 11 7 positive positive ADJ cana-1164 11 8 integers integer NOUN cana-1164 11 9 ๐‘› ๐‘› VERB cana-1164 11 10 and and CCONJ cana-1164 11 11 ๐‘˜ ๐‘˜ X cana-1164 11 12 , , PUNCT cana-1164 11 13 we we PRON cana-1164 11 14 consider consider VERB cana-1164 11 15 a a DET cana-1164 11 16 power power NOUN cana-1164 11 17 digraph digraph NOUN cana-1164 11 18 modulo modulo NOUN cana-1164 12 1 ๐‘› ๐‘› ADP cana-1164 12 2 denoted denote VERB cana-1164 12 3 by by ADP cana-1164 12 4 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 12 5 , , PUNCT cana-1164 12 6 ๐‘˜ ๐‘˜ NOUN cana-1164 12 7 ) ) PUNCT cana-1164 12 8 whose whose DET cana-1164 12 9 vertex vertex NOUN cana-1164 12 10 set set NOUN cana-1164 12 11 is be AUX cana-1164 12 12 โ„ค๐‘› โ„ค๐‘› NOUN cana-1164 12 13 = = PUNCT cana-1164 12 14 { { PUNCT cana-1164 12 15 0,1,2,โ‹ฏ 0,1,2,โ‹ฏ NUM cana-1164 12 16 , , PUNCT cana-1164 12 17 ๐‘› ๐‘› DET cana-1164 12 18 โˆ’ โˆ’ PROPN cana-1164 12 19 1 1 NUM cana-1164 12 20 } } PUNCT cana-1164 12 21 and and CCONJ cana-1164 12 22 the the DET cana-1164 12 23 ordered order VERB cana-1164 12 24 pair pair NOUN cana-1164 12 25 ( ( PUNCT cana-1164 12 26 ๐‘ฅ ๐‘ฅ NOUN cana-1164 12 27 , , PUNCT cana-1164 12 28 ๐‘ฆ ๐‘ฆ NOUN cana-1164 12 29 ) ) PUNCT cana-1164 12 30 is be AUX cana-1164 12 31 a a DET cana-1164 12 32 directed direct VERB cana-1164 12 33 arc arc NOUN cana-1164 12 34 ( ( PUNCT cana-1164 12 35 or or CCONJ cana-1164 12 36 directed direct VERB cana-1164 12 37 edge edge NOUN cana-1164 12 38 ) ) PUNCT cana-1164 12 39 of of ADP cana-1164 12 40 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 12 41 , , PUNCT cana-1164 12 42 ๐‘˜ ๐‘˜ NOUN cana-1164 12 43 ) ) PUNCT cana-1164 12 44 from from ADP cana-1164 12 45 ๐‘ฅ ๐‘ฅ PRON cana-1164 12 46 to to ADP cana-1164 12 47 ๐‘ฆ ๐‘ฆ PRON cana-1164 12 48 iff iff PROPN cana-1164 12 49 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ X cana-1164 12 50 โ‰ก โ‰ก PROPN cana-1164 12 51 ๐‘ฆ(๐‘š๐‘œ๐‘‘ ๐‘ฆ(๐‘š๐‘œ๐‘‘ PROPN cana-1164 12 52 ๐‘› ๐‘› PROPN cana-1164 12 53 ) ) PUNCT cana-1164 12 54 , , PUNCT cana-1164 12 55 where where SCONJ cana-1164 12 56 ๐‘ฅ ๐‘ฅ NOUN cana-1164 12 57 , , PUNCT cana-1164 12 58 ๐‘ฆ ๐‘ฆ NOUN cana-1164 12 59 โˆˆ โˆˆ PROPN cana-1164 12 60 โ„ค๐‘›. โ„ค๐‘›. PROPN cana-1164 12 61 in in ADP cana-1164 12 62 [ [ X cana-1164 12 63 3 3 NUM cana-1164 12 64 , , PUNCT cana-1164 12 65 6 6 NUM cana-1164 12 66 , , PUNCT cana-1164 12 67 9 9 NUM cana-1164 12 68 , , PUNCT cana-1164 12 69 12 12 NUM cana-1164 12 70 , , PUNCT cana-1164 12 71 14 14 NUM cana-1164 12 72 , , PUNCT cana-1164 12 73 17 17 NUM cana-1164 12 74 , , PUNCT cana-1164 12 75 18 18 NUM cana-1164 12 76 , , PUNCT cana-1164 12 77 21 21 NUM cana-1164 12 78 ] ] PUNCT cana-1164 12 79 some some DET cana-1164 12 80 properties property NOUN cana-1164 12 81 of of ADP cana-1164 12 82 the the DET cana-1164 12 83 power power NOUN cana-1164 12 84 digraph digraph NOUN cana-1164 12 85 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 12 86 , , PUNCT cana-1164 12 87 ๐‘˜ ๐‘˜ NOUN cana-1164 12 88 ) ) PUNCT cana-1164 12 89 were be AUX cana-1164 12 90 studied study VERB cana-1164 12 91 . . PUNCT cana-1164 13 1 the the DET cana-1164 13 2 adjacency adjacency NOUN cana-1164 13 3 matrix matrix NOUN cana-1164 13 4 is be AUX cana-1164 13 5 a a DET cana-1164 13 6 commonly commonly ADV cana-1164 13 7 used use VERB cana-1164 13 8 matrix matrix NOUN cana-1164 13 9 representation representation NOUN cana-1164 13 10 for for ADP cana-1164 13 11 graphs graph NOUN cana-1164 13 12 , , PUNCT cana-1164 13 13 and and CCONJ cana-1164 13 14 numerous numerous ADJ cana-1164 13 15 researchers researcher NOUN cana-1164 13 16 have have AUX cana-1164 13 17 investigated investigate VERB cana-1164 13 18 the the DET cana-1164 13 19 connection connection NOUN cana-1164 13 20 between between ADP cana-1164 13 21 the the DET cana-1164 13 22 eigenvalues eigenvalue NOUN cana-1164 13 23 of of ADP cana-1164 13 24 the the DET cana-1164 13 25 adjacency adjacency NOUN cana-1164 13 26 matrix matrix NOUN cana-1164 13 27 and and CCONJ cana-1164 13 28 the the DET cana-1164 13 29 graph graph NOUN cana-1164 13 30 โ€™s โ€™s PART cana-1164 13 31 structures structure NOUN cana-1164 13 32 in in ADP cana-1164 13 33 the the DET cana-1164 13 34 past past NOUN cana-1164 13 35 , , PUNCT cana-1164 13 36 for for ADP cana-1164 13 37 example example NOUN cana-1164 13 38 , , PUNCT cana-1164 13 39 [ [ X cana-1164 13 40 1 1 NUM cana-1164 13 41 , , PUNCT cana-1164 13 42 2 2 NUM cana-1164 13 43 , , PUNCT cana-1164 13 44 7 7 NUM cana-1164 13 45 ] ] PUNCT cana-1164 13 46 . . PUNCT cana-1164 14 1 in in ADP cana-1164 14 2 the the DET cana-1164 14 3 case case NOUN cana-1164 14 4 of of ADP cana-1164 14 5 a a DET cana-1164 14 6 multidigraph multidigraph NOUN cana-1164 14 7 ๐บ ๐บ NOUN cana-1164 14 8 with with ADP cana-1164 14 9 ๐‘› ๐‘› PROPN cana-1164 14 10 vertices vertex NOUN cana-1164 14 11 , , PUNCT cana-1164 14 12 the the DET cana-1164 14 13 adjacency adjacency NOUN cana-1164 14 14 matrix matrix NOUN cana-1164 14 15 of of ADP cana-1164 14 16 ๐บ ๐บ PROPN cana-1164 14 17 defined define VERB cana-1164 14 18 in in ADP cana-1164 14 19 [ [ X cana-1164 14 20 1 1 NUM cana-1164 14 21 ] ] PUNCT cana-1164 14 22 as as ADP cana-1164 14 23 the the DET cana-1164 14 24 ๐‘› ๐‘› PROPN cana-1164 14 25 ร— ร— NOUN cana-1164 14 26 ๐‘› ๐‘› PRON cana-1164 14 27 matrix matrix NOUN cana-1164 14 28 ๐ด(๐บ ๐ด(๐บ NOUN cana-1164 14 29 ) ) PUNCT cana-1164 14 30 = = PUNCT cana-1164 15 1 [ [ X cana-1164 15 2 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 15 3 ] ] PUNCT cana-1164 15 4 , , PUNCT cana-1164 15 5 where where SCONJ cana-1164 15 6 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 15 7 represents represent VERB cana-1164 15 8 the the DET cana-1164 15 9 number number NOUN cana-1164 15 10 of of ADP cana-1164 15 11 directed direct VERB cana-1164 15 12 edges edge NOUN cana-1164 15 13 that that PRON cana-1164 15 14 start start VERB cana-1164 15 15 at at ADP cana-1164 15 16 the the DET cana-1164 15 17 vertex vertex NOUN cana-1164 15 18 ๐‘– ๐‘– NOUN cana-1164 15 19 and and CCONJ cana-1164 15 20 ends end VERB cana-1164 15 21 at at ADP cana-1164 15 22 the the DET cana-1164 15 23 vertex vertex NOUN cana-1164 15 24 ๐‘—. ๐‘—. NOUN cana-1164 15 25 it it PRON cana-1164 15 26 is be AUX cana-1164 15 27 important important ADJ cana-1164 15 28 to to PART cana-1164 15 29 note note VERB cana-1164 15 30 that that SCONJ cana-1164 15 31 based base VERB cana-1164 15 32 on on ADP cana-1164 15 33 this this DET cana-1164 15 34 definition definition NOUN cana-1164 15 35 , , PUNCT cana-1164 15 36 the the DET cana-1164 15 37 adjacency adjacency NOUN cana-1164 15 38 matrix matrix NOUN cana-1164 15 39 of of ADP cana-1164 15 40 a a DET cana-1164 15 41 multidigraph multidigraph NOUN cana-1164 15 42 is be AUX cana-1164 15 43 not not PART cana-1164 15 44 symmetric symmetric ADJ cana-1164 15 45 in in ADP cana-1164 15 46 general general ADJ cana-1164 15 47 . . PUNCT cana-1164 16 1 so so ADV cana-1164 16 2 , , PUNCT cana-1164 16 3 it it PRON cana-1164 16 4 may may AUX cana-1164 16 5 have have VERB cana-1164 16 6 complex complex ADJ cana-1164 16 7 eigenvalues eigenvalue NOUN cana-1164 16 8 . . PUNCT cana-1164 17 1 furthermore furthermore ADV cana-1164 17 2 , , PUNCT cana-1164 17 3 a a DET cana-1164 17 4 graph graph NOUN cana-1164 17 5 is be AUX cana-1164 17 6 completely completely ADV cana-1164 17 7 determined determine VERB cana-1164 17 8 by by ADP cana-1164 17 9 its its PRON cana-1164 17 10 adjacency adjacency NOUN cana-1164 17 11 eigenvalues eigenvalue NOUN cana-1164 17 12 and and CCONJ cana-1164 17 13 corresponding corresponding ADJ cana-1164 17 14 eigenvectors eigenvector NOUN cana-1164 17 15 . . PUNCT cana-1164 18 1 this this PRON cana-1164 18 2 is be AUX cana-1164 18 3 evident evident ADJ cana-1164 18 4 from from ADP cana-1164 18 5 the the DET cana-1164 18 6 fact fact NOUN cana-1164 18 7 that that SCONJ cana-1164 18 8 a a DET cana-1164 18 9 graph graph NOUN cana-1164 18 10 ๐บ ๐บ NOUN cana-1164 18 11 can can AUX cana-1164 18 12 be be AUX cana-1164 18 13 uniquely uniquely ADV cana-1164 18 14 determined determine VERB cana-1164 18 15 by by ADP cana-1164 18 16 ๐ด(๐บ ๐ด(๐บ NOUN cana-1164 18 17 ) ) PUNCT cana-1164 18 18 . . PUNCT cana-1164 19 1 in in ADP cana-1164 19 2 the the DET cana-1164 19 3 case case NOUN cana-1164 19 4 of of ADP cana-1164 19 5 an an DET cana-1164 19 6 undirected undirected ADJ cana-1164 19 7 simple simple ADJ cana-1164 19 8 graph graph NOUN cana-1164 19 9 ๐บ ๐บ PROPN cana-1164 19 10 , , PUNCT cana-1164 19 11 ๐ด(๐บ ๐ด(๐บ PROPN cana-1164 19 12 ) ) PUNCT cana-1164 19 13 is be AUX cana-1164 19 14 symmetric symmetric ADJ cana-1164 19 15 . . PUNCT cana-1164 20 1 it it PRON cana-1164 20 2 is be AUX cana-1164 20 3 important important ADJ cana-1164 20 4 to to PART cana-1164 20 5 mention mention VERB cana-1164 20 6 that that SCONJ cana-1164 20 7 the the DET cana-1164 20 8 study study NOUN cana-1164 20 9 of of ADP cana-1164 20 10 adjacency adjacency NOUN cana-1164 20 11 matrices matrix NOUN cana-1164 20 12 of of ADP cana-1164 20 13 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 20 14 , , PUNCT cana-1164 20 15 ๐‘˜ ๐‘˜ NOUN cana-1164 20 16 ) ) PUNCT cana-1164 20 17 , , PUNCT cana-1164 20 18 the the DET cana-1164 20 19 power power NOUN cana-1164 20 20 digraph digraph NOUN cana-1164 20 21 modulo modulo VERB cana-1164 20 22 ๐‘› ๐‘› PRON cana-1164 20 23 is be AUX cana-1164 20 24 still still ADV cana-1164 20 25 open open ADJ cana-1164 20 26 . . PUNCT cana-1164 21 1 in in ADP cana-1164 21 2 this this DET cana-1164 21 3 paper paper NOUN cana-1164 21 4 , , PUNCT cana-1164 21 5 we we PRON cana-1164 21 6 aim aim VERB cana-1164 21 7 to to PART cana-1164 21 8 define define VERB cana-1164 21 9 the the DET cana-1164 21 10 adjacency adjacency NOUN cana-1164 21 11 matrices matrix NOUN cana-1164 21 12 of of ADP cana-1164 21 13 the the DET cana-1164 21 14 digraph digraph ADJ cana-1164 21 15 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 21 16 , , PUNCT cana-1164 21 17 ๐‘˜ ๐‘˜ NOUN cana-1164 21 18 ) ) PUNCT cana-1164 21 19 and and CCONJ cana-1164 21 20 try try VERB cana-1164 21 21 to to PART cana-1164 21 22 explore explore VERB cana-1164 21 23 some some DET cana-1164 21 24 properties property NOUN cana-1164 21 25 associated associate VERB cana-1164 21 26 with with ADP cana-1164 21 27 them they PRON cana-1164 21 28 . . PUNCT cana-1164 22 1 communications communication NOUN cana-1164 22 2 on on ADP cana-1164 22 3 applied apply VERB cana-1164 22 4 nonlinear nonlinear ADJ cana-1164 22 5 analysis analysis NOUN cana-1164 22 6 issn issn NOUN cana-1164 22 7 : : PUNCT cana-1164 22 8 1074 1074 NUM cana-1164 22 9 - - PUNCT cana-1164 22 10 133x 133x NUM cana-1164 22 11 vol vol NOUN cana-1164 22 12 31 31 NUM cana-1164 22 13 no no NOUN cana-1164 22 14 . . PUNCT cana-1164 23 1 6s 6s NUM cana-1164 23 2 ( ( PUNCT cana-1164 23 3 2024 2024 NUM cana-1164 23 4 ) ) PUNCT cana-1164 23 5 98 98 NUM cana-1164 23 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 24 1 we we PRON cana-1164 24 2 organize organize VERB cana-1164 24 3 the the DET cana-1164 24 4 rest rest NOUN cana-1164 24 5 of of ADP cana-1164 24 6 the the DET cana-1164 24 7 paper paper NOUN cana-1164 24 8 as as SCONJ cana-1164 24 9 follows follow VERB cana-1164 24 10 : : PUNCT cana-1164 24 11 in in ADP cana-1164 24 12 section section NOUN cana-1164 24 13 2 2 NUM cana-1164 24 14 , , PUNCT cana-1164 24 15 we we PRON cana-1164 24 16 provide provide VERB cana-1164 24 17 some some DET cana-1164 24 18 definitions definition NOUN cana-1164 24 19 and and CCONJ cana-1164 24 20 results result NOUN cana-1164 24 21 from from ADP cana-1164 24 22 graph graph NOUN cana-1164 24 23 theory theory NOUN cana-1164 24 24 and and CCONJ cana-1164 24 25 matrix matrix NOUN cana-1164 24 26 theory theory NOUN cana-1164 24 27 . . PUNCT cana-1164 25 1 in in ADP cana-1164 25 2 section section NOUN cana-1164 25 3 3 3 NUM cana-1164 25 4 , , PUNCT cana-1164 25 5 we we PRON cana-1164 25 6 define define VERB cana-1164 25 7 the the DET cana-1164 25 8 out out NOUN cana-1164 25 9 - - PUNCT cana-1164 25 10 adjacency adjacency NOUN cana-1164 25 11 ( ( PUNCT cana-1164 25 12 ๐ดฮณ ๐ดฮณ PROPN cana-1164 25 13 + + PROPN cana-1164 25 14 ) ) PUNCT cana-1164 25 15 and and CCONJ cana-1164 25 16 the the DET cana-1164 25 17 inadjacency inadjacency NOUN cana-1164 25 18 ( ( PUNCT cana-1164 25 19 ๐ดฮณ ๐ดฮณ PROPN cana-1164 25 20 โˆ’ โˆ’ PROPN cana-1164 25 21 ) ) PUNCT cana-1164 25 22 matrices matrix NOUN cana-1164 25 23 of of ADP cana-1164 25 24 the the DET cana-1164 25 25 digraph digraph ADJ cana-1164 25 26 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 25 27 , , PUNCT cana-1164 25 28 ๐‘˜ ๐‘˜ NOUN cana-1164 25 29 ) ) PUNCT cana-1164 25 30 and and CCONJ cana-1164 25 31 some some DET cana-1164 25 32 results result NOUN cana-1164 25 33 on on ADP cana-1164 25 34 ๐ดฮณ ๐ดฮณ PROPN cana-1164 25 35 + + CCONJ cana-1164 25 36 and and CCONJ cana-1164 25 37 ๐ดฮณ ๐ดฮณ PROPN cana-1164 25 38 โˆ’ โˆ’ PROPN cana-1164 25 39 are be AUX cana-1164 25 40 discussed discuss VERB cana-1164 25 41 . . PUNCT cana-1164 26 1 it it PRON cana-1164 26 2 is be AUX cana-1164 26 3 proved prove VERB cana-1164 26 4 that that SCONJ cana-1164 26 5 the the DET cana-1164 26 6 matrices matrix NOUN cana-1164 26 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 26 8 + + CCONJ cana-1164 26 9 and and CCONJ cana-1164 26 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 26 11 โˆ’ โˆ’ PROPN cana-1164 26 12 are be AUX cana-1164 26 13 singular singular ADJ cana-1164 26 14 if if SCONJ cana-1164 26 15 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 26 16 ) ) PUNCT cana-1164 26 17 or or CCONJ cana-1164 26 18 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 26 19 , , PUNCT cana-1164 26 20 for for ADP cana-1164 26 21 some some DET cana-1164 26 22 prime prime ADJ cana-1164 26 23 ๐‘. ๐‘. NOUN cana-1164 26 24 in in ADP cana-1164 26 25 section section NOUN cana-1164 26 26 4 4 NUM cana-1164 26 27 , , PUNCT cana-1164 26 28 some some DET cana-1164 26 29 spectral spectral ADJ cana-1164 26 30 properties property NOUN cana-1164 26 31 of of ADP cana-1164 26 32 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 26 33 , , PUNCT cana-1164 26 34 ๐‘˜ ๐‘˜ NOUN cana-1164 26 35 ) ) PUNCT cana-1164 26 36 are be AUX cana-1164 26 37 presented present VERB cana-1164 26 38 . . PUNCT cana-1164 27 1 it it PRON cana-1164 27 2 is be AUX cana-1164 27 3 also also ADV cana-1164 27 4 proved prove VERB cana-1164 27 5 that that SCONJ cana-1164 27 6 the the DET cana-1164 27 7 algebraic algebraic ADJ cana-1164 27 8 multiplicity multiplicity NOUN cana-1164 27 9 of of ADP cana-1164 27 10 1 1 NUM cana-1164 27 11 as as ADP cana-1164 27 12 an an DET cana-1164 27 13 eigenvalue eigenvalue NOUN cana-1164 27 14 of of ADP cana-1164 27 15 ๐ดฮณ ๐ดฮณ PROPN cana-1164 27 16 + + CCONJ cana-1164 27 17 is be AUX cana-1164 27 18 the the DET cana-1164 27 19 number number NOUN cana-1164 27 20 of of ADP cana-1164 27 21 components component NOUN cana-1164 27 22 of of ADP cana-1164 27 23 the the DET cana-1164 27 24 digraph digraph ADJ cana-1164 27 25 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 27 26 , , PUNCT cana-1164 27 27 ๐‘˜ ๐‘˜ NOUN cana-1164 27 28 ) ) PUNCT cana-1164 27 29 . . PUNCT cana-1164 28 1 2 2 X cana-1164 28 2 . . X cana-1164 28 3 preliminaries preliminary NOUN cana-1164 28 4 for for ADP cana-1164 28 5 each each DET cana-1164 28 6 positive positive ADJ cana-1164 28 7 integers integer NOUN cana-1164 28 8 ๐‘› ๐‘› VERB cana-1164 28 9 and and CCONJ cana-1164 28 10 ๐‘˜ ๐‘˜ X cana-1164 28 11 , , PUNCT cana-1164 28 12 we we PRON cana-1164 28 13 consider consider VERB cana-1164 28 14 a a DET cana-1164 28 15 power power NOUN cana-1164 28 16 digraph digraph NOUN cana-1164 28 17 modulo modulo NOUN cana-1164 29 1 ๐‘› ๐‘› ADP cana-1164 29 2 denoted denote VERB cana-1164 29 3 by by ADP cana-1164 29 4 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 29 5 , , PUNCT cana-1164 29 6 ๐‘˜ ๐‘˜ NOUN cana-1164 29 7 ) ) PUNCT cana-1164 29 8 ( ( PUNCT cana-1164 29 9 in in ADP cana-1164 29 10 short short ADJ cana-1164 29 11 , , PUNCT cana-1164 29 12 directed direct VERB cana-1164 29 13 graph graph NOUN cana-1164 29 14 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 29 15 , , PUNCT cana-1164 29 16 ๐‘˜ ๐‘˜ NOUN cana-1164 29 17 ) ) PUNCT cana-1164 29 18 or or CCONJ cana-1164 29 19 digraph digraph ADJ cana-1164 29 20 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 29 21 , , PUNCT cana-1164 29 22 ๐‘˜ ๐‘˜ NOUN cana-1164 29 23 ) ) PUNCT cana-1164 29 24 ) ) PUNCT cana-1164 30 1 whose whose DET cana-1164 30 2 vertex vertex NOUN cana-1164 30 3 set set NOUN cana-1164 30 4 is be AUX cana-1164 30 5 โ„ค๐‘› โ„ค๐‘› PROPN cana-1164 30 6 and and CCONJ cana-1164 30 7 any any DET cana-1164 30 8 two two NUM cana-1164 30 9 vertices vertex NOUN cana-1164 30 10 ๐‘ฅ ๐‘ฅ NOUN cana-1164 30 11 , , PUNCT cana-1164 30 12 ๐‘ฆ ๐‘ฆ NOUN cana-1164 30 13 โˆˆ โˆˆ NOUN cana-1164 31 1 โ„ค๐‘› โ„ค๐‘› NOUN cana-1164 31 2 are be AUX cana-1164 31 3 connected connect VERB cana-1164 31 4 by by ADP cana-1164 31 5 a a DET cana-1164 31 6 directed direct VERB cana-1164 31 7 arc arc NOUN cana-1164 31 8 from from ADP cana-1164 31 9 ๐‘ฅ ๐‘ฅ PRON cana-1164 31 10 to to ADP cana-1164 31 11 ๐‘ฆ ๐‘ฆ PRON cana-1164 31 12 if if SCONJ cana-1164 32 1 and and CCONJ cana-1164 32 2 only only ADV cana-1164 32 3 if if SCONJ cana-1164 32 4 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ X cana-1164 32 5 โ‰ก โ‰ก PROPN cana-1164 32 6 ๐‘ฆ(๐‘š๐‘œ๐‘‘ ๐‘ฆ(๐‘š๐‘œ๐‘‘ PROPN cana-1164 32 7 ๐‘› ๐‘› PROPN cana-1164 32 8 ) ) PUNCT cana-1164 32 9 . . PUNCT cana-1164 33 1 we we PRON cana-1164 33 2 denote denote VERB cana-1164 33 3 the the DET cana-1164 33 4 vertex vertex NOUN cana-1164 33 5 set set NOUN cana-1164 33 6 of of ADP cana-1164 33 7 the the DET cana-1164 33 8 digraph digraph ADJ cana-1164 33 9 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 33 10 , , PUNCT cana-1164 33 11 ๐‘˜ ๐‘˜ NOUN cana-1164 33 12 ) ) PUNCT cana-1164 33 13 by by ADP cana-1164 33 14 ๐‘‰(ฮณ(๐‘› ๐‘‰(ฮณ(๐‘› PROPN cana-1164 33 15 , , PUNCT cana-1164 33 16 ๐‘˜ ๐‘˜ NOUN cana-1164 33 17 ) ) PUNCT cana-1164 33 18 ) ) PUNCT cana-1164 33 19 or or CCONJ cana-1164 33 20 by by ADP cana-1164 33 21 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 33 22 ) ) PUNCT cana-1164 33 23 (= (= NOUN cana-1164 34 1 โ„ค๐‘› โ„ค๐‘› NOUN cana-1164 34 2 ) ) PUNCT cana-1164 34 3 and and CCONJ cana-1164 34 4 the the DET cana-1164 34 5 arc arc NOUN cana-1164 34 6 set set VERB cana-1164 34 7 by by ADP cana-1164 34 8 ๐ด(ฮณ(๐‘› ๐ด(ฮณ(๐‘› PROPN cana-1164 34 9 , , PUNCT cana-1164 34 10 ๐‘˜ ๐‘˜ NOUN cana-1164 34 11 ) ) PUNCT cana-1164 34 12 ) ) PUNCT cana-1164 34 13 or or CCONJ cana-1164 34 14 by by ADP cana-1164 34 15 ๐ด(ฮณ ๐ด(ฮณ NOUN cana-1164 34 16 ) ) PUNCT cana-1164 34 17 . . PUNCT cana-1164 35 1 the the DET cana-1164 35 2 distinct distinct ADJ cana-1164 35 3 vertices vertex NOUN cana-1164 35 4 ๐‘ฃ1 ๐‘ฃ1 PROPN cana-1164 35 5 , , PUNCT cana-1164 35 6 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 35 7 , , PUNCT cana-1164 35 8 ๐‘ฃ3 ๐‘ฃ3 ADJ cana-1164 35 9 , , PUNCT cana-1164 35 10 โ€ฆ โ€ฆ PUNCT cana-1164 35 11 , , PUNCT cana-1164 35 12 ๐‘ฃ๐‘ก ๐‘ฃ๐‘ก ADP cana-1164 35 13 in in ADP cana-1164 35 14 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 35 15 ) ) PUNCT cana-1164 35 16 will will AUX cana-1164 35 17 form form VERB cana-1164 35 18 a a DET cana-1164 35 19 cycle cycle NOUN cana-1164 35 20 of of ADP cana-1164 35 21 length length NOUN cana-1164 35 22 ๐‘ก ๐‘ก PROPN cana-1164 35 23 if if SCONJ cana-1164 35 24 ๐‘ฃ1 ๐‘ฃ1 PROPN cana-1164 35 25 ๐‘˜ ๐‘˜ PROPN cana-1164 35 26 โ‰ก โ‰ก PROPN cana-1164 35 27 ๐‘ฃ2(๐‘š๐‘œ๐‘‘ ๐‘ฃ2(๐‘š๐‘œ๐‘‘ ADP cana-1164 35 28 ๐‘› ๐‘› PROPN cana-1164 35 29 ) ) PUNCT cana-1164 35 30 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 35 31 ๐‘˜ ๐‘˜ ADP cana-1164 35 32 โ‰ก โ‰ก PROPN cana-1164 35 33 ๐‘ฃ3(๐‘š๐‘œ๐‘‘ ๐‘ฃ3(๐‘š๐‘œ๐‘‘ PROPN cana-1164 35 34 ๐‘› ๐‘› NOUN cana-1164 35 35 ) ) PUNCT cana-1164 35 36 ๐‘ฃ3 ๐‘ฃ3 PROPN cana-1164 35 37 ๐‘˜ ๐‘˜ ADP cana-1164 35 38 โ‰ก โ‰ก PROPN cana-1164 35 39 ๐‘ฃ4(๐‘š๐‘œ๐‘‘ ๐‘ฃ4(๐‘š๐‘œ๐‘‘ NUM cana-1164 35 40 ๐‘› ๐‘› NOUN cana-1164 35 41 ) ) PUNCT cana-1164 35 42 โ‹ฎ โ‹ฎ NOUN cana-1164 35 43 ๐‘ฃ๐‘ก ๐‘ฃ๐‘ก ADP cana-1164 35 44 ๐‘˜ ๐‘˜ DET cana-1164 35 45 โ‰ก โ‰ก PROPN cana-1164 35 46 ๐‘ฃ1(๐‘š๐‘œ๐‘‘ ๐‘ฃ1(๐‘š๐‘œ๐‘‘ NUM cana-1164 35 47 ๐‘› ๐‘› NOUN cana-1164 35 48 ) ) PUNCT cana-1164 35 49 we we PRON cana-1164 35 50 call call VERB cana-1164 35 51 a a DET cana-1164 35 52 cycle cycle NOUN cana-1164 35 53 of of ADP cana-1164 35 54 length length NOUN cana-1164 35 55 ๐‘ก ๐‘ก PROPN cana-1164 35 56 as as ADP cana-1164 35 57 a a DET cana-1164 35 58 tcycle tcycle NOUN cana-1164 35 59 and and CCONJ cana-1164 35 60 a a DET cana-1164 35 61 cycle cycle NOUN cana-1164 35 62 of of ADP cana-1164 35 63 length length NOUN cana-1164 35 64 1 1 NUM cana-1164 35 65 is be AUX cana-1164 35 66 named name VERB cana-1164 35 67 as as ADP cana-1164 35 68 a a DET cana-1164 35 69 fixed fix VERB cana-1164 35 70 point point NOUN cana-1164 35 71 ( ( PUNCT cana-1164 35 72 or or CCONJ cana-1164 35 73 a a DET cana-1164 35 74 selfloop selfloop NOUN cana-1164 35 75 ) ) PUNCT cana-1164 35 76 . . PUNCT cana-1164 36 1 a a DET cana-1164 36 2 vertex vertex NOUN cana-1164 36 3 is be AUX cana-1164 36 4 isolated isolate VERB cana-1164 36 5 if if SCONJ cana-1164 36 6 it it PRON cana-1164 36 7 is be AUX cana-1164 36 8 not not PART cana-1164 36 9 connected connect VERB cana-1164 36 10 to to ADP cana-1164 36 11 any any DET cana-1164 36 12 other other ADJ cana-1164 36 13 vertex vertex NOUN cana-1164 36 14 in in ADP cana-1164 36 15 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 36 16 , , PUNCT cana-1164 36 17 ๐‘˜ ๐‘˜ NOUN cana-1164 36 18 ) ) PUNCT cana-1164 36 19 . . PUNCT cana-1164 37 1 some some DET cana-1164 37 2 researchers researcher NOUN cana-1164 37 3 have have AUX cana-1164 37 4 developed develop VERB cana-1164 37 5 theorems theorem NOUN cana-1164 37 6 to to PART cana-1164 37 7 find find VERB cana-1164 37 8 the the DET cana-1164 37 9 number number NOUN cana-1164 37 10 of of ADP cana-1164 37 11 fixed fix VERB cana-1164 37 12 points point NOUN cana-1164 37 13 of of ADP cana-1164 37 14 the the DET cana-1164 37 15 digraph digraph ADJ cana-1164 37 16 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 37 17 , , PUNCT cana-1164 37 18 ๐‘˜ ๐‘˜ NOUN cana-1164 37 19 ) ) PUNCT cana-1164 37 20 , , PUNCT cana-1164 37 21 denoted denote VERB cana-1164 37 22 by by ADP cana-1164 37 23 ๐ฟ(๐‘› ๐ฟ(๐‘› NOUN cana-1164 37 24 ) ) PUNCT cana-1164 37 25 for for ADP cana-1164 37 26 some some DET cana-1164 37 27 values value NOUN cana-1164 37 28 of of ADP cana-1164 37 29 ๐‘˜ ๐‘˜ DET cana-1164 37 30 see see NOUN cana-1164 37 31 [ [ X cana-1164 37 32 5 5 NUM cana-1164 37 33 , , PUNCT cana-1164 37 34 15 15 NUM cana-1164 37 35 , , PUNCT cana-1164 37 36 16 16 NUM cana-1164 37 37 , , PUNCT cana-1164 37 38 19 19 NUM cana-1164 37 39 , , PUNCT cana-1164 37 40 20 20 NUM cana-1164 37 41 ] ] PUNCT cana-1164 37 42 . . PUNCT cana-1164 38 1 from from ADP cana-1164 38 2 these these DET cana-1164 38 3 theorems theorem NOUN cana-1164 38 4 , , PUNCT cana-1164 38 5 it it PRON cana-1164 38 6 is be AUX cana-1164 38 7 clear clear ADJ cana-1164 38 8 that that SCONJ cana-1164 38 9 0 0 NUM cana-1164 38 10 is be AUX cana-1164 38 11 always always ADV cana-1164 38 12 a a DET cana-1164 38 13 fixed fix VERB cana-1164 38 14 point point NOUN cana-1164 38 15 of of ADP cana-1164 38 16 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 38 17 , , PUNCT cana-1164 38 18 ๐‘˜ ๐‘˜ NOUN cana-1164 38 19 ) ) PUNCT cana-1164 38 20 and and CCONJ cana-1164 38 21 so so ADV cana-1164 38 22 the the DET cana-1164 38 23 number number NOUN cana-1164 38 24 of of ADP cana-1164 38 25 fixed fix VERB cana-1164 38 26 points point NOUN cana-1164 38 27 , , PUNCT cana-1164 38 28 ๐ฟ(๐‘› ๐ฟ(๐‘› NUM cana-1164 38 29 ) ) PUNCT cana-1164 38 30 > > X cana-1164 38 31 0 0 X cana-1164 38 32 . . PUNCT cana-1164 39 1 the the DET cana-1164 39 2 in in ADP cana-1164 39 3 - - PUNCT cana-1164 39 4 degree degree NOUN cana-1164 39 5 of of ADP cana-1164 39 6 a a DET cana-1164 39 7 vertex vertex NOUN cana-1164 39 8 ๐‘ฃ ๐‘ฃ ADP cana-1164 39 9 โˆˆ โˆˆ PROPN cana-1164 39 10 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 39 11 ) ) PUNCT cana-1164 39 12 , , PUNCT cana-1164 39 13 denoted denote VERB cana-1164 39 14 by by ADP cana-1164 39 15 ๐‘‘ฮณ ๐‘‘ฮณ PROPN cana-1164 39 16 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 39 17 ) ) PUNCT cana-1164 39 18 is be AUX cana-1164 39 19 the the DET cana-1164 39 20 number number NOUN cana-1164 39 21 of of ADP cana-1164 39 22 directed direct VERB cana-1164 39 23 arcs arcs PROPN cana-1164 39 24 incident incident NOUN cana-1164 39 25 into into ADP cana-1164 39 26 the the DET cana-1164 39 27 vertex vertex NOUN cana-1164 39 28 ๐‘ฃ ๐‘ฃ PROPN cana-1164 39 29 and and CCONJ cana-1164 39 30 the the DET cana-1164 39 31 out out ADJ cana-1164 39 32 - - PUNCT cana-1164 39 33 degree degree NOUN cana-1164 39 34 of of ADP cana-1164 39 35 a a DET cana-1164 39 36 vertex vertex NOUN cana-1164 39 37 ๐‘ฃ ๐‘ฃ NOUN cana-1164 39 38 , , PUNCT cana-1164 39 39 denoted denote VERB cana-1164 39 40 by by ADP cana-1164 39 41 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 39 42 + + PROPN cana-1164 39 43 ( ( PUNCT cana-1164 39 44 ๐‘ฃ ๐‘ฃ NOUN cana-1164 39 45 ) ) PUNCT cana-1164 39 46 is be AUX cana-1164 39 47 the the DET cana-1164 39 48 number number NOUN cana-1164 39 49 of of ADP cana-1164 39 50 directed direct VERB cana-1164 39 51 arcs arcs PROPN cana-1164 39 52 incident incident NOUN cana-1164 39 53 out out ADP cana-1164 39 54 of of ADP cana-1164 39 55 the the DET cana-1164 39 56 vertex vertex NOUN cana-1164 39 57 ๐‘ฃ. ๐‘ฃ. NOUN cana-1164 39 58 since since SCONJ cana-1164 39 59 the the DET cana-1164 39 60 residue residue NOUN cana-1164 39 61 of of ADP cana-1164 39 62 a a DET cana-1164 39 63 number number NOUN cana-1164 39 64 modulo modulo VERB cana-1164 39 65 ๐‘› ๐‘› NOUN cana-1164 39 66 is be AUX cana-1164 39 67 unique unique ADJ cana-1164 39 68 , , PUNCT cana-1164 39 69 so so SCONJ cana-1164 39 70 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 39 71 + + PROPN cana-1164 39 72 ( ( PUNCT cana-1164 39 73 ๐‘ฃ ๐‘ฃ NOUN cana-1164 39 74 ) ) PUNCT cana-1164 39 75 = = SYM cana-1164 39 76 1 1 NUM cana-1164 39 77 and and CCONJ cana-1164 39 78 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 39 79 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 39 80 ) ) PUNCT cana-1164 39 81 โ‰ฅ โ‰ฅ NOUN cana-1164 39 82 0 0 NUM cana-1164 39 83 for for ADP cana-1164 39 84 each each DET cana-1164 39 85 vertex vertex NOUN cana-1164 39 86 ๐‘ฃ ๐‘ฃ ADP cana-1164 39 87 โˆˆ โˆˆ NOUN cana-1164 39 88 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 39 89 ) ) PUNCT cana-1164 39 90 . . PUNCT cana-1164 40 1 also also ADV cana-1164 40 2 , , PUNCT cana-1164 40 3 for for ADP cana-1164 40 4 an an DET cana-1164 40 5 isolated isolated ADJ cana-1164 40 6 fixed fix VERB cana-1164 40 7 point point NOUN cana-1164 40 8 ๐‘ฃ ๐‘ฃ ADP cana-1164 40 9 โˆˆ โˆˆ PROPN cana-1164 40 10 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 40 11 ) ) PUNCT cana-1164 40 12 , , PUNCT cana-1164 40 13 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 40 14 + + PROPN cana-1164 40 15 ( ( PUNCT cana-1164 40 16 ๐‘ฃ ๐‘ฃ NOUN cana-1164 40 17 ) ) PUNCT cana-1164 40 18 = = PUNCT cana-1164 40 19 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 40 20 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 40 21 ) ) PUNCT cana-1164 40 22 = = PUNCT cana-1164 41 1 1 1 X cana-1164 41 2 . . PUNCT cana-1164 42 1 the the DET cana-1164 42 2 total total ADJ cana-1164 42 3 degree degree NOUN cana-1164 42 4 ( ( PUNCT cana-1164 42 5 or or CCONJ cana-1164 42 6 simply simply ADV cana-1164 42 7 degree degree NOUN cana-1164 42 8 ) ) PUNCT cana-1164 42 9 of of ADP cana-1164 42 10 a a DET cana-1164 42 11 vertex vertex NOUN cana-1164 42 12 ๐‘ฃ ๐‘ฃ ADP cana-1164 42 13 โˆˆ โˆˆ PROPN cana-1164 42 14 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 42 15 ) ) PUNCT cana-1164 42 16 , , PUNCT cana-1164 42 17 denoted denote VERB cana-1164 42 18 by by ADP cana-1164 42 19 ๐‘‘ฮณ(๐‘ฃ ๐‘‘ฮณ(๐‘ฃ PROPN cana-1164 42 20 ) ) PUNCT cana-1164 42 21 is be AUX cana-1164 42 22 the the DET cana-1164 42 23 sum sum NOUN cana-1164 42 24 of of ADP cana-1164 42 25 out out ADJ cana-1164 42 26 - - PUNCT cana-1164 42 27 degree degree NOUN cana-1164 42 28 and and CCONJ cana-1164 42 29 indegree indegree NOUN cana-1164 42 30 of of ADP cana-1164 42 31 ๐‘ฃ ๐‘ฃ PRON cana-1164 42 32 i.e. i.e. X cana-1164 42 33 ๐‘‘ฮณ(๐‘ฃ ๐‘‘ฮณ(๐‘ฃ NOUN cana-1164 42 34 ) ) PUNCT cana-1164 42 35 = = PUNCT cana-1164 42 36 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 42 37 + + PROPN cana-1164 42 38 ( ( PUNCT cana-1164 42 39 ๐‘ฃ ๐‘ฃ NOUN cana-1164 42 40 ) ) PUNCT cana-1164 42 41 + + CCONJ cana-1164 42 42 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 42 43 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 42 44 ) ) PUNCT cana-1164 42 45 . . PUNCT cana-1164 43 1 a a DET cana-1164 43 2 component component NOUN cana-1164 43 3 of of ADP cana-1164 43 4 a a DET cana-1164 43 5 digraph digraph NOUN cana-1164 43 6 is be AUX cana-1164 43 7 a a DET cana-1164 43 8 subdigraph subdigraph NOUN cana-1164 43 9 which which PRON cana-1164 43 10 is be AUX cana-1164 43 11 a a DET cana-1164 43 12 maximal maximal ADJ cana-1164 43 13 connected connected ADJ cana-1164 43 14 subgraph subgraph NOUN cana-1164 43 15 of of ADP cana-1164 43 16 the the DET cana-1164 43 17 associated associated ADJ cana-1164 43 18 nondirected nondirecte VERB cana-1164 43 19 graph graph NOUN cana-1164 43 20 . . PUNCT cana-1164 44 1 as as SCONJ cana-1164 44 2 the the DET cana-1164 44 3 out out ADJ cana-1164 44 4 - - PUNCT cana-1164 44 5 degree degree NOUN cana-1164 44 6 of of ADP cana-1164 44 7 each each DET cana-1164 44 8 vertex vertex NOUN cana-1164 44 9 of of ADP cana-1164 44 10 the the DET cana-1164 44 11 digraph digraph ADJ cana-1164 44 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 44 13 , , PUNCT cana-1164 44 14 ๐‘˜ ๐‘˜ NOUN cana-1164 44 15 ) ) PUNCT cana-1164 44 16 is be AUX cana-1164 44 17 equal equal ADJ cana-1164 44 18 to to ADP cana-1164 44 19 1 1 NUM cana-1164 44 20 , , PUNCT cana-1164 44 21 the the DET cana-1164 44 22 number number NOUN cana-1164 44 23 of of ADP cana-1164 44 24 components component NOUN cana-1164 44 25 of of ADP cana-1164 44 26 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 44 27 , , PUNCT cana-1164 44 28 ๐‘˜ ๐‘˜ NOUN cana-1164 44 29 ) ) PUNCT cana-1164 44 30 equals equal VERB cana-1164 44 31 the the DET cana-1164 44 32 number number NOUN cana-1164 44 33 of of ADP cana-1164 44 34 all all DET cana-1164 44 35 cycles cycle NOUN cana-1164 44 36 . . PUNCT cana-1164 45 1 the the DET cana-1164 45 2 cycles cycle NOUN cana-1164 45 3 may may AUX cana-1164 45 4 or or CCONJ cana-1164 45 5 may may AUX cana-1164 45 6 not not PART cana-1164 45 7 be be AUX cana-1164 45 8 isolated isolate VERB cana-1164 45 9 . . PUNCT cana-1164 46 1 we we PRON cana-1164 46 2 call call VERB cana-1164 46 3 a a DET cana-1164 46 4 digraph digraph NOUN cana-1164 46 5 regular regular NOUN cana-1164 46 6 if if SCONJ cana-1164 46 7 the the DET cana-1164 46 8 in in NOUN cana-1164 46 9 - - PUNCT cana-1164 46 10 degree degree NOUN cana-1164 46 11 of of ADP cana-1164 46 12 each each DET cana-1164 46 13 vertex vertex NOUN cana-1164 46 14 is be AUX cana-1164 46 15 equal equal ADJ cana-1164 46 16 to to ADP cana-1164 46 17 1 1 NUM cana-1164 46 18 . . PUNCT cana-1164 47 1 every every DET cana-1164 47 2 component component NOUN cana-1164 47 3 of of ADP cana-1164 47 4 such such DET cana-1164 47 5 a a DET cana-1164 47 6 digraph digraph NOUN cana-1164 47 7 is be AUX cana-1164 47 8 a a DET cana-1164 47 9 cycle cycle NOUN cana-1164 47 10 . . PUNCT cana-1164 48 1 a a DET cana-1164 48 2 digraph digraph NOUN cana-1164 48 3 is be AUX cana-1164 48 4 semi semi ADJ cana-1164 48 5 - - ADJ cana-1164 48 6 regular regular ADJ cana-1164 48 7 if if SCONJ cana-1164 48 8 there there PRON cana-1164 48 9 exists exist VERB cana-1164 48 10 a a DET cana-1164 48 11 positive positive ADJ cana-1164 48 12 integer integer NOUN cana-1164 48 13 ๐‘‘ ๐‘‘ PROPN cana-1164 48 14 such such ADJ cana-1164 48 15 that that SCONJ cana-1164 48 16 each each DET cana-1164 48 17 vertex vertex NOUN cana-1164 48 18 either either CCONJ cana-1164 48 19 has have VERB cana-1164 48 20 in in ADP cana-1164 48 21 - - PUNCT cana-1164 48 22 degree degree NOUN cana-1164 48 23 0 0 NUM cana-1164 48 24 or or CCONJ cana-1164 48 25 ๐‘‘. ๐‘‘. ADJ cana-1164 48 26 communications communication NOUN cana-1164 48 27 on on ADP cana-1164 48 28 applied apply VERB cana-1164 48 29 nonlinear nonlinear ADJ cana-1164 48 30 analysis analysis NOUN cana-1164 48 31 issn issn NOUN cana-1164 48 32 : : PUNCT cana-1164 48 33 1074 1074 NUM cana-1164 48 34 - - PUNCT cana-1164 48 35 133x 133x NUM cana-1164 48 36 vol vol NOUN cana-1164 48 37 31 31 NUM cana-1164 48 38 no no NOUN cana-1164 48 39 . . PUNCT cana-1164 49 1 6s 6s NUM cana-1164 49 2 ( ( PUNCT cana-1164 49 3 2024 2024 NUM cana-1164 49 4 ) ) PUNCT cana-1164 49 5 99 99 NUM cana-1164 49 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 49 7 for for ADP cana-1164 49 8 ๐‘› ๐‘› PROPN cana-1164 49 9 > > SYM cana-1164 49 10 1 1 NUM cana-1164 49 11 , , PUNCT cana-1164 49 12 let let VERB cana-1164 49 13 us we PRON cana-1164 49 14 divide divide VERB cana-1164 49 15 the the DET cana-1164 49 16 digraph digraph ADJ cana-1164 49 17 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 49 18 , , PUNCT cana-1164 49 19 ๐‘˜ ๐‘˜ NOUN cana-1164 49 20 ) ) PUNCT cana-1164 49 21 into into ADP cana-1164 49 22 two two NUM cana-1164 49 23 subdigraphs subdigraph NOUN cana-1164 49 24 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 49 25 , , PUNCT cana-1164 49 26 ๐‘˜ ๐‘˜ NOUN cana-1164 49 27 ) ) PUNCT cana-1164 49 28 and and CCONJ cana-1164 49 29 ฮณ2(๐‘› ฮณ2(๐‘› PROPN cana-1164 49 30 , , PUNCT cana-1164 49 31 ๐‘˜ ๐‘˜ NOUN cana-1164 49 32 ) ) PUNCT cana-1164 49 33 , , PUNCT cana-1164 49 34 where where SCONJ cana-1164 49 35 ฮณ1(๐‘› ฮณ1(๐‘› ADP cana-1164 49 36 , , PUNCT cana-1164 49 37 ๐‘˜ ๐‘˜ NOUN cana-1164 49 38 ) ) PUNCT cana-1164 49 39 is be AUX cana-1164 49 40 the the DET cana-1164 49 41 subdigraph subdigraph NOUN cana-1164 49 42 induced induce VERB cana-1164 49 43 on on ADP cana-1164 49 44 the the DET cana-1164 49 45 set set NOUN cana-1164 49 46 of of ADP cana-1164 49 47 the the DET cana-1164 49 48 vertices vertex NOUN cana-1164 49 49 ๐‘ฃ ๐‘ฃ ADP cana-1164 49 50 โˆˆ โˆˆ PROPN cana-1164 50 1 โ„ค๐‘› โ„ค๐‘› ADP cana-1164 50 2 such such ADJ cana-1164 50 3 that that SCONJ cana-1164 50 4 gcd(๐‘ฃ gcd(๐‘ฃ NOUN cana-1164 50 5 , , PUNCT cana-1164 50 6 ๐‘› ๐‘› NOUN cana-1164 50 7 ) ) PUNCT cana-1164 50 8 = = SYM cana-1164 50 9 1 1 NUM cana-1164 50 10 and and CCONJ cana-1164 50 11 ฮณ2(๐‘› ฮณ2(๐‘› PROPN cana-1164 50 12 , , PUNCT cana-1164 50 13 ๐‘˜ ๐‘˜ NOUN cana-1164 50 14 ) ) PUNCT cana-1164 50 15 is be AUX cana-1164 50 16 the the DET cana-1164 50 17 subdigraph subdigraph NOUN cana-1164 50 18 induced induce VERB cana-1164 50 19 on on ADP cana-1164 50 20 the the DET cana-1164 50 21 set set NOUN cana-1164 50 22 of of ADP cana-1164 50 23 the the DET cana-1164 50 24 vertices vertex NOUN cana-1164 50 25 ๐‘ฃ ๐‘ฃ ADP cana-1164 50 26 โˆˆ โˆˆ PROPN cana-1164 51 1 โ„ค๐‘› โ„ค๐‘› ADP cana-1164 51 2 such such ADJ cana-1164 51 3 that that SCONJ cana-1164 51 4 gcd(๐‘ฃ gcd(๐‘ฃ NOUN cana-1164 51 5 , , PUNCT cana-1164 51 6 ๐‘› ๐‘› NOUN cana-1164 51 7 ) ) PUNCT cana-1164 51 8 โ‰  โ‰  PROPN cana-1164 51 9 1 1 NUM cana-1164 51 10 . . PUNCT cana-1164 52 1 clearly clearly ADV cana-1164 52 2 , , PUNCT cana-1164 52 3 the the DET cana-1164 52 4 vertex vertex NOUN cana-1164 52 5 set set NOUN cana-1164 52 6 of of ADP cana-1164 52 7 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 52 8 , , PUNCT cana-1164 52 9 ๐‘˜ ๐‘˜ NOUN cana-1164 52 10 ) ) PUNCT cana-1164 52 11 is be AUX cana-1164 52 12 the the DET cana-1164 52 13 unit unit NOUN cana-1164 52 14 group group NOUN cana-1164 52 15 โ„ค๐‘› โ„ค๐‘› PROPN cana-1164 52 16 โˆ— โˆ— VERB cana-1164 52 17 with with ADP cana-1164 52 18 order order NOUN cana-1164 52 19 ๐œ™(๐‘› ๐œ™(๐‘› ADJ cana-1164 52 20 ) ) PUNCT cana-1164 52 21 , , PUNCT cana-1164 52 22 where where SCONJ cana-1164 52 23 ๐œ™(๐‘› ๐œ™(๐‘› ADJ cana-1164 52 24 ) ) PUNCT cana-1164 52 25 denotes denote VERB cana-1164 52 26 the the DET cana-1164 52 27 euler euler PROPN cana-1164 52 28 โ€™s โ€™s PART cana-1164 52 29 totient totient PROPN cana-1164 52 30 function function NOUN cana-1164 52 31 . . PUNCT cana-1164 53 1 also also ADV cana-1164 53 2 , , PUNCT cana-1164 53 3 1 1 NUM cana-1164 53 4 and and CCONJ cana-1164 53 5 ( ( PUNCT cana-1164 53 6 ๐‘› ๐‘› PRON cana-1164 53 7 โˆ’ โˆ’ NOUN cana-1164 53 8 1 1 NUM cana-1164 53 9 ) ) PUNCT cana-1164 53 10 are be AUX cana-1164 53 11 vertices vertex NOUN cana-1164 53 12 of of ADP cana-1164 53 13 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 53 14 , , PUNCT cana-1164 53 15 ๐‘˜ ๐‘˜ NOUN cana-1164 53 16 ) ) PUNCT cana-1164 53 17 and and CCONJ cana-1164 53 18 0 0 NUM cana-1164 53 19 is be AUX cana-1164 53 20 always always ADV cana-1164 53 21 a a DET cana-1164 53 22 vertex vertex NOUN cana-1164 53 23 of of ADP cana-1164 53 24 ฮณ2(๐‘› ฮณ2(๐‘› PROPN cana-1164 53 25 , , PUNCT cana-1164 53 26 ๐‘˜ ๐‘˜ NOUN cana-1164 53 27 ) ) PUNCT cana-1164 53 28 . . PUNCT cana-1164 54 1 one one PRON cana-1164 54 2 can can AUX cana-1164 54 3 easily easily ADV cana-1164 54 4 observe observe VERB cana-1164 54 5 that that SCONJ cana-1164 54 6 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 54 7 , , PUNCT cana-1164 54 8 ๐‘˜ ๐‘˜ NOUN cana-1164 54 9 ) ) PUNCT cana-1164 54 10 โˆช โˆช NOUN cana-1164 54 11 ฮณ2(๐‘› ฮณ2(๐‘› PROPN cana-1164 54 12 , , PUNCT cana-1164 54 13 ๐‘˜ ๐‘˜ NOUN cana-1164 54 14 ) ) PUNCT cana-1164 55 1 = = SYM cana-1164 55 2 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 55 3 , , PUNCT cana-1164 55 4 ๐‘˜ ๐‘˜ NOUN cana-1164 55 5 ) ) PUNCT cana-1164 55 6 and and CCONJ cana-1164 55 7 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 55 8 , , PUNCT cana-1164 55 9 ๐‘˜ ๐‘˜ NOUN cana-1164 55 10 ) ) PUNCT cana-1164 55 11 โˆฉ โˆฉ PROPN cana-1164 55 12 ฮณ2(๐‘› ฮณ2(๐‘› PROPN cana-1164 55 13 , , PUNCT cana-1164 55 14 ๐‘˜ ๐‘˜ NOUN cana-1164 55 15 ) ) PUNCT cana-1164 55 16 = = SYM cana-1164 55 17 ๐œ™. ๐œ™. NOUN cana-1164 55 18 from from ADP cana-1164 55 19 definition definition NOUN cana-1164 55 20 of of ADP cana-1164 55 21 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 55 22 , , PUNCT cana-1164 55 23 ๐‘˜ ๐‘˜ NOUN cana-1164 55 24 ) ) PUNCT cana-1164 55 25 , , PUNCT cana-1164 55 26 it it PRON cana-1164 55 27 is be AUX cana-1164 55 28 clear clear ADJ cana-1164 55 29 that that SCONJ cana-1164 55 30 |๐ด(ฮณ)| |๐ด(ฮณ)| PROPN cana-1164 55 31 = = PUNCT cana-1164 55 32 ๐‘›. ๐‘›. NOUN cana-1164 55 33 since since SCONJ cana-1164 55 34 the the DET cana-1164 55 35 number number NOUN cana-1164 55 36 of of ADP cana-1164 55 37 arcs arc NOUN cana-1164 55 38 in in ADP cana-1164 55 39 a a DET cana-1164 55 40 directed direct VERB cana-1164 55 41 graph graph NOUN cana-1164 55 42 equals equal VERB cana-1164 55 43 the the DET cana-1164 55 44 number number NOUN cana-1164 55 45 of of ADP cana-1164 55 46 their their PRON cana-1164 55 47 tails tail NOUN cana-1164 55 48 ( ( PUNCT cana-1164 55 49 or or CCONJ cana-1164 55 50 their their PRON cana-1164 55 51 heads head NOUN cana-1164 55 52 ) ) PUNCT cana-1164 55 53 , , PUNCT cana-1164 55 54 we we PRON cana-1164 55 55 have have VERB cana-1164 55 56 the the DET cana-1164 55 57 following follow VERB cana-1164 55 58 theorem theorem ADJ cana-1164 55 59 . . PUNCT cana-1164 55 60 theorem theorem VERB cana-1164 55 61 2.1 2.1 NUM cana-1164 55 62 . . PUNCT cana-1164 56 1 [ [ X cana-1164 56 2 22 22 NUM cana-1164 56 3 ] ] PUNCT cana-1164 56 4 ( ( PUNCT cana-1164 56 5 handshaking handshake VERB cana-1164 56 6 theorem theorem NOUN cana-1164 56 7 ) ) PUNCT cana-1164 56 8 in in ADP cana-1164 56 9 the the DET cana-1164 56 10 digraph digraph ADJ cana-1164 56 11 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 56 12 , , PUNCT cana-1164 56 13 ๐‘˜ ๐‘˜ NOUN cana-1164 56 14 ) ) PUNCT cana-1164 56 15 , , PUNCT cana-1164 56 16 โˆ‘๐‘ฃโˆˆ โˆ‘๐‘ฃโˆˆ PROPN cana-1164 56 17 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 56 18 ) ) PUNCT cana-1164 56 19 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 56 20 + + PROPN cana-1164 56 21 ( ( PUNCT cana-1164 56 22 ๐‘ฃ ๐‘ฃ NOUN cana-1164 56 23 ) ) PUNCT cana-1164 56 24 = = SYM cana-1164 56 25 โˆ‘๐‘ฃโˆˆ โˆ‘๐‘ฃโˆˆ PROPN cana-1164 56 26 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 56 27 ) ) PUNCT cana-1164 56 28 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 56 29 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 56 30 ) ) PUNCT cana-1164 56 31 = = PUNCT cana-1164 57 1 |๐ด(ฮณ)| |๐ด(ฮณ)| PROPN cana-1164 57 2 a a DET cana-1164 57 3 directed direct VERB cana-1164 57 4 walk walk NOUN cana-1164 57 5 in in ADP cana-1164 57 6 a a DET cana-1164 57 7 digraph digraph NOUN cana-1164 57 8 d d NOUN cana-1164 57 9 is be AUX cana-1164 57 10 an an DET cana-1164 57 11 alternating alternate VERB cana-1164 57 12 sequence sequence NOUN cana-1164 57 13 ๐‘ฃ1 ๐‘ฃ1 NOUN cana-1164 57 14 , , PUNCT cana-1164 57 15 ๐‘’1 ๐‘’1 NOUN cana-1164 57 16 , , PUNCT cana-1164 57 17 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 57 18 , , PUNCT cana-1164 57 19 ๐‘’2 ๐‘’2 PROPN cana-1164 57 20 , , PUNCT cana-1164 57 21 ๐‘ฃ3 ๐‘ฃ3 ADJ cana-1164 57 22 , , PUNCT cana-1164 57 23 โ€ฆ โ€ฆ PUNCT cana-1164 57 24 , , PUNCT cana-1164 57 25 ๐‘’๐‘›โˆ’1 ๐‘’๐‘›โˆ’1 PROPN cana-1164 57 26 , , PUNCT cana-1164 57 27 ๐‘ฃ๐‘› ๐‘ฃ๐‘› ADP cana-1164 57 28 of of ADP cana-1164 57 29 vertices vertex NOUN cana-1164 57 30 and and CCONJ cana-1164 57 31 arcs arc NOUN cana-1164 57 32 in in ADP cana-1164 57 33 which which PRON cana-1164 57 34 each each DET cana-1164 57 35 arc arc NOUN cana-1164 57 36 ๐‘’๐‘– ๐‘’๐‘– NOUN cana-1164 57 37 is be AUX cana-1164 57 38 ๐‘ฃ๐‘–๐‘ฃ๐‘–+1 ๐‘ฃ๐‘–๐‘ฃ๐‘–+1 PROPN cana-1164 57 39 . . PUNCT cana-1164 58 1 a a DET cana-1164 58 2 directed direct VERB cana-1164 58 3 path path NOUN cana-1164 58 4 is be AUX cana-1164 58 5 a a DET cana-1164 58 6 walk walk NOUN cana-1164 58 7 in in ADP cana-1164 58 8 which which PRON cana-1164 58 9 all all DET cana-1164 58 10 vertices vertex NOUN cana-1164 58 11 are be AUX cana-1164 58 12 distinct distinct ADJ cana-1164 58 13 . . PUNCT cana-1164 59 1 if if SCONJ cana-1164 59 2 there there PRON cana-1164 59 3 is be VERB cana-1164 59 4 a a DET cana-1164 59 5 directed direct VERB cana-1164 59 6 path path NOUN cana-1164 59 7 from from ADP cana-1164 59 8 a a DET cana-1164 59 9 vertex vertex NOUN cana-1164 59 10 ๐‘ข ๐‘ข NOUN cana-1164 59 11 to to ADP cana-1164 59 12 a a DET cana-1164 59 13 vertex vertex NOUN cana-1164 59 14 ๐‘ฃ ๐‘ฃ ADP cana-1164 59 15 , , PUNCT cana-1164 59 16 then then ADV cana-1164 59 17 ๐‘ฃ ๐‘ฃ PROPN cana-1164 59 18 is be AUX cana-1164 59 19 said say VERB cana-1164 59 20 to to PART cana-1164 59 21 be be AUX cana-1164 59 22 reachable reachable ADJ cana-1164 59 23 from from ADP cana-1164 59 24 ๐‘ข. ๐‘ข. NOUN cana-1164 59 25 in in ADP cana-1164 59 26 a a DET cana-1164 59 27 digraph digraph NOUN cana-1164 59 28 d d NOUN cana-1164 59 29 , , PUNCT cana-1164 59 30 a a DET cana-1164 59 31 semi semi ADJ cana-1164 59 32 - - ADJ cana-1164 59 33 walk walk NOUN cana-1164 59 34 is be AUX cana-1164 59 35 an an DET cana-1164 59 36 alternating alternate VERB cana-1164 59 37 sequence sequence NOUN cana-1164 59 38 ๐‘ฃ1 ๐‘ฃ1 NOUN cana-1164 59 39 , , PUNCT cana-1164 59 40 ๐‘’1 ๐‘’1 NOUN cana-1164 59 41 , , PUNCT cana-1164 59 42 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 59 43 , , PUNCT cana-1164 59 44 ๐‘’2 ๐‘’2 PROPN cana-1164 59 45 , , PUNCT cana-1164 59 46 ๐‘ฃ3 ๐‘ฃ3 ADJ cana-1164 59 47 , , PUNCT cana-1164 59 48 โ€ฆ โ€ฆ PUNCT cana-1164 59 49 , , PUNCT cana-1164 59 50 ๐‘’๐‘›โˆ’1 ๐‘’๐‘›โˆ’1 PROPN cana-1164 59 51 , , PUNCT cana-1164 59 52 ๐‘ฃ๐‘› ๐‘ฃ๐‘› ADP cana-1164 59 53 of of ADP cana-1164 59 54 vertices vertex NOUN cana-1164 59 55 and and CCONJ cana-1164 59 56 arcs arc NOUN cana-1164 59 57 in in ADP cana-1164 59 58 which which PRON cana-1164 59 59 each each DET cana-1164 59 60 arc arc NOUN cana-1164 59 61 ๐‘’๐‘– ๐‘’๐‘– NOUN cana-1164 59 62 may may AUX cana-1164 59 63 be be AUX cana-1164 59 64 either either CCONJ cana-1164 59 65 ๐‘ฃ๐‘–๐‘ฃ๐‘–+1 ๐‘ฃ๐‘–๐‘ฃ๐‘–+1 PROPN cana-1164 59 66 or or CCONJ cana-1164 59 67 ๐‘ฃ๐‘–+1๐‘ฃ๐‘–. ๐‘ฃ๐‘–+1๐‘ฃ๐‘–. NOUN cana-1164 59 68 a a DET cana-1164 59 69 semi semi NOUN cana-1164 59 70 - - NOUN cana-1164 59 71 path path NOUN cana-1164 59 72 is be AUX cana-1164 59 73 a a DET cana-1164 59 74 semi semi ADJ cana-1164 59 75 - - NOUN cana-1164 59 76 walk walk NOUN cana-1164 59 77 in in ADP cana-1164 59 78 which which PRON cana-1164 59 79 all all DET cana-1164 59 80 vertices vertex NOUN cana-1164 59 81 are be AUX cana-1164 59 82 distinct distinct ADJ cana-1164 59 83 . . PUNCT cana-1164 60 1 a a DET cana-1164 60 2 digraph digraph NOUN cana-1164 60 3 is be AUX cana-1164 60 4 strongly strongly ADV cana-1164 60 5 connected connect VERB cana-1164 60 6 ( ( PUNCT cana-1164 60 7 or or CCONJ cana-1164 60 8 strong strong ADJ cana-1164 60 9 ) ) PUNCT cana-1164 60 10 if if SCONJ cana-1164 60 11 every every DET cana-1164 60 12 two two NUM cana-1164 60 13 vertices vertex NOUN cana-1164 60 14 are be AUX cana-1164 60 15 mutually mutually ADV cana-1164 60 16 reachable reachable ADJ cana-1164 60 17 . . PUNCT cana-1164 61 1 a a DET cana-1164 61 2 digraph digraph NOUN cana-1164 61 3 is be AUX cana-1164 61 4 unilaterally unilaterally ADV cana-1164 61 5 connected connect VERB cana-1164 61 6 ( ( PUNCT cana-1164 61 7 or or CCONJ cana-1164 61 8 unilateral unilateral ADJ cana-1164 61 9 ) ) PUNCT cana-1164 61 10 if if SCONJ cana-1164 61 11 for for ADP cana-1164 61 12 any any DET cana-1164 61 13 two two NUM cana-1164 61 14 vertices vertex NOUN cana-1164 61 15 at at ADP cana-1164 61 16 least least ADJ cana-1164 61 17 one one NUM cana-1164 61 18 is be AUX cana-1164 61 19 reachable reachable ADJ cana-1164 61 20 from from ADP cana-1164 61 21 the the DET cana-1164 61 22 other other ADJ cana-1164 61 23 . . PUNCT cana-1164 62 1 a a DET cana-1164 62 2 digraph digraph NOUN cana-1164 62 3 is be AUX cana-1164 62 4 weakly weakly ADV cana-1164 62 5 connected connected ADJ cana-1164 62 6 ( ( PUNCT cana-1164 62 7 or or CCONJ cana-1164 62 8 weak weak ADJ cana-1164 62 9 ) ) PUNCT cana-1164 62 10 if if SCONJ cana-1164 62 11 every every DET cana-1164 62 12 two two NUM cana-1164 62 13 vertices vertex NOUN cana-1164 62 14 are be AUX cana-1164 62 15 joined join VERB cana-1164 62 16 by by ADP cana-1164 62 17 a a DET cana-1164 62 18 semi semi NOUN cana-1164 62 19 - - NOUN cana-1164 62 20 path path NOUN cana-1164 62 21 . . PUNCT cana-1164 63 1 every every DET cana-1164 63 2 strongly strongly ADV cana-1164 63 3 connected connect VERB cana-1164 63 4 ( ( PUNCT cana-1164 63 5 or or CCONJ cana-1164 63 6 strong strong ADJ cana-1164 63 7 ) ) PUNCT cana-1164 63 8 digraph digraph NOUN cana-1164 63 9 is be AUX cana-1164 63 10 a a DET cana-1164 63 11 unilateral unilateral ADJ cana-1164 63 12 digraph digraph NOUN cana-1164 63 13 and and CCONJ cana-1164 63 14 every every DET cana-1164 63 15 unilateral unilateral ADJ cana-1164 63 16 digraph digraph NOUN cana-1164 63 17 is be AUX cana-1164 63 18 weak weak ADJ cana-1164 63 19 . . PUNCT cana-1164 64 1 but but CCONJ cana-1164 64 2 the the DET cana-1164 64 3 converse converse NOUN cana-1164 64 4 statements statement NOUN cana-1164 64 5 are be AUX cana-1164 64 6 not not PART cana-1164 64 7 true true ADJ cana-1164 64 8 . . PUNCT cana-1164 65 1 a a DET cana-1164 65 2 digraph digraph NOUN cana-1164 65 3 is be AUX cana-1164 65 4 disconnected disconnect VERB cana-1164 65 5 if if SCONJ cana-1164 65 6 it it PRON cana-1164 65 7 is be AUX cana-1164 65 8 not not PART cana-1164 65 9 even even ADV cana-1164 65 10 weak weak ADJ cana-1164 65 11 . . PUNCT cana-1164 66 1 note note VERB cana-1164 66 2 2.1 2.1 NUM cana-1164 66 3 . . PUNCT cana-1164 67 1 from from ADP cana-1164 67 2 the the DET cana-1164 67 3 definition definition NOUN cana-1164 67 4 of of ADP cana-1164 67 5 the the DET cana-1164 67 6 digraph digraph ADJ cana-1164 67 7 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 67 8 , , PUNCT cana-1164 67 9 ๐‘˜ ๐‘˜ NOUN cana-1164 67 10 ) ) PUNCT cana-1164 67 11 , , PUNCT cana-1164 67 12 it it PRON cana-1164 67 13 is be AUX cana-1164 67 14 clear clear ADJ cana-1164 67 15 that that SCONJ cana-1164 67 16 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 67 17 , , PUNCT cana-1164 67 18 ๐‘˜ ๐‘˜ NOUN cana-1164 67 19 ) ) PUNCT cana-1164 67 20 is be AUX cana-1164 67 21 a a DET cana-1164 67 22 disconnected disconnected ADJ cana-1164 67 23 graph graph NOUN cana-1164 67 24 , , PUNCT cana-1164 67 25 and and CCONJ cana-1164 67 26 the the DET cana-1164 67 27 components component NOUN cana-1164 67 28 of of ADP cana-1164 67 29 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 67 30 , , PUNCT cana-1164 67 31 ๐‘˜ ๐‘˜ NOUN cana-1164 67 32 ) ) PUNCT cana-1164 67 33 are be AUX cana-1164 67 34 weakly weakly ADV cana-1164 67 35 connected connect VERB cana-1164 67 36 . . PUNCT cana-1164 68 1 a a DET cana-1164 68 2 tree tree NOUN cana-1164 68 3 is be AUX cana-1164 68 4 a a DET cana-1164 68 5 connected connected ADJ cana-1164 68 6 acyclic acyclic ADJ cana-1164 68 7 graph graph NOUN cana-1164 68 8 . . PUNCT cana-1164 69 1 a a DET cana-1164 69 2 tree tree NOUN cana-1164 69 3 in in ADP cana-1164 69 4 which which PRON cana-1164 69 5 one one NUM cana-1164 69 6 vertex vertex NOUN cana-1164 69 7 has have AUX cana-1164 69 8 been be AUX cana-1164 69 9 designated designate VERB cana-1164 69 10 as as SCONJ cana-1164 69 11 the the DET cana-1164 69 12 root root NOUN cana-1164 69 13 is be AUX cana-1164 69 14 a a DET cana-1164 69 15 rooted rooted ADJ cana-1164 69 16 tree tree NOUN cana-1164 69 17 . . PUNCT cana-1164 70 1 the the DET cana-1164 70 2 edges edge NOUN cana-1164 70 3 of of ADP cana-1164 70 4 a a DET cana-1164 70 5 rooted rooted ADJ cana-1164 70 6 tree tree NOUN cana-1164 70 7 can can AUX cana-1164 70 8 be be AUX cana-1164 70 9 assigned assign VERB cana-1164 70 10 a a DET cana-1164 70 11 natural natural ADJ cana-1164 70 12 orientation orientation NOUN cana-1164 70 13 , , PUNCT cana-1164 70 14 either either CCONJ cana-1164 70 15 away away ADV cana-1164 70 16 from from ADP cana-1164 70 17 or or CCONJ cana-1164 70 18 towards towards ADP cana-1164 70 19 the the DET cana-1164 70 20 root root NOUN cana-1164 70 21 , , PUNCT cana-1164 70 22 in in ADP cana-1164 70 23 which which DET cana-1164 70 24 case case NOUN cana-1164 70 25 the the DET cana-1164 70 26 structure structure NOUN cana-1164 70 27 becomes become VERB cana-1164 70 28 a a DET cana-1164 70 29 directed direct VERB cana-1164 70 30 rooted rooted ADJ cana-1164 70 31 tree tree NOUN cana-1164 70 32 . . PUNCT cana-1164 71 1 when when SCONJ cana-1164 71 2 a a DET cana-1164 71 3 directed direct VERB cana-1164 71 4 rooted rooted ADJ cana-1164 71 5 tree tree NOUN cana-1164 71 6 has have VERB cana-1164 71 7 an an DET cana-1164 71 8 orientation orientation NOUN cana-1164 71 9 away away ADV cana-1164 71 10 from from ADP cana-1164 71 11 the the DET cana-1164 71 12 root root NOUN cana-1164 71 13 , , PUNCT cana-1164 71 14 it it PRON cana-1164 71 15 is be AUX cana-1164 71 16 called call VERB cana-1164 71 17 an an DET cana-1164 71 18 arborescence arborescence NOUN cana-1164 71 19 or or CCONJ cana-1164 71 20 out out NOUN cana-1164 71 21 - - PUNCT cana-1164 71 22 tree tree NOUN cana-1164 71 23 and and CCONJ cana-1164 71 24 when when SCONJ cana-1164 71 25 it it PRON cana-1164 71 26 has have VERB cana-1164 71 27 an an DET cana-1164 71 28 orientation orientation NOUN cana-1164 71 29 towards towards ADP cana-1164 71 30 the the DET cana-1164 71 31 root root NOUN cana-1164 71 32 , , PUNCT cana-1164 71 33 it it PRON cana-1164 71 34 is be AUX cana-1164 71 35 called call VERB cana-1164 71 36 an an DET cana-1164 71 37 anti anti ADJ cana-1164 71 38 - - ADJ cana-1164 71 39 arborescence arborescence ADJ cana-1164 71 40 or or CCONJ cana-1164 71 41 in in ADP cana-1164 71 42 - - PUNCT cana-1164 71 43 tree tree NOUN cana-1164 71 44 . . PUNCT cana-1164 72 1 a a DET cana-1164 72 2 vertex vertex NOUN cana-1164 72 3 in in ADP cana-1164 72 4 a a DET cana-1164 72 5 rooted rooted ADJ cana-1164 72 6 tree tree NOUN cana-1164 72 7 is be AUX cana-1164 72 8 called call VERB cana-1164 72 9 a a DET cana-1164 72 10 leaf leaf NOUN cana-1164 72 11 if if SCONJ cana-1164 72 12 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 72 13 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 72 14 ) ) PUNCT cana-1164 73 1 = = PUNCT cana-1164 74 1 0 0 X cana-1164 74 2 . . PUNCT cana-1164 75 1 a a DET cana-1164 75 2 block block NOUN cana-1164 75 3 diagonal diagonal ADJ cana-1164 75 4 matrix matrix NOUN cana-1164 75 5 is be AUX cana-1164 75 6 a a DET cana-1164 75 7 square square ADJ cana-1164 75 8 matrix matrix NOUN cana-1164 75 9 of of ADP cana-1164 75 10 the the DET cana-1164 75 11 form form NOUN cana-1164 75 12 b b NOUN cana-1164 75 13 = = PUNCT cana-1164 75 14 [ [ PUNCT cana-1164 75 15 ๐ด11 ๐ด11 PROPN cana-1164 75 16 0 0 NUM cana-1164 75 17 0 0 PUNCT cana-1164 75 18 โ‹ฏ โ‹ฏ X cana-1164 75 19 0 0 NOUN cana-1164 75 20 0 0 NUM cana-1164 75 21 ๐ด22 ๐ด22 X cana-1164 75 22 0 0 NUM cana-1164 75 23 โ‹ฏ โ‹ฏ NOUN cana-1164 75 24 0 0 NUM cana-1164 75 25 โ‹ฎ โ‹ฎ NOUN cana-1164 75 26 โ‹ฎ โ‹ฎ ADJ cana-1164 75 27 โ‹ฎ โ‹ฎ NOUN cana-1164 75 28 โ‹ฑ โ‹ฑ PUNCT cana-1164 75 29 โ‹ฎ โ‹ฎ NOUN cana-1164 75 30 0 0 NUM cana-1164 75 31 0 0 NUM cana-1164 75 32 0 0 PUNCT cana-1164 75 33 โ‹ฏ โ‹ฏ VERB cana-1164 76 1 ๐ด๐‘š๐‘š ๐ด๐‘š๐‘š PROPN cana-1164 76 2 ] ] PUNCT cana-1164 76 3 where where SCONJ cana-1164 76 4 ๐ด11 ๐ด11 PROPN cana-1164 76 5 , , PUNCT cana-1164 76 6 ๐ด22 ๐ด22 PROPN cana-1164 76 7 , , PUNCT cana-1164 76 8 โ‹ฏ โ‹ฏ PROPN cana-1164 76 9 , , PUNCT cana-1164 76 10 ๐ด๐‘š๐‘š ๐ด๐‘š๐‘š PROPN cana-1164 76 11 are be AUX cana-1164 76 12 square square ADJ cana-1164 76 13 matrices matrix NOUN cana-1164 76 14 lying lie VERB cana-1164 76 15 along along ADP cana-1164 76 16 the the DET cana-1164 76 17 diagonal diagonal ADJ cana-1164 76 18 and and CCONJ cana-1164 76 19 all all DET cana-1164 76 20 other other ADJ cana-1164 76 21 entries entry NOUN cana-1164 76 22 of of ADP cana-1164 76 23 the the DET cana-1164 76 24 matrix matrix NOUN cana-1164 76 25 is be AUX cana-1164 76 26 0 0 NUM cana-1164 76 27 ( ( PUNCT cana-1164 76 28 zero zero NUM cana-1164 76 29 matrices matrix NOUN cana-1164 76 30 ) ) PUNCT cana-1164 76 31 . . PUNCT cana-1164 77 1 determinant determinant ADJ cana-1164 77 2 of of ADP cana-1164 77 3 b b PROPN cana-1164 77 4 is be AUX cana-1164 77 5 given give VERB cana-1164 77 6 by by ADP cana-1164 77 7 communications communication NOUN cana-1164 77 8 on on ADP cana-1164 77 9 applied apply VERB cana-1164 77 10 nonlinear nonlinear ADJ cana-1164 77 11 analysis analysis NOUN cana-1164 77 12 issn issn NOUN cana-1164 77 13 : : PUNCT cana-1164 77 14 1074 1074 NUM cana-1164 77 15 - - PUNCT cana-1164 77 16 133x 133x NUM cana-1164 77 17 vol vol NOUN cana-1164 77 18 31 31 NUM cana-1164 77 19 no no NOUN cana-1164 77 20 . . PUNCT cana-1164 78 1 6s 6s NUM cana-1164 78 2 ( ( PUNCT cana-1164 78 3 2024 2024 NUM cana-1164 78 4 ) ) PUNCT cana-1164 78 5 100 100 NUM cana-1164 79 1 https://internationalpubls.com https://internationalpubls.com X cana-1164 79 2 det(๐ต det(๐ต NUM cana-1164 79 3 ) ) PUNCT cana-1164 79 4 = = SYM cana-1164 79 5 det(๐ด11 det(๐ด11 PROPN cana-1164 79 6 ) ) PUNCT cana-1164 79 7 ร— ร— NOUN cana-1164 79 8 det(๐ด22 det(๐ด22 NOUN cana-1164 79 9 ) ) PUNCT cana-1164 79 10 ร— ร— PROPN cana-1164 79 11 โ‹ฏร— โ‹ฏร— PROPN cana-1164 79 12 det(๐ด๐‘š๐‘š det(๐ด๐‘š๐‘š NOUN cana-1164 79 13 ) ) PUNCT cana-1164 79 14 . . PUNCT cana-1164 80 1 3 3 X cana-1164 80 2 . . X cana-1164 80 3 adjacency adjacency NOUN cana-1164 80 4 matrices matrix NOUN cana-1164 80 5 of of ADP cana-1164 80 6 the the DET cana-1164 80 7 digraph digraph NOUN cana-1164 80 8 ๐šช(๐’ ๐šช(๐’ PROPN cana-1164 80 9 , , PUNCT cana-1164 80 10 ๐’Œ ๐’Œ ADJ cana-1164 80 11 ) ) PUNCT cana-1164 80 12 in in ADP cana-1164 80 13 this this DET cana-1164 80 14 section section NOUN cana-1164 80 15 , , PUNCT cana-1164 80 16 we we PRON cana-1164 80 17 try try VERB cana-1164 80 18 to to PART cana-1164 80 19 define define VERB cana-1164 80 20 adjacency adjacency NOUN cana-1164 80 21 matrices matrix NOUN cana-1164 80 22 of of ADP cana-1164 80 23 the the DET cana-1164 80 24 digraph digraph ADJ cana-1164 80 25 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 80 26 , , PUNCT cana-1164 80 27 ๐‘˜ ๐‘˜ NOUN cana-1164 80 28 ) ) PUNCT cana-1164 80 29 and and CCONJ cana-1164 80 30 try try VERB cana-1164 80 31 to to PART cana-1164 80 32 study study VERB cana-1164 80 33 some some DET cana-1164 80 34 properties property NOUN cana-1164 80 35 associated associate VERB cana-1164 80 36 with with ADP cana-1164 80 37 them they PRON cana-1164 80 38 . . PUNCT cana-1164 81 1 definition definition NOUN cana-1164 81 2 3.1 3.1 NUM cana-1164 81 3 . . PUNCT cana-1164 82 1 we we PRON cana-1164 82 2 define define VERB cana-1164 82 3 the the DET cana-1164 82 4 out out ADJ cana-1164 82 5 - - PUNCT cana-1164 82 6 adjacency adjacency NOUN cana-1164 82 7 matrix matrix NOUN cana-1164 82 8 of of ADP cana-1164 82 9 the the DET cana-1164 82 10 digraph digraph ADJ cana-1164 82 11 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 82 12 , , PUNCT cana-1164 82 13 ๐‘˜ ๐‘˜ NOUN cana-1164 82 14 ) ) PUNCT cana-1164 82 15 as as ADP cana-1164 82 16 an an DET cana-1164 82 17 ๐‘› ๐‘› PROPN cana-1164 82 18 ร— ร— NOUN cana-1164 82 19 ๐‘› ๐‘› DET cana-1164 82 20 matrix matrix NOUN cana-1164 82 21 [ [ X cana-1164 82 22 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 82 23 ] ] PUNCT cana-1164 82 24 such such ADJ cana-1164 82 25 that that SCONJ cana-1164 82 26 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 82 27 = = SYM cana-1164 82 28 { { PUNCT cana-1164 82 29 1 1 NUM cana-1164 82 30 , , PUNCT cana-1164 82 31 if if SCONJ cana-1164 82 32 ( ( PUNCT cana-1164 82 33 vi vi NOUN cana-1164 82 34 , , PUNCT cana-1164 82 35 vj vj ADJ cana-1164 82 36 ) ) PUNCT cana-1164 82 37 โˆˆ โˆˆ PROPN cana-1164 82 38 a(ฮณ a(ฮณ NOUN cana-1164 82 39 ) ) PUNCT cana-1164 82 40 0 0 NUM cana-1164 82 41 , , PUNCT cana-1164 82 42 otherwise otherwise ADV cana-1164 82 43 we we PRON cana-1164 82 44 denote denote VERB cana-1164 82 45 this this DET cana-1164 82 46 matrix matrix NOUN cana-1164 82 47 by by ADP cana-1164 82 48 ๐ด+(ฮณ(๐‘› ๐ด+(ฮณ(๐‘› PROPN cana-1164 82 49 , , PUNCT cana-1164 82 50 ๐‘˜ ๐‘˜ NOUN cana-1164 82 51 ) ) PUNCT cana-1164 82 52 ) ) PUNCT cana-1164 82 53 or or CCONJ cana-1164 82 54 by by ADP cana-1164 82 55 ๐ดฮณ ๐ดฮณ PROPN cana-1164 82 56 + + PROPN cana-1164 82 57 . . PUNCT cana-1164 82 58 definition definition NOUN cana-1164 82 59 3.2 3.2 NUM cana-1164 82 60 . . PUNCT cana-1164 83 1 we we PRON cana-1164 83 2 define define VERB cana-1164 83 3 the the DET cana-1164 83 4 in in ADP cana-1164 83 5 - - PUNCT cana-1164 83 6 adjacency adjacency NOUN cana-1164 83 7 matrix matrix NOUN cana-1164 83 8 of of ADP cana-1164 83 9 the the DET cana-1164 83 10 digraph digraph ADJ cana-1164 83 11 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 83 12 , , PUNCT cana-1164 83 13 ๐‘˜ ๐‘˜ NOUN cana-1164 83 14 ) ) PUNCT cana-1164 83 15 as as ADP cana-1164 83 16 an an DET cana-1164 83 17 ๐‘› ๐‘› PROPN cana-1164 83 18 ร— ร— NOUN cana-1164 83 19 ๐‘› ๐‘› DET cana-1164 83 20 matrix matrix NOUN cana-1164 83 21 [ [ X cana-1164 83 22 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 83 23 ] ] PUNCT cana-1164 83 24 such such ADJ cana-1164 83 25 that that SCONJ cana-1164 83 26 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 83 27 = = SYM cana-1164 83 28 { { PUNCT cana-1164 83 29 1 1 NUM cana-1164 83 30 , , PUNCT cana-1164 83 31 if if SCONJ cana-1164 83 32 ( ( PUNCT cana-1164 83 33 vj vj INTJ cana-1164 83 34 , , PUNCT cana-1164 83 35 vi vi NOUN cana-1164 83 36 ) ) PUNCT cana-1164 83 37 โˆˆ โˆˆ PROPN cana-1164 83 38 a(ฮณ a(ฮณ NOUN cana-1164 83 39 ) ) PUNCT cana-1164 83 40 0 0 NUM cana-1164 83 41 , , PUNCT cana-1164 83 42 otherwise otherwise ADV cana-1164 83 43 we we PRON cana-1164 83 44 denote denote VERB cana-1164 83 45 this this DET cana-1164 83 46 matrix matrix NOUN cana-1164 83 47 by by ADP cana-1164 83 48 ๐ดโˆ’(ฮณ(๐‘› ๐ดโˆ’(ฮณ(๐‘› PROPN cana-1164 83 49 , , PUNCT cana-1164 83 50 ๐‘˜ ๐‘˜ NOUN cana-1164 83 51 ) ) PUNCT cana-1164 83 52 ) ) PUNCT cana-1164 83 53 or or CCONJ cana-1164 83 54 by by ADP cana-1164 83 55 ๐ดฮณ ๐ดฮณ PROPN cana-1164 83 56 โˆ’. โˆ’. ADV cana-1164 83 57 from from ADP cana-1164 83 58 definition definition NOUN cana-1164 83 59 of of ADP cana-1164 83 60 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 83 61 , , PUNCT cana-1164 83 62 ๐‘˜ ๐‘˜ NOUN cana-1164 83 63 ) ) PUNCT cana-1164 83 64 it it PRON cana-1164 83 65 is be AUX cana-1164 83 66 clear clear ADJ cana-1164 83 67 that that SCONJ cana-1164 83 68 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 83 69 , , PUNCT cana-1164 83 70 ๐‘˜ ๐‘˜ NOUN cana-1164 83 71 ) ) PUNCT cana-1164 83 72 is be AUX cana-1164 83 73 a a DET cana-1164 83 74 disconnected disconnected ADJ cana-1164 83 75 graph graph NOUN cana-1164 83 76 , , PUNCT cana-1164 83 77 so so SCONJ cana-1164 83 78 the the DET cana-1164 83 79 out out ADJ cana-1164 83 80 - - PUNCT cana-1164 83 81 adjacency adjacency NOUN cana-1164 83 82 matrix matrix NOUN cana-1164 83 83 ๐ดฮณ ๐ดฮณ PROPN cana-1164 83 84 + + CCONJ cana-1164 83 85 can can AUX cana-1164 83 86 also also ADV cana-1164 83 87 be be AUX cana-1164 83 88 defined define VERB cana-1164 83 89 as as ADP cana-1164 83 90 a a DET cana-1164 83 91 block block NOUN cana-1164 83 92 diagonal diagonal ADJ cana-1164 83 93 matrix matrix NOUN cana-1164 84 1 [ [ X cana-1164 84 2 ๐ด๐‘–๐‘—]๐‘šร—๐‘š ๐ด๐‘–๐‘—]๐‘šร—๐‘š NOUN cana-1164 84 3 i.e. i.e. X cana-1164 84 4 ๐ดฮณ ๐ดฮณ PROPN cana-1164 84 5 + + NOUN cana-1164 84 6 = = SYM cana-1164 85 1 [ [ X cana-1164 85 2 ๐ด๐‘–๐‘—]๐‘šร—๐‘š ๐ด๐‘–๐‘—]๐‘šร—๐‘š PROPN cana-1164 85 3 , , PUNCT cana-1164 85 4 such such ADJ cana-1164 85 5 that that SCONJ cana-1164 85 6 ๐ด๐‘–๐‘— ๐ด๐‘–๐‘— PROPN cana-1164 85 7 = = PUNCT cana-1164 85 8 { { PUNCT cana-1164 85 9 [ [ X cana-1164 85 10 ๐‘Ž๐‘ข๐‘ฃ]๐‘žร—๐‘ž ๐‘Ž๐‘ข๐‘ฃ]๐‘žร—๐‘ž PROPN cana-1164 85 11 , , PUNCT cana-1164 85 12 for for ADP cana-1164 85 13 i i PROPN cana-1164 85 14 = = SYM cana-1164 85 15 j j PROPN cana-1164 85 16 ; ; PUNCT cana-1164 85 17 q q PROPN cana-1164 85 18 โ‰ค โ‰ค NUM cana-1164 85 19 m m VERB cana-1164 85 20 โ‰ค โ‰ค NOUN cana-1164 85 21 n n PRON cana-1164 85 22 0 0 NUM cana-1164 85 23 , , PUNCT cana-1164 85 24 for for ADP cana-1164 85 25 i i PRON cana-1164 85 26 โ‰  โ‰  PROPN cana-1164 85 27 j. j. PROPN cana-1164 85 28 where where SCONJ cana-1164 85 29 , , PUNCT cana-1164 85 30 ๐‘Ž๐‘ข๐‘ฃ ๐‘Ž๐‘ข๐‘ฃ X cana-1164 85 31 = = PUNCT cana-1164 85 32 { { PUNCT cana-1164 85 33 1 1 NUM cana-1164 85 34 , , PUNCT cana-1164 85 35 if if SCONJ cana-1164 85 36 there there PRON cana-1164 85 37 is be VERB cana-1164 85 38 a a DET cana-1164 85 39 directed direct VERB cana-1164 85 40 arc arc NOUN cana-1164 85 41 from from ADP cana-1164 85 42 uth uth NOUN cana-1164 85 43 vertex vertex NOUN cana-1164 85 44 to to PART cana-1164 85 45 vth vth VERB cana-1164 85 46 vertex vertex NOUN cana-1164 85 47 . . PUNCT cana-1164 85 48 0 0 NUM cana-1164 85 49 , , PUNCT cana-1164 85 50 otherwise otherwise ADV cana-1164 85 51 . . PUNCT cana-1164 86 1 similarly similarly ADV cana-1164 86 2 , , PUNCT cana-1164 86 3 the the DET cana-1164 86 4 in in ADP cana-1164 86 5 - - PUNCT cana-1164 86 6 adjacency adjacency NOUN cana-1164 86 7 matrix matrix NOUN cana-1164 86 8 ๐ดฮณ ๐ดฮณ PROPN cana-1164 86 9 โˆ’ โˆ’ PROPN cana-1164 86 10 can can AUX cana-1164 86 11 be be AUX cana-1164 86 12 defined define VERB cana-1164 86 13 as as ADP cana-1164 86 14 a a DET cana-1164 86 15 block block NOUN cana-1164 86 16 diagonal diagonal ADJ cana-1164 86 17 matrix matrix NOUN cana-1164 86 18 [ [ X cana-1164 86 19 ๐ด๐‘–๐‘—]๐‘šร—๐‘š ๐ด๐‘–๐‘—]๐‘šร—๐‘š NOUN cana-1164 86 20 i.e. i.e. X cana-1164 87 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 87 2 โˆ’ โˆ’ PROPN cana-1164 88 1 = = PUNCT cana-1164 89 1 [ [ X cana-1164 89 2 ๐ด๐‘–๐‘—]๐‘šร—๐‘š ๐ด๐‘–๐‘—]๐‘šร—๐‘š PROPN cana-1164 89 3 , , PUNCT cana-1164 89 4 such such ADJ cana-1164 89 5 that that SCONJ cana-1164 89 6 ๐ด๐‘–๐‘— ๐ด๐‘–๐‘— PROPN cana-1164 89 7 = = PUNCT cana-1164 89 8 { { PUNCT cana-1164 89 9 [ [ X cana-1164 89 10 ๐‘Ž๐‘ข๐‘ฃ]๐‘žร—๐‘ž ๐‘Ž๐‘ข๐‘ฃ]๐‘žร—๐‘ž PROPN cana-1164 89 11 , , PUNCT cana-1164 89 12 for for ADP cana-1164 89 13 i i PROPN cana-1164 89 14 = = SYM cana-1164 89 15 j j PROPN cana-1164 89 16 ; ; PUNCT cana-1164 89 17 q q PROPN cana-1164 89 18 โ‰ค โ‰ค NUM cana-1164 89 19 m m VERB cana-1164 89 20 โ‰ค โ‰ค NOUN cana-1164 89 21 n n PRON cana-1164 89 22 0 0 NUM cana-1164 89 23 , , PUNCT cana-1164 89 24 for for ADP cana-1164 89 25 i i PRON cana-1164 89 26 โ‰  โ‰  PROPN cana-1164 89 27 j. j. PROPN cana-1164 89 28 where where SCONJ cana-1164 89 29 , , PUNCT cana-1164 89 30 ๐‘Ž๐‘ข๐‘ฃ ๐‘Ž๐‘ข๐‘ฃ X cana-1164 89 31 = = PUNCT cana-1164 89 32 { { PUNCT cana-1164 89 33 1 1 NUM cana-1164 89 34 , , PUNCT cana-1164 89 35 if if SCONJ cana-1164 89 36 there there PRON cana-1164 89 37 is be VERB cana-1164 89 38 a a DET cana-1164 89 39 directed direct VERB cana-1164 89 40 arc arc NOUN cana-1164 89 41 from from ADP cana-1164 89 42 vth vth NOUN cana-1164 89 43 vertex vertex NOUN cana-1164 89 44 to to AUX cana-1164 89 45 uth uth VERB cana-1164 89 46 vertex vertex NOUN cana-1164 89 47 . . PUNCT cana-1164 89 48 0 0 NUM cana-1164 89 49 , , PUNCT cana-1164 89 50 otherwise otherwise ADV cana-1164 89 51 . . PUNCT cana-1164 90 1 we we PRON cana-1164 90 2 have have VERB cana-1164 90 3 the the DET cana-1164 90 4 following follow VERB cana-1164 90 5 observations observation NOUN cana-1164 90 6 about about ADP cana-1164 90 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 90 8 + + CCONJ cana-1164 90 9 and and CCONJ cana-1164 90 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 90 11 โˆ’ โˆ’ PROPN cana-1164 90 12 of of ADP cana-1164 90 13 a a DET cana-1164 90 14 digraph digraph ADJ cana-1164 90 15 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 90 16 , , PUNCT cana-1164 90 17 ๐‘˜ ๐‘˜ NOUN cana-1164 90 18 ): ): PUNCT cana-1164 90 19 i. i. NOUN cana-1164 90 20 each each DET cana-1164 90 21 non non ADJ cana-1164 90 22 - - ADJ cana-1164 90 23 zero zero NUM cana-1164 90 24 element element NOUN cana-1164 90 25 on on ADP cana-1164 90 26 the the DET cana-1164 90 27 main main ADJ cana-1164 90 28 diagonal diagonal NOUN cana-1164 90 29 of of ADP cana-1164 90 30 ๐ดฮณ ๐ดฮณ PROPN cana-1164 90 31 + + CCONJ cana-1164 90 32 and and CCONJ cana-1164 90 33 ๐ดฮณ ๐ดฮณ PROPN cana-1164 90 34 โˆ’ โˆ’ PROPN cana-1164 90 35 represents represent VERB cana-1164 90 36 a a DET cana-1164 90 37 loop loop NOUN cana-1164 90 38 at at ADP cana-1164 90 39 the the DET cana-1164 90 40 corresponding corresponding ADJ cana-1164 90 41 vertex vertex NOUN cana-1164 90 42 . . PUNCT cana-1164 91 1 ii ii PROPN cana-1164 91 2 . . PUNCT cana-1164 92 1 the the DET cana-1164 92 2 number number NOUN cana-1164 92 3 of of ADP cana-1164 92 4 non non ADJ cana-1164 92 5 - - ADJ cana-1164 92 6 zero zero NUM cana-1164 92 7 entries entry NOUN cana-1164 92 8 of of ADP cana-1164 92 9 either either CCONJ cana-1164 92 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 92 11 + + CCONJ cana-1164 92 12 or or CCONJ cana-1164 92 13 ๐ดฮณ ๐ดฮณ PROPN cana-1164 92 14 โˆ’ โˆ’ PROPN cana-1164 92 15 equals equal VERB cana-1164 92 16 the the DET cana-1164 92 17 number number NOUN cana-1164 92 18 of of ADP cana-1164 92 19 directed direct VERB cana-1164 92 20 arcs arc NOUN cana-1164 92 21 in in ADP cana-1164 92 22 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 92 23 , , PUNCT cana-1164 92 24 ๐‘˜ ๐‘˜ NOUN cana-1164 92 25 ) ) PUNCT cana-1164 92 26 . . PUNCT cana-1164 93 1 communications communication NOUN cana-1164 93 2 on on ADP cana-1164 93 3 applied apply VERB cana-1164 93 4 nonlinear nonlinear ADJ cana-1164 93 5 analysis analysis NOUN cana-1164 93 6 issn issn NOUN cana-1164 93 7 : : PUNCT cana-1164 93 8 1074 1074 NUM cana-1164 93 9 - - PUNCT cana-1164 93 10 133x 133x NUM cana-1164 93 11 vol vol NOUN cana-1164 93 12 31 31 NUM cana-1164 93 13 no no NOUN cana-1164 93 14 . . PUNCT cana-1164 94 1 6s 6s NUM cana-1164 94 2 ( ( PUNCT cana-1164 94 3 2024 2024 NUM cana-1164 94 4 ) ) PUNCT cana-1164 94 5 101 101 NUM cana-1164 94 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 94 7 iii iii X cana-1164 94 8 . . PUNCT cana-1164 95 1 permutations permutation NOUN cana-1164 95 2 of of ADP cana-1164 95 3 any any DET cana-1164 95 4 rows row NOUN cana-1164 95 5 together together ADV cana-1164 95 6 with with ADP cana-1164 95 7 a a DET cana-1164 95 8 permutation permutation NOUN cana-1164 95 9 of of ADP cana-1164 95 10 the the DET cana-1164 95 11 corresponding correspond VERB cana-1164 95 12 columns column NOUN cana-1164 95 13 do do AUX cana-1164 95 14 not not PART cana-1164 95 15 alter alter VERB cana-1164 95 16 the the DET cana-1164 95 17 power power NOUN cana-1164 95 18 digraph digraph NOUN cana-1164 95 19 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 95 20 , , PUNCT cana-1164 95 21 ๐‘˜ ๐‘˜ NOUN cana-1164 95 22 ) ) PUNCT cana-1164 95 23 ; ; PUNCT cana-1164 95 24 indicating indicate VERB cana-1164 95 25 that that SCONJ cana-1164 95 26 the the DET cana-1164 95 27 permutation permutation NOUN cana-1164 95 28 simply simply ADV cana-1164 95 29 rearranges rearrange VERB cana-1164 95 30 the the DET cana-1164 95 31 vertices vertex NOUN cana-1164 95 32 . . PUNCT cana-1164 96 1 iv iv X cana-1164 96 2 . . PUNCT cana-1164 97 1 the the DET cana-1164 97 2 out out ADJ cana-1164 97 3 - - PUNCT cana-1164 97 4 adjacency adjacency NOUN cana-1164 97 5 matrix matrix NOUN cana-1164 97 6 ๐ดฮณ ๐ดฮณ PROPN cana-1164 97 7 + + CCONJ cana-1164 97 8 ( ( PUNCT cana-1164 97 9 or or CCONJ cana-1164 97 10 in in ADP cana-1164 97 11 - - PUNCT cana-1164 97 12 adjacency adjacency NOUN cana-1164 97 13 matrix matrix NOUN cana-1164 97 14 ๐ดฮณ ๐ดฮณ VERB cana-1164 97 15 โˆ’ โˆ’ PROPN cana-1164 97 16 ) ) PUNCT cana-1164 97 17 is be AUX cana-1164 97 18 not not PART cana-1164 97 19 unique unique ADJ cana-1164 97 20 ( ( PUNCT cana-1164 97 21 follows follow VERB cana-1164 97 22 from from ADP cana-1164 97 23 iii iii PROPN cana-1164 97 24 . . PUNCT cana-1164 97 25 ) ) PUNCT cana-1164 97 26 . . PUNCT cana-1164 98 1 v. v. ADP cana-1164 98 2 the the DET cana-1164 98 3 out out NOUN cana-1164 98 4 - - PUNCT cana-1164 98 5 adjacency adjacency NOUN cana-1164 98 6 ( ( PUNCT cana-1164 98 7 or or CCONJ cana-1164 98 8 in in ADP cana-1164 98 9 - - PUNCT cana-1164 98 10 adjacency adjacency NOUN cana-1164 98 11 ) ) PUNCT cana-1164 98 12 matrix matrix NOUN cana-1164 99 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 99 2 + + CCONJ cana-1164 99 3 ( ( PUNCT cana-1164 99 4 or or CCONJ cana-1164 99 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 99 6 โˆ’ โˆ’ PROPN cana-1164 99 7 ) ) PUNCT cana-1164 99 8 of of ADP cana-1164 99 9 the the DET cana-1164 99 10 digraph digraph ADJ cana-1164 99 11 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 99 12 , , PUNCT cana-1164 99 13 ๐‘˜ ๐‘˜ NOUN cana-1164 99 14 ) ) PUNCT cana-1164 99 15 can can AUX cana-1164 99 16 be be AUX cana-1164 99 17 written write VERB cana-1164 99 18 as as ADP cana-1164 99 19 a a DET cana-1164 99 20 block block NOUN cana-1164 99 21 - - PUNCT cana-1164 99 22 diagonal diagonal ADJ cana-1164 99 23 matrix matrix NOUN cana-1164 99 24 with with ADP cana-1164 99 25 diagonal diagonal ADJ cana-1164 99 26 elements element NOUN cana-1164 99 27 as as SCONJ cana-1164 99 28 the the DET cana-1164 99 29 out out NOUN cana-1164 99 30 - - PUNCT cana-1164 99 31 adjacency adjacency NOUN cana-1164 99 32 ( ( PUNCT cana-1164 99 33 or or CCONJ cana-1164 99 34 in in ADP cana-1164 99 35 - - PUNCT cana-1164 99 36 adjacency adjacency NOUN cana-1164 99 37 ) ) PUNCT cana-1164 99 38 matrices matrix NOUN cana-1164 99 39 of of ADP cana-1164 99 40 the the DET cana-1164 99 41 component component NOUN cana-1164 99 42 digraphs digraph VERB cana-1164 99 43 of of ADP cana-1164 99 44 the the DET cana-1164 99 45 digraph digraph ADJ cana-1164 99 46 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 99 47 , , PUNCT cana-1164 99 48 ๐‘˜ ๐‘˜ NOUN cana-1164 99 49 ) ) PUNCT cana-1164 99 50 . . PUNCT cana-1164 100 1 example example NOUN cana-1164 100 2 3.1 3.1 NUM cana-1164 100 3 . . PUNCT cana-1164 101 1 let let VERB cana-1164 101 2 us we PRON cana-1164 101 3 consider consider VERB cana-1164 101 4 the the DET cana-1164 101 5 digraph digraph ADJ cana-1164 101 6 ฮณ(6 ฮณ(6 PROPN cana-1164 101 7 , , PUNCT cana-1164 101 8 2 2 NUM cana-1164 101 9 ) ) PUNCT cana-1164 101 10 . . PUNCT cana-1164 102 1 figure figure VERB cana-1164 102 2 1 1 NUM cana-1164 102 3 : : PUNCT cana-1164 102 4 digraph digraph ADJ cana-1164 102 5 ฮณ(6 ฮณ(6 PROPN cana-1164 102 6 , , PUNCT cana-1164 102 7 2 2 NUM cana-1164 102 8 ) ) PUNCT cana-1164 102 9 with with ADP cana-1164 102 10 components component NOUN cana-1164 102 11 ฮณ1 ฮณ1 PROPN cana-1164 102 12 , , PUNCT cana-1164 102 13 ฮณ2 ฮณ2 PROPN cana-1164 102 14 , , PUNCT cana-1164 102 15 ฮณ3 ฮณ3 NOUN cana-1164 102 16 , , PUNCT cana-1164 102 17 ฮณ4 ฮณ4 PROPN cana-1164 102 18 . . PUNCT cana-1164 103 1 here here ADV cana-1164 103 2 , , PUNCT cana-1164 103 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 103 4 + + CCONJ cana-1164 103 5 = = SYM cana-1164 103 6 0 0 NUM cana-1164 103 7 1 1 NUM cana-1164 103 8 2 2 NUM cana-1164 103 9 3 3 NUM cana-1164 103 10 4 4 NUM cana-1164 103 11 5 5 NUM cana-1164 103 12 0 0 NUM cana-1164 103 13 1 1 NUM cana-1164 103 14 0 0 NUM cana-1164 103 15 0 0 NUM cana-1164 103 16 0 0 NUM cana-1164 103 17 0 0 NUM cana-1164 103 18 0 0 NUM cana-1164 103 19 1 1 NUM cana-1164 103 20 0 0 NUM cana-1164 103 21 1 1 NUM cana-1164 103 22 0 0 NUM cana-1164 103 23 0 0 NUM cana-1164 103 24 0 0 NUM cana-1164 103 25 0 0 NUM cana-1164 103 26 2 2 NUM cana-1164 103 27 0 0 NUM cana-1164 103 28 0 0 NUM cana-1164 103 29 0 0 NUM cana-1164 103 30 0 0 NUM cana-1164 103 31 1 1 NUM cana-1164 103 32 0 0 NUM cana-1164 103 33 3 3 NUM cana-1164 103 34 0 0 NUM cana-1164 103 35 0 0 NUM cana-1164 103 36 0 0 NUM cana-1164 103 37 1 1 NUM cana-1164 103 38 0 0 NUM cana-1164 103 39 0 0 NUM cana-1164 103 40 4 4 NUM cana-1164 103 41 0 0 NUM cana-1164 103 42 0 0 NUM cana-1164 103 43 0 0 NUM cana-1164 103 44 0 0 NUM cana-1164 103 45 1 1 NUM cana-1164 103 46 0 0 NUM cana-1164 103 47 5 5 NUM cana-1164 103 48 0 0 NUM cana-1164 103 49 1 1 NUM cana-1164 103 50 0 0 NUM cana-1164 103 51 0 0 NUM cana-1164 103 52 0 0 NUM cana-1164 103 53 0 0 NUM cana-1164 103 54 ๏ƒฉ ๏ƒฉ NUM cana-1164 103 55 ๏ƒน ๏ƒน ADJ cana-1164 103 56 ๏ƒช ๏ƒช NOUN cana-1164 103 57 ๏ƒบ ๏ƒบ PROPN cana-1164 103 58 ๏ƒช ๏ƒช NOUN cana-1164 103 59 ๏ƒบ ๏ƒบ NOUN cana-1164 103 60 ๏ƒช ๏ƒช NOUN cana-1164 103 61 ๏ƒบ ๏ƒบ NOUN cana-1164 103 62 ๏ƒช ๏ƒช NOUN cana-1164 103 63 ๏ƒบ ๏ƒบ NOUN cana-1164 103 64 ๏ƒช ๏ƒช NOUN cana-1164 103 65 ๏ƒบ ๏ƒบ NOUN cana-1164 103 66 ๏ƒช ๏ƒช NOUN cana-1164 103 67 ๏ƒบ ๏ƒบ NOUN cana-1164 103 68 ๏ƒช ๏ƒช NOUN cana-1164 103 69 ๏ƒบ ๏ƒบ NOUN cana-1164 103 70 ๏ƒซ ๏ƒซ NOUN cana-1164 103 71 ๏ƒป ๏ƒป PROPN cana-1164 103 72 , , PUNCT cana-1164 103 73 ๐ดฮณ ๐ดฮณ PROPN cana-1164 103 74 โˆ’ โˆ’ PROPN cana-1164 104 1 = = SYM cana-1164 104 2 0 0 NUM cana-1164 104 3 1 1 NUM cana-1164 104 4 2 2 NUM cana-1164 104 5 3 3 NUM cana-1164 104 6 4 4 NUM cana-1164 104 7 5 5 NUM cana-1164 104 8 0 0 NUM cana-1164 104 9 1 1 NUM cana-1164 104 10 0 0 NUM cana-1164 104 11 0 0 NUM cana-1164 104 12 0 0 NUM cana-1164 104 13 0 0 NUM cana-1164 104 14 0 0 NUM cana-1164 104 15 1 1 NUM cana-1164 104 16 0 0 NUM cana-1164 104 17 1 1 NUM cana-1164 104 18 0 0 NUM cana-1164 104 19 0 0 NUM cana-1164 104 20 0 0 NUM cana-1164 104 21 1 1 NUM cana-1164 104 22 2 2 NUM cana-1164 104 23 0 0 NUM cana-1164 104 24 0 0 NUM cana-1164 104 25 0 0 NUM cana-1164 104 26 0 0 NUM cana-1164 104 27 0 0 NUM cana-1164 104 28 0 0 NUM cana-1164 104 29 3 3 NUM cana-1164 104 30 0 0 NUM cana-1164 104 31 0 0 NUM cana-1164 104 32 0 0 NUM cana-1164 104 33 1 1 NUM cana-1164 104 34 0 0 NUM cana-1164 104 35 0 0 NUM cana-1164 104 36 4 4 NUM cana-1164 104 37 0 0 NUM cana-1164 104 38 0 0 NUM cana-1164 104 39 1 1 NUM cana-1164 104 40 0 0 NUM cana-1164 104 41 1 1 NUM cana-1164 104 42 0 0 NUM cana-1164 104 43 5 5 NUM cana-1164 104 44 0 0 NUM cana-1164 104 45 0 0 NUM cana-1164 104 46 0 0 NUM cana-1164 104 47 0 0 NUM cana-1164 104 48 0 0 NUM cana-1164 104 49 0 0 NUM cana-1164 104 50 ๏ƒฉ ๏ƒฉ NUM cana-1164 104 51 ๏ƒน ๏ƒน ADJ cana-1164 104 52 ๏ƒช ๏ƒช NOUN cana-1164 104 53 ๏ƒบ ๏ƒบ PROPN cana-1164 104 54 ๏ƒช ๏ƒช NOUN cana-1164 104 55 ๏ƒบ ๏ƒบ NOUN cana-1164 104 56 ๏ƒช ๏ƒช NOUN cana-1164 104 57 ๏ƒบ ๏ƒบ NOUN cana-1164 104 58 ๏ƒช ๏ƒช NOUN cana-1164 104 59 ๏ƒบ ๏ƒบ NOUN cana-1164 104 60 ๏ƒช ๏ƒช NOUN cana-1164 104 61 ๏ƒบ ๏ƒบ NOUN cana-1164 104 62 ๏ƒช ๏ƒช NOUN cana-1164 104 63 ๏ƒบ ๏ƒบ NOUN cana-1164 104 64 ๏ƒช ๏ƒช NOUN cana-1164 104 65 ๏ƒบ ๏ƒบ NOUN cana-1164 104 66 ๏ƒซ ๏ƒซ NOUN cana-1164 104 67 ๏ƒป ๏ƒป PROPN cana-1164 104 68 , , PUNCT cana-1164 104 69 and and CCONJ cana-1164 104 70 ( ( PUNCT cana-1164 104 71 ๐ดฮณ ๐ดฮณ PROPN cana-1164 104 72 + + NOUN cana-1164 104 73 ) ) PUNCT cana-1164 104 74 ๐‘ก ๐‘ก NOUN cana-1164 104 75 = = NOUN cana-1164 104 76 0 0 NUM cana-1164 104 77 1 1 NUM cana-1164 104 78 2 2 NUM cana-1164 104 79 3 3 NUM cana-1164 104 80 4 4 NUM cana-1164 104 81 5 5 NUM cana-1164 104 82 0 0 NUM cana-1164 104 83 1 1 NUM cana-1164 104 84 0 0 NUM cana-1164 104 85 0 0 NUM cana-1164 104 86 0 0 NUM cana-1164 104 87 0 0 NUM cana-1164 104 88 0 0 NUM cana-1164 104 89 1 1 NUM cana-1164 104 90 0 0 NUM cana-1164 104 91 1 1 NUM cana-1164 104 92 0 0 NUM cana-1164 104 93 0 0 NUM cana-1164 104 94 0 0 NUM cana-1164 104 95 1 1 NUM cana-1164 104 96 2 2 NUM cana-1164 104 97 0 0 NUM cana-1164 104 98 0 0 NUM cana-1164 104 99 0 0 NUM cana-1164 104 100 0 0 NUM cana-1164 104 101 0 0 NUM cana-1164 104 102 0 0 NUM cana-1164 104 103 3 3 NUM cana-1164 104 104 0 0 NUM cana-1164 104 105 0 0 NUM cana-1164 104 106 0 0 NUM cana-1164 104 107 1 1 NUM cana-1164 104 108 0 0 NUM cana-1164 104 109 0 0 NUM cana-1164 104 110 4 4 NUM cana-1164 104 111 0 0 NUM cana-1164 104 112 0 0 NUM cana-1164 104 113 1 1 NUM cana-1164 104 114 0 0 NUM cana-1164 104 115 1 1 NUM cana-1164 104 116 0 0 NUM cana-1164 104 117 5 5 NUM cana-1164 104 118 0 0 NUM cana-1164 104 119 0 0 NUM cana-1164 104 120 0 0 NUM cana-1164 104 121 0 0 NUM cana-1164 104 122 0 0 NUM cana-1164 104 123 0 0 NUM cana-1164 104 124 ๏ƒฉ ๏ƒฉ NUM cana-1164 104 125 ๏ƒน ๏ƒน ADJ cana-1164 104 126 ๏ƒช ๏ƒช NOUN cana-1164 104 127 ๏ƒบ ๏ƒบ PROPN cana-1164 104 128 ๏ƒช ๏ƒช NOUN cana-1164 104 129 ๏ƒบ ๏ƒบ NOUN cana-1164 104 130 ๏ƒช ๏ƒช NOUN cana-1164 104 131 ๏ƒบ ๏ƒบ NOUN cana-1164 104 132 ๏ƒช ๏ƒช NOUN cana-1164 104 133 ๏ƒบ ๏ƒบ NOUN cana-1164 104 134 ๏ƒช ๏ƒช NOUN cana-1164 104 135 ๏ƒบ ๏ƒบ NOUN cana-1164 104 136 ๏ƒช ๏ƒช NOUN cana-1164 104 137 ๏ƒบ ๏ƒบ NOUN cana-1164 104 138 ๏ƒช ๏ƒช NOUN cana-1164 104 139 ๏ƒบ ๏ƒบ NOUN cana-1164 104 140 ๏ƒซ ๏ƒซ NOUN cana-1164 104 141 ๏ƒป ๏ƒป PROPN cana-1164 104 142 let let VERB cana-1164 104 143 us we PRON cana-1164 104 144 apply apply VERB cana-1164 104 145 the the DET cana-1164 104 146 elementary elementary ADJ cana-1164 104 147 operations operation NOUN cana-1164 104 148 ๐‘…2 ๐‘…2 VERB cana-1164 104 149 โ†” โ†” NOUN cana-1164 104 150 ๐‘…6 ๐‘…6 NOUN cana-1164 104 151 and and CCONJ cana-1164 104 152 then then ADV cana-1164 104 153 ๐ถ2 ๐ถ2 INTJ cana-1164 104 154 โ†” โ†” PROPN cana-1164 104 155 ๐ถ6 ๐ถ6 VERB cana-1164 104 156 ; ; PUNCT cana-1164 104 157 ๐‘…3 ๐‘…3 PROPN cana-1164 104 158 โ†” โ†” PROPN cana-1164 104 159 ๐‘…6 ๐‘…6 NOUN cana-1164 104 160 and and CCONJ cana-1164 104 161 then then ADV cana-1164 104 162 ๐ถ3 ๐ถ3 PROPN cana-1164 104 163 โ†” โ†” PROPN cana-1164 104 164 ๐ถ6 ๐ถ6 VERB cana-1164 104 165 and and CCONJ cana-1164 104 166 finally finally ADV cana-1164 104 167 ๐‘…5 ๐‘…5 PRON cana-1164 104 168 โ†” โ†” VERB cana-1164 104 169 ๐‘…6 ๐‘…6 NOUN cana-1164 105 1 and and CCONJ cana-1164 105 2 then then ADV cana-1164 105 3 ๐ถ5 ๐ถ5 VERB cana-1164 105 4 โ†” โ†” PROPN cana-1164 105 5 ๐ถ6 ๐ถ6 VERB cana-1164 105 6 to to ADP cana-1164 105 7 the the DET cana-1164 105 8 matrix matrix NOUN cana-1164 105 9 ๐ดฮณ ๐ดฮณ PROPN cana-1164 105 10 + + ADV cana-1164 105 11 , , PUNCT cana-1164 105 12 we we PRON cana-1164 105 13 get get VERB cana-1164 105 14 the the DET cana-1164 105 15 following follow VERB cana-1164 105 16 matrix matrix NOUN cana-1164 105 17 0 0 NUM cana-1164 105 18 5 5 NUM cana-1164 105 19 1 1 NUM cana-1164 105 20 3 3 NUM cana-1164 105 21 2 2 NUM cana-1164 105 22 4 4 NUM cana-1164 105 23 0 0 NUM cana-1164 105 24 1 1 NUM cana-1164 105 25 0 0 NUM cana-1164 105 26 0 0 NUM cana-1164 105 27 0 0 NUM cana-1164 105 28 0 0 NUM cana-1164 105 29 0 0 NUM cana-1164 105 30 5 5 NUM cana-1164 105 31 0 0 NUM cana-1164 105 32 0 0 NUM cana-1164 105 33 1 1 NUM cana-1164 105 34 0 0 NUM cana-1164 105 35 0 0 NUM cana-1164 105 36 0 0 NUM cana-1164 105 37 1 1 NUM cana-1164 105 38 0 0 NUM cana-1164 105 39 0 0 NUM cana-1164 105 40 1 1 NUM cana-1164 105 41 0 0 NUM cana-1164 105 42 0 0 NUM cana-1164 105 43 0 0 NUM cana-1164 105 44 3 3 NUM cana-1164 105 45 0 0 NUM cana-1164 105 46 0 0 NUM cana-1164 105 47 0 0 NUM cana-1164 105 48 1 1 NUM cana-1164 105 49 0 0 NUM cana-1164 105 50 0 0 NUM cana-1164 105 51 2 2 NUM cana-1164 105 52 0 0 NUM cana-1164 105 53 0 0 NUM cana-1164 105 54 0 0 NUM cana-1164 105 55 0 0 NUM cana-1164 105 56 0 0 NUM cana-1164 105 57 1 1 NUM cana-1164 105 58 4 4 NUM cana-1164 105 59 0 0 NUM cana-1164 105 60 0 0 NUM cana-1164 105 61 0 0 NUM cana-1164 105 62 0 0 NUM cana-1164 105 63 0 0 NUM cana-1164 105 64 1 1 NUM cana-1164 105 65 ๏ƒฉ ๏ƒฉ NUM cana-1164 105 66 ๏ƒน ๏ƒน ADJ cana-1164 105 67 ๏ƒช ๏ƒช NOUN cana-1164 105 68 ๏ƒบ ๏ƒบ PROPN cana-1164 105 69 ๏ƒช ๏ƒช NOUN cana-1164 105 70 ๏ƒบ ๏ƒบ NOUN cana-1164 105 71 ๏ƒช ๏ƒช NOUN cana-1164 105 72 ๏ƒบ ๏ƒบ NOUN cana-1164 105 73 ๏ƒช ๏ƒช NOUN cana-1164 105 74 ๏ƒบ ๏ƒบ NOUN cana-1164 105 75 ๏ƒช ๏ƒช NOUN cana-1164 105 76 ๏ƒบ ๏ƒบ NOUN cana-1164 105 77 ๏ƒช ๏ƒช NOUN cana-1164 105 78 ๏ƒบ ๏ƒบ NOUN cana-1164 105 79 ๏ƒช ๏ƒช NOUN cana-1164 105 80 ๏ƒบ ๏ƒบ NOUN cana-1164 105 81 ๏ƒซ ๏ƒซ NOUN cana-1164 105 82 ๏ƒป ๏ƒป PROPN cana-1164 105 83 , , PUNCT cana-1164 105 84 which which PRON cana-1164 105 85 gives give VERB cana-1164 105 86 us we PRON cana-1164 105 87 the the DET cana-1164 105 88 digraph digraph ADJ cana-1164 105 89 ฮณ(6 ฮณ(6 PROPN cana-1164 105 90 , , PUNCT cana-1164 105 91 2 2 NUM cana-1164 105 92 ) ) PUNCT cana-1164 105 93 . . PUNCT cana-1164 106 1 hence hence ADV cana-1164 106 2 , , PUNCT cana-1164 106 3 permuting permuting NOUN cana-1164 106 4 rows row VERB cana-1164 106 5 together together ADV cana-1164 106 6 with with ADP cana-1164 106 7 the the DET cana-1164 106 8 corresponding correspond VERB cana-1164 106 9 columns column NOUN cana-1164 106 10 , , PUNCT cana-1164 106 11 the the DET cana-1164 106 12 matrix matrix NOUN cana-1164 107 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 107 2 + + CCONJ cana-1164 107 3 can can AUX cana-1164 107 4 be be AUX cana-1164 107 5 written write VERB cana-1164 107 6 as as SCONJ cana-1164 107 7 communications communication NOUN cana-1164 107 8 on on ADP cana-1164 107 9 applied apply VERB cana-1164 107 10 nonlinear nonlinear ADJ cana-1164 107 11 analysis analysis NOUN cana-1164 107 12 issn issn NOUN cana-1164 107 13 : : PUNCT cana-1164 107 14 1074 1074 NUM cana-1164 107 15 - - PUNCT cana-1164 107 16 133x 133x NUM cana-1164 107 17 vol vol NOUN cana-1164 107 18 31 31 NUM cana-1164 107 19 no no NOUN cana-1164 107 20 . . PUNCT cana-1164 108 1 6s 6s NUM cana-1164 108 2 ( ( PUNCT cana-1164 108 3 2024 2024 NUM cana-1164 108 4 ) ) PUNCT cana-1164 108 5 102 102 NUM cana-1164 108 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 109 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 109 2 + + CCONJ cana-1164 109 3 = = SYM cana-1164 109 4 0 0 SYM cana-1164 109 5 5 5 NUM cana-1164 109 6 1 1 NUM cana-1164 109 7 3 3 NUM cana-1164 109 8 2 2 NUM cana-1164 109 9 4 4 NUM cana-1164 109 10 0 0 NUM cana-1164 109 11 1 1 NUM cana-1164 109 12 0 0 NUM cana-1164 109 13 0 0 NUM cana-1164 109 14 0 0 NUM cana-1164 109 15 0 0 NUM cana-1164 109 16 0 0 NUM cana-1164 109 17 5 5 NUM cana-1164 109 18 0 0 NUM cana-1164 109 19 0 0 NUM cana-1164 109 20 1 1 NUM cana-1164 109 21 0 0 NUM cana-1164 109 22 0 0 NUM cana-1164 109 23 0 0 NUM cana-1164 109 24 1 1 NUM cana-1164 109 25 0 0 NUM cana-1164 109 26 0 0 NUM cana-1164 109 27 1 1 NUM cana-1164 109 28 0 0 NUM cana-1164 109 29 0 0 NUM cana-1164 109 30 0 0 NUM cana-1164 109 31 3 3 NUM cana-1164 109 32 0 0 NUM cana-1164 109 33 0 0 NUM cana-1164 109 34 0 0 NUM cana-1164 109 35 1 1 NUM cana-1164 109 36 0 0 NUM cana-1164 109 37 0 0 NUM cana-1164 109 38 2 2 NUM cana-1164 109 39 0 0 NUM cana-1164 109 40 0 0 NUM cana-1164 109 41 0 0 NUM cana-1164 109 42 0 0 NUM cana-1164 109 43 0 0 NUM cana-1164 109 44 1 1 NUM cana-1164 109 45 4 4 NUM cana-1164 109 46 0 0 NUM cana-1164 109 47 0 0 NUM cana-1164 109 48 0 0 NUM cana-1164 109 49 0 0 NUM cana-1164 109 50 0 0 NUM cana-1164 109 51 1 1 NUM cana-1164 109 52 ๏ƒฉ ๏ƒฉ NUM cana-1164 109 53 ๏ƒน ๏ƒน ADJ cana-1164 109 54 ๏ƒช ๏ƒช NOUN cana-1164 109 55 ๏ƒบ ๏ƒบ PROPN cana-1164 109 56 ๏ƒช ๏ƒช NOUN cana-1164 109 57 ๏ƒบ ๏ƒบ NOUN cana-1164 109 58 ๏ƒช ๏ƒช NOUN cana-1164 109 59 ๏ƒบ ๏ƒบ NOUN cana-1164 109 60 ๏ƒช ๏ƒช NOUN cana-1164 109 61 ๏ƒบ ๏ƒบ NOUN cana-1164 109 62 ๏ƒช ๏ƒช NOUN cana-1164 109 63 ๏ƒบ ๏ƒบ NOUN cana-1164 109 64 ๏ƒช ๏ƒช NOUN cana-1164 109 65 ๏ƒบ ๏ƒบ NOUN cana-1164 109 66 ๏ƒช ๏ƒช NOUN cana-1164 109 67 ๏ƒบ ๏ƒบ NOUN cana-1164 109 68 ๏ƒซ ๏ƒซ VERB cana-1164 109 69 ๏ƒป ๏ƒป PROPN cana-1164 109 70 = = PRON cana-1164 109 71 [ [ PUNCT cana-1164 109 72 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 109 73 + + X cana-1164 109 74 0 0 NUM cana-1164 109 75 0 0 NUM cana-1164 109 76 0 0 NUM cana-1164 109 77 0 0 NUM cana-1164 109 78 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 109 79 + + X cana-1164 109 80 0 0 NUM cana-1164 109 81 0 0 NUM cana-1164 109 82 0 0 NUM cana-1164 109 83 0 0 NUM cana-1164 109 84 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 110 1 + + X cana-1164 110 2 0 0 NUM cana-1164 110 3 0 0 NUM cana-1164 110 4 0 0 NUM cana-1164 110 5 0 0 NUM cana-1164 110 6 ๐ดฮณ4 ๐ดฮณ4 NOUN cana-1164 111 1 + + X cana-1164 111 2 ] ] PUNCT cana-1164 111 3 where where SCONJ cana-1164 111 4 , , PUNCT cana-1164 111 5 ๐ดฮณ1 ๐ดฮณ1 VERB cana-1164 111 6 + + X cana-1164 112 1 = = SYM cana-1164 113 1 [ [ X cana-1164 113 2 1 1 NUM cana-1164 113 3 ] ] PUNCT cana-1164 113 4 , , PUNCT cana-1164 113 5 ๐ดฮณ2 ๐ดฮณ2 VERB cana-1164 113 6 + + X cana-1164 114 1 = = X cana-1164 114 2 [ [ PUNCT cana-1164 114 3 0 0 NUM cana-1164 114 4 1 1 NUM cana-1164 114 5 0 0 NUM cana-1164 114 6 1 1 NUM cana-1164 114 7 ] ] PUNCT cana-1164 114 8 , , PUNCT cana-1164 114 9 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 114 10 + + X cana-1164 115 1 = = SYM cana-1164 116 1 [ [ X cana-1164 116 2 1 1 NUM cana-1164 116 3 ] ] PUNCT cana-1164 116 4 , , PUNCT cana-1164 116 5 and and CCONJ cana-1164 116 6 ๐ดฮณ4 ๐ดฮณ4 VERB cana-1164 117 1 + + X cana-1164 117 2 = = X cana-1164 117 3 [ [ PUNCT cana-1164 117 4 0 0 NUM cana-1164 117 5 1 1 NUM cana-1164 117 6 0 0 NUM cana-1164 117 7 1 1 NUM cana-1164 117 8 ] ] PUNCT cana-1164 117 9 are be AUX cana-1164 117 10 out out ADV cana-1164 117 11 - - PUNCT cana-1164 117 12 adjacency adjacency NOUN cana-1164 117 13 matrices matrix NOUN cana-1164 117 14 of of ADP cana-1164 117 15 the the DET cana-1164 117 16 component component NOUN cana-1164 117 17 digraphs digraph NOUN cana-1164 117 18 ฮณ1 ฮณ1 NOUN cana-1164 117 19 , , PUNCT cana-1164 117 20 ฮณ2 ฮณ2 PROPN cana-1164 117 21 , , PUNCT cana-1164 117 22 ฮณ3 ฮณ3 NOUN cana-1164 117 23 , , PUNCT cana-1164 117 24 and and CCONJ cana-1164 117 25 ฮณ4 ฮณ4 NOUN cana-1164 117 26 respectively respectively ADV cana-1164 117 27 . . PUNCT cana-1164 118 1 thus thus ADV cana-1164 118 2 , , PUNCT cana-1164 118 3 the the DET cana-1164 118 4 out out ADJ cana-1164 118 5 - - PUNCT cana-1164 118 6 adjacency adjacency NOUN cana-1164 118 7 matrix matrix NOUN cana-1164 119 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 119 2 + + NUM cana-1164 119 3 of of ADP cana-1164 119 4 the the DET cana-1164 119 5 digraph digraph ADJ cana-1164 119 6 ฮณ(6 ฮณ(6 PROPN cana-1164 119 7 , , PUNCT cana-1164 119 8 2 2 NUM cana-1164 119 9 ) ) PUNCT cana-1164 119 10 can can AUX cana-1164 119 11 be be AUX cana-1164 119 12 written write VERB cana-1164 119 13 as as ADP cana-1164 119 14 a a DET cana-1164 119 15 block block NOUN cana-1164 119 16 - - PUNCT cana-1164 119 17 diagonal diagonal ADJ cana-1164 119 18 matrix matrix NOUN cana-1164 119 19 with with ADP cana-1164 119 20 diagonal diagonal ADJ cana-1164 119 21 elements element NOUN cana-1164 119 22 as as SCONJ cana-1164 119 23 the the DET cana-1164 119 24 out out ADJ cana-1164 119 25 - - PUNCT cana-1164 119 26 adjacency adjacency NOUN cana-1164 119 27 matrices matrix NOUN cana-1164 119 28 of of ADP cana-1164 119 29 the the DET cana-1164 119 30 component component NOUN cana-1164 119 31 digraphs digraph VERB cana-1164 119 32 of of ADP cana-1164 119 33 the the DET cana-1164 119 34 digraph digraph ADJ cana-1164 119 35 ฮณ(6 ฮณ(6 PROPN cana-1164 119 36 , , PUNCT cana-1164 119 37 2 2 NUM cana-1164 119 38 ) ) PUNCT cana-1164 119 39 . . PUNCT cana-1164 120 1 similarly similarly ADV cana-1164 120 2 , , PUNCT cana-1164 120 3 the the DET cana-1164 120 4 in in ADP cana-1164 120 5 - - PUNCT cana-1164 120 6 adjacency adjacency NOUN cana-1164 120 7 matrix matrix NOUN cana-1164 120 8 ๐ดฮณ ๐ดฮณ ADP cana-1164 120 9 โˆ’ โˆ’ PROPN cana-1164 120 10 of of ADP cana-1164 120 11 the the DET cana-1164 120 12 digraph digraph ADJ cana-1164 120 13 ฮณ(6 ฮณ(6 PROPN cana-1164 120 14 , , PUNCT cana-1164 120 15 2 2 NUM cana-1164 120 16 ) ) PUNCT cana-1164 120 17 can can AUX cana-1164 120 18 be be AUX cana-1164 120 19 written write VERB cana-1164 120 20 as as ADP cana-1164 120 21 a a DET cana-1164 120 22 block block NOUN cana-1164 120 23 - - PUNCT cana-1164 120 24 diagonal diagonal ADJ cana-1164 120 25 matrix matrix NOUN cana-1164 120 26 with with ADP cana-1164 120 27 diagonal diagonal ADJ cana-1164 120 28 elements element NOUN cana-1164 120 29 as as SCONJ cana-1164 120 30 the the DET cana-1164 120 31 in in ADP cana-1164 120 32 - - PUNCT cana-1164 120 33 adjacency adjacency NOUN cana-1164 120 34 matrices matrix NOUN cana-1164 120 35 of of ADP cana-1164 120 36 the the DET cana-1164 120 37 component component NOUN cana-1164 120 38 digraphs digraph VERB cana-1164 120 39 of of ADP cana-1164 120 40 the the DET cana-1164 120 41 digraph digraph ADJ cana-1164 120 42 ฮณ(6 ฮณ(6 PROPN cana-1164 120 43 , , PUNCT cana-1164 120 44 2 2 NUM cana-1164 120 45 ) ) PUNCT cana-1164 120 46 ๐‘–. ๐‘–. PUNCT cana-1164 120 47 ๐‘’. ๐‘’. PUNCT cana-1164 121 1 ๐ดฮณ ๐ดฮณ ADP cana-1164 121 2 โˆ’ โˆ’ PROPN cana-1164 122 1 = = SYM cana-1164 122 2 0 0 NUM cana-1164 122 3 5 5 NUM cana-1164 122 4 1 1 NUM cana-1164 122 5 3 3 NUM cana-1164 122 6 2 2 NUM cana-1164 122 7 4 4 NUM cana-1164 122 8 0 0 NUM cana-1164 122 9 1 1 NUM cana-1164 122 10 0 0 NUM cana-1164 122 11 0 0 NUM cana-1164 122 12 0 0 NUM cana-1164 122 13 0 0 NUM cana-1164 122 14 0 0 NUM cana-1164 122 15 5 5 NUM cana-1164 122 16 0 0 NUM cana-1164 122 17 0 0 NUM cana-1164 122 18 0 0 NUM cana-1164 122 19 0 0 NUM cana-1164 122 20 0 0 NUM cana-1164 122 21 0 0 NUM cana-1164 122 22 1 1 NUM cana-1164 122 23 0 0 NUM cana-1164 122 24 1 1 NUM cana-1164 122 25 1 1 NUM cana-1164 122 26 0 0 NUM cana-1164 122 27 0 0 NUM cana-1164 122 28 0 0 NUM cana-1164 122 29 3 3 NUM cana-1164 122 30 0 0 NUM cana-1164 122 31 0 0 NUM cana-1164 122 32 0 0 NUM cana-1164 122 33 1 1 NUM cana-1164 122 34 0 0 NUM cana-1164 122 35 0 0 NUM cana-1164 122 36 2 2 NUM cana-1164 122 37 0 0 NUM cana-1164 122 38 0 0 NUM cana-1164 122 39 0 0 NUM cana-1164 122 40 0 0 NUM cana-1164 122 41 0 0 NUM cana-1164 122 42 0 0 NUM cana-1164 122 43 4 4 NUM cana-1164 122 44 0 0 NUM cana-1164 122 45 0 0 NUM cana-1164 122 46 0 0 NUM cana-1164 122 47 0 0 NUM cana-1164 122 48 1 1 NUM cana-1164 122 49 1 1 NUM cana-1164 122 50 ๏ƒฉ ๏ƒฉ NUM cana-1164 122 51 ๏ƒน ๏ƒน ADJ cana-1164 122 52 ๏ƒช ๏ƒช NOUN cana-1164 122 53 ๏ƒบ ๏ƒบ PROPN cana-1164 122 54 ๏ƒช ๏ƒช NOUN cana-1164 122 55 ๏ƒบ ๏ƒบ NOUN cana-1164 122 56 ๏ƒช ๏ƒช NOUN cana-1164 122 57 ๏ƒบ ๏ƒบ NOUN cana-1164 122 58 ๏ƒช ๏ƒช NOUN cana-1164 122 59 ๏ƒบ ๏ƒบ NOUN cana-1164 122 60 ๏ƒช ๏ƒช NOUN cana-1164 122 61 ๏ƒบ ๏ƒบ NOUN cana-1164 122 62 ๏ƒช ๏ƒช NOUN cana-1164 122 63 ๏ƒบ ๏ƒบ NOUN cana-1164 122 64 ๏ƒช ๏ƒช NOUN cana-1164 122 65 ๏ƒบ ๏ƒบ NOUN cana-1164 122 66 ๏ƒซ ๏ƒซ VERB cana-1164 122 67 ๏ƒป ๏ƒป PROPN cana-1164 122 68 = = SYM cana-1164 122 69 [ [ PUNCT cana-1164 122 70 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 122 71 โˆ’ โˆ’ NOUN cana-1164 122 72 0 0 NUM cana-1164 122 73 0 0 NUM cana-1164 122 74 0 0 NUM cana-1164 122 75 0 0 NUM cana-1164 122 76 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 122 77 โˆ’ โˆ’ PROPN cana-1164 122 78 0 0 NUM cana-1164 122 79 0 0 NUM cana-1164 122 80 0 0 NUM cana-1164 122 81 0 0 NUM cana-1164 122 82 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 122 83 โˆ’ โˆ’ NOUN cana-1164 122 84 0 0 NUM cana-1164 122 85 0 0 NUM cana-1164 122 86 0 0 NUM cana-1164 122 87 0 0 NUM cana-1164 122 88 ๐ดฮณ4 ๐ดฮณ4 NOUN cana-1164 122 89 โˆ’ โˆ’ PROPN cana-1164 122 90 ] ] PUNCT cana-1164 122 91 where where SCONJ cana-1164 122 92 , , PUNCT cana-1164 122 93 ๐ดฮณ1 ๐ดฮณ1 VERB cana-1164 122 94 โˆ’ โˆ’ PROPN cana-1164 123 1 = = PUNCT cana-1164 124 1 [ [ X cana-1164 124 2 1 1 NUM cana-1164 124 3 ] ] PUNCT cana-1164 124 4 , , PUNCT cana-1164 124 5 ๐ดฮณ2 ๐ดฮณ2 VERB cana-1164 124 6 โˆ’ โˆ’ PROPN cana-1164 125 1 = = PUNCT cana-1164 126 1 [ [ PUNCT cana-1164 126 2 0 0 NUM cana-1164 126 3 0 0 NUM cana-1164 126 4 1 1 NUM cana-1164 126 5 1 1 NUM cana-1164 126 6 ] ] PUNCT cana-1164 126 7 , , PUNCT cana-1164 126 8 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 126 9 โˆ’ โˆ’ PROPN cana-1164 127 1 = = PUNCT cana-1164 128 1 [ [ X cana-1164 128 2 1 1 NUM cana-1164 128 3 ] ] PUNCT cana-1164 128 4 , , PUNCT cana-1164 128 5 and and CCONJ cana-1164 128 6 ๐ดฮณ4 ๐ดฮณ4 VERB cana-1164 128 7 โˆ’ โˆ’ PROPN cana-1164 129 1 = = PUNCT cana-1164 130 1 [ [ PUNCT cana-1164 130 2 0 0 NUM cana-1164 130 3 0 0 NUM cana-1164 130 4 1 1 NUM cana-1164 130 5 1 1 NUM cana-1164 130 6 ] ] PUNCT cana-1164 130 7 are be AUX cana-1164 130 8 in in ADP cana-1164 130 9 - - PUNCT cana-1164 130 10 adjacency adjacency NOUN cana-1164 130 11 matrices matrix NOUN cana-1164 130 12 of of ADP cana-1164 130 13 the the DET cana-1164 130 14 component component NOUN cana-1164 130 15 digraphs digraph NOUN cana-1164 130 16 ฮณ1 ฮณ1 NOUN cana-1164 130 17 , , PUNCT cana-1164 130 18 ฮณ2 ฮณ2 PROPN cana-1164 130 19 , , PUNCT cana-1164 130 20 ฮณ3 ฮณ3 NOUN cana-1164 130 21 , , PUNCT cana-1164 130 22 and and CCONJ cana-1164 130 23 ฮณ4 ฮณ4 NOUN cana-1164 130 24 respectively respectively ADV cana-1164 130 25 . . PUNCT cana-1164 131 1 result result VERB cana-1164 131 2 3.1 3.1 NUM cana-1164 131 3 . . PUNCT cana-1164 132 1 ( ( PUNCT cana-1164 132 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 132 3 + + NOUN cana-1164 132 4 ) ) PUNCT cana-1164 132 5 ๐‘ก ๐‘ก NOUN cana-1164 132 6 = = PUNCT cana-1164 133 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 133 2 โˆ’ โˆ’ PROPN cana-1164 133 3 and and CCONJ cana-1164 133 4 ( ( PUNCT cana-1164 133 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 133 6 โˆ’)๐‘ก โˆ’)๐‘ก NOUN cana-1164 134 1 = = PUNCT cana-1164 134 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 134 3 + + PROPN cana-1164 134 4 . . PUNCT cana-1164 134 5 proof proof NOUN cana-1164 134 6 . . PUNCT cana-1164 135 1 clearly clearly ADV cana-1164 135 2 , , PUNCT cana-1164 135 3 the the DET cana-1164 135 4 matrices matrix NOUN cana-1164 135 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 135 6 + + NOUN cana-1164 135 7 , , PUNCT cana-1164 135 8 ๐ดฮณ ๐ดฮณ PROPN cana-1164 135 9 โˆ’ โˆ’ PROPN cana-1164 135 10 , , PUNCT cana-1164 135 11 and and CCONJ cana-1164 135 12 ( ( PUNCT cana-1164 135 13 ๐ดฮณ ๐ดฮณ PROPN cana-1164 135 14 + + PROPN cana-1164 135 15 ) ) PUNCT cana-1164 135 16 ๐‘ก ๐‘ก NOUN cana-1164 135 17 are be AUX cana-1164 135 18 of of ADP cana-1164 135 19 the the DET cana-1164 135 20 same same ADJ cana-1164 135 21 order order NOUN cana-1164 135 22 ๐‘› ๐‘› DET cana-1164 135 23 ร— ร— ADJ cana-1164 135 24 ๐‘›. ๐‘›. NOUN cana-1164 135 25 also also ADV cana-1164 135 26 , , PUNCT cana-1164 135 27 the the DET cana-1164 135 28 ( ( PUNCT cana-1164 135 29 ๐‘– ๐‘– PROPN cana-1164 135 30 , , PUNCT cana-1164 135 31 ๐‘—)๐‘กโ„Ž ๐‘—)๐‘กโ„Ž PROPN cana-1164 135 32 element element NOUN cana-1164 135 33 of of ADP cana-1164 135 34 ( ( PUNCT cana-1164 135 35 ๐ดฮณ ๐ดฮณ PROPN cana-1164 135 36 + + PROPN cana-1164 135 37 ) ) PUNCT cana-1164 135 38 ๐‘ก ๐‘ก NOUN cana-1164 135 39 = = SYM cana-1164 135 40 the the DET cana-1164 135 41 ( ( PUNCT cana-1164 135 42 ๐‘— ๐‘— NOUN cana-1164 135 43 , , PUNCT cana-1164 135 44 ๐‘–)๐‘กโ„Ž ๐‘–)๐‘กโ„Ž PROPN cana-1164 135 45 element element NOUN cana-1164 135 46 of of ADP cana-1164 135 47 ๐ดฮณ ๐ดฮณ PROPN cana-1164 135 48 + + CCONJ cana-1164 135 49 = = PUNCT cana-1164 135 50 the the DET cana-1164 135 51 ( ( PUNCT cana-1164 135 52 ๐‘– ๐‘– PROPN cana-1164 135 53 , , PUNCT cana-1164 135 54 ๐‘—)๐‘กโ„Ž ๐‘—)๐‘กโ„Ž PROPN cana-1164 135 55 element element NOUN cana-1164 135 56 of of ADP cana-1164 135 57 ๐ดฮณ ๐ดฮณ PROPN cana-1164 135 58 โˆ’. โˆ’. PROPN cana-1164 136 1 [ [ X cana-1164 136 2 by by ADP cana-1164 136 3 definition definition NOUN cana-1164 136 4 of of ADP cana-1164 136 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 136 6 โˆ’ โˆ’ PROPN cana-1164 136 7 ] ] PUNCT cana-1164 136 8 hence hence ADV cana-1164 136 9 , , PUNCT cana-1164 136 10 ( ( PUNCT cana-1164 136 11 ๐ดฮณ ๐ดฮณ PROPN cana-1164 136 12 + + PROPN cana-1164 136 13 ) ) PUNCT cana-1164 136 14 ๐‘ก ๐‘ก NOUN cana-1164 136 15 = = SYM cana-1164 136 16 ๐ดฮณ ๐ดฮณ PROPN cana-1164 136 17 โˆ’. โˆ’. ADV cana-1164 136 18 similarly similarly ADV cana-1164 136 19 , , PUNCT cana-1164 136 20 it it PRON cana-1164 136 21 can can AUX cana-1164 136 22 be be AUX cana-1164 136 23 shown show VERB cana-1164 136 24 that that SCONJ cana-1164 136 25 ( ( PUNCT cana-1164 136 26 ๐ดฮณ ๐ดฮณ PROPN cana-1164 136 27 โˆ’)๐‘ก โˆ’)๐‘ก NOUN cana-1164 136 28 = = PUNCT cana-1164 136 29 ๐ดฮณ ๐ดฮณ PROPN cana-1164 136 30 + + PROPN cana-1164 136 31 . . PROPN cana-1164 136 32 result result PROPN cana-1164 136 33 3.2 3.2 NUM cana-1164 136 34 . . PUNCT cana-1164 137 1 let let VERB cana-1164 137 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 137 3 be be AUX cana-1164 137 4 the the DET cana-1164 137 5 adjacency adjacency NOUN cana-1164 137 6 matrix matrix NOUN cana-1164 137 7 of of ADP cana-1164 137 8 the the DET cana-1164 137 9 underlying underlie VERB cana-1164 137 10 graph graph NOUN cana-1164 137 11 ๐บ ๐บ PROPN cana-1164 137 12 of of ADP cana-1164 137 13 the the DET cana-1164 137 14 digraph digraph ADJ cana-1164 137 15 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 137 16 , , PUNCT cana-1164 137 17 ๐‘˜ ๐‘˜ NOUN cana-1164 137 18 ) ) PUNCT cana-1164 137 19 , , PUNCT cana-1164 137 20 then then ADV cana-1164 138 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 138 2 = = PUNCT cana-1164 139 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 139 2 + + PROPN cana-1164 140 1 + + CCONJ cana-1164 140 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 140 3 โˆ’. โˆ’. ADV cana-1164 140 4 proof proof NOUN cana-1164 140 5 . . PUNCT cana-1164 141 1 let let VERB cana-1164 142 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 142 2 + + NOUN cana-1164 142 3 = = PUNCT cana-1164 143 1 [ [ X cana-1164 143 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ADJ cana-1164 143 3 , , PUNCT cana-1164 143 4 communications communication NOUN cana-1164 143 5 on on ADP cana-1164 143 6 applied apply VERB cana-1164 143 7 nonlinear nonlinear ADJ cana-1164 143 8 analysis analysis NOUN cana-1164 143 9 issn issn NOUN cana-1164 143 10 : : PUNCT cana-1164 143 11 1074 1074 NUM cana-1164 143 12 - - PUNCT cana-1164 143 13 133x 133x NUM cana-1164 143 14 vol vol NOUN cana-1164 143 15 31 31 NUM cana-1164 143 16 no no NOUN cana-1164 143 17 . . PUNCT cana-1164 144 1 6s 6s NUM cana-1164 144 2 ( ( PUNCT cana-1164 144 3 2024 2024 NUM cana-1164 144 4 ) ) PUNCT cana-1164 144 5 103 103 NUM cana-1164 144 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 144 7 where where SCONJ cana-1164 144 8 , , PUNCT cana-1164 144 9 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 144 10 = = SYM cana-1164 144 11 { { PUNCT cana-1164 144 12 1 1 NUM cana-1164 144 13 , , PUNCT cana-1164 144 14 if if SCONJ cana-1164 144 15 there there PRON cana-1164 144 16 is be VERB cana-1164 144 17 a a DET cana-1164 144 18 directed direct VERB cana-1164 144 19 arc arc NOUN cana-1164 144 20 from from ADP cana-1164 144 21 the the DET cana-1164 144 22 vertex vertex NOUN cana-1164 144 23 vi vi NOUN cana-1164 144 24 to to ADP cana-1164 144 25 the the DET cana-1164 144 26 vertex vertex NOUN cana-1164 144 27 vj vj PROPN cana-1164 144 28 . . PROPN cana-1164 144 29 0 0 NUM cana-1164 144 30 , , PUNCT cana-1164 144 31 otherwise otherwise ADV cana-1164 144 32 . . PUNCT cana-1164 145 1 and and CCONJ cana-1164 145 2 , , PUNCT cana-1164 146 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 146 2 โˆ’ โˆ’ PROPN cana-1164 147 1 = = PUNCT cana-1164 148 1 [ [ X cana-1164 148 2 ๐‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘๐‘–๐‘—]๐‘›ร—๐‘› PROPN cana-1164 148 3 , , PUNCT cana-1164 148 4 where where SCONJ cana-1164 148 5 , , PUNCT cana-1164 148 6 ๐‘๐‘–๐‘— ๐‘๐‘–๐‘— NOUN cana-1164 148 7 = = SYM cana-1164 148 8 { { PUNCT cana-1164 148 9 1 1 NUM cana-1164 148 10 , , PUNCT cana-1164 148 11 if if SCONJ cana-1164 148 12 there there PRON cana-1164 148 13 is be VERB cana-1164 148 14 a a DET cana-1164 148 15 directed direct VERB cana-1164 148 16 arc arc NOUN cana-1164 148 17 from from ADP cana-1164 148 18 the the DET cana-1164 148 19 vertex vertex NOUN cana-1164 148 20 vj vj INTJ cana-1164 148 21 to to ADP cana-1164 148 22 the the DET cana-1164 148 23 vertex vertex NOUN cana-1164 148 24 vi vi PROPN cana-1164 148 25 . . PROPN cana-1164 148 26 0 0 NUM cana-1164 148 27 , , PUNCT cana-1164 148 28 otherwise otherwise ADV cana-1164 148 29 . . PUNCT cana-1164 149 1 also also ADV cana-1164 149 2 , , PUNCT cana-1164 149 3 let let VERB cana-1164 149 4 ๐ดฮณ ๐ดฮณ PROPN cana-1164 149 5 = = PUNCT cana-1164 150 1 [ [ X cana-1164 150 2 ๐‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘๐‘–๐‘—]๐‘›ร—๐‘› PROPN cana-1164 150 3 , , PUNCT cana-1164 150 4 where where SCONJ cana-1164 150 5 , , PUNCT cana-1164 150 6 ๐‘๐‘–๐‘— ๐‘๐‘–๐‘— PROPN cana-1164 150 7 = = PUNCT cana-1164 150 8 { { PUNCT cana-1164 150 9 2 2 NUM cana-1164 150 10 , , PUNCT cana-1164 150 11 if if SCONJ cana-1164 150 12 there there PRON cana-1164 150 13 is be VERB cana-1164 150 14 a a DET cana-1164 150 15 loop loop NOUN cana-1164 150 16 at at ADP cana-1164 150 17 the the DET cana-1164 150 18 vertex vertex NOUN cana-1164 150 19 vi vi PROPN cana-1164 150 20 . . PROPN cana-1164 150 21 1 1 NUM cana-1164 150 22 , , PUNCT cana-1164 150 23 if if SCONJ cana-1164 150 24 there there PRON cana-1164 150 25 is be VERB cana-1164 150 26 an an DET cana-1164 150 27 edge edge NOUN cana-1164 150 28 between between ADP cana-1164 150 29 the the DET cana-1164 150 30 vertices vertex NOUN cana-1164 150 31 vi vi NOUN cana-1164 150 32 and and CCONJ cana-1164 150 33 vj vj PROPN cana-1164 150 34 . . PROPN cana-1164 150 35 0 0 NUM cana-1164 150 36 , , PUNCT cana-1164 150 37 otherwise otherwise ADV cana-1164 150 38 . . PUNCT cana-1164 151 1 clearly clearly ADV cana-1164 151 2 , , PUNCT cana-1164 151 3 the the DET cana-1164 151 4 matrices matrix NOUN cana-1164 151 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 151 6 and and CCONJ cana-1164 151 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 151 8 + + PROPN cana-1164 152 1 + + CCONJ cana-1164 152 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 152 3 โˆ’ โˆ’ PROPN cana-1164 152 4 are be AUX cana-1164 152 5 of of ADP cana-1164 152 6 the the DET cana-1164 152 7 same same ADJ cana-1164 152 8 order order NOUN cana-1164 152 9 ๐‘› ๐‘› DET cana-1164 152 10 ร— ร— ADJ cana-1164 152 11 ๐‘›. ๐‘›. NOUN cana-1164 152 12 we we PRON cana-1164 152 13 have have VERB cana-1164 152 14 , , PUNCT cana-1164 153 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 153 2 + + PROPN cana-1164 154 1 + + CCONJ cana-1164 154 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 154 3 โˆ’ โˆ’ PROPN cana-1164 155 1 = = PUNCT cana-1164 156 1 [ [ X cana-1164 156 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› PROPN cana-1164 156 3 + + X cana-1164 156 4 [ [ X cana-1164 156 5 ๐‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘๐‘–๐‘—]๐‘›ร—๐‘› X cana-1164 156 6 = = PUNCT cana-1164 156 7 [ [ X cana-1164 156 8 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 156 9 + + X cana-1164 156 10 ๐‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘๐‘–๐‘—]๐‘›ร—๐‘› PROPN cana-1164 156 11 = = PUNCT cana-1164 157 1 [ [ X cana-1164 157 2 ๐‘‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘‘๐‘–๐‘—]๐‘›ร—๐‘› PROPN cana-1164 157 3 , , PUNCT cana-1164 157 4 where where SCONJ cana-1164 157 5 , , PUNCT cana-1164 157 6 ๐‘‘๐‘–๐‘— ๐‘‘๐‘–๐‘— PROPN cana-1164 157 7 = = PUNCT cana-1164 157 8 { { PUNCT cana-1164 157 9 2 2 NUM cana-1164 157 10 , , PUNCT cana-1164 157 11 if if SCONJ cana-1164 157 12 there there PRON cana-1164 157 13 is be VERB cana-1164 157 14 a a DET cana-1164 157 15 loop loop NOUN cana-1164 157 16 at at ADP cana-1164 157 17 the the DET cana-1164 157 18 vertex vertex NOUN cana-1164 157 19 vi vi PROPN cana-1164 157 20 . . PROPN cana-1164 157 21 1 1 NUM cana-1164 157 22 , , PUNCT cana-1164 157 23 if if SCONJ cana-1164 157 24 there there PRON cana-1164 157 25 is be VERB cana-1164 157 26 an an DET cana-1164 157 27 edge edge NOUN cana-1164 157 28 between between ADP cana-1164 157 29 the the DET cana-1164 157 30 vertices vertex NOUN cana-1164 157 31 vi vi NOUN cana-1164 157 32 and and CCONJ cana-1164 157 33 vj vj PROPN cana-1164 157 34 . . PROPN cana-1164 157 35 0 0 NUM cana-1164 157 36 , , PUNCT cana-1164 157 37 otherwise otherwise ADV cana-1164 157 38 . . PUNCT cana-1164 158 1 = = PUNCT cana-1164 159 1 [ [ X cana-1164 159 2 ๐‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘๐‘–๐‘—]๐‘›ร—๐‘› X cana-1164 159 3 = = SYM cana-1164 159 4 ๐ดฮณ ๐ดฮณ PROPN cana-1164 159 5 hence hence ADV cana-1164 159 6 , , PUNCT cana-1164 160 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 160 2 = = PUNCT cana-1164 160 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 160 4 + + PROPN cana-1164 161 1 + + CCONJ cana-1164 161 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 161 3 โˆ’. โˆ’. ADV cana-1164 161 4 remark remark VERB cana-1164 161 5 3.1 3.1 NUM cana-1164 161 6 . . PUNCT cana-1164 162 1 i. i. PROPN cana-1164 162 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 162 3 is be AUX cana-1164 162 4 symmetric symmetric ADJ cana-1164 162 5 i.e. i.e. X cana-1164 162 6 ( ( PUNCT cana-1164 162 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 162 8 ) ) PUNCT cana-1164 162 9 ๐‘ก ๐‘ก PROPN cana-1164 162 10 = = SYM cana-1164 162 11 ๐ดฮณ ๐ดฮณ PROPN cana-1164 162 12 ii ii NOUN cana-1164 162 13 . . PUNCT cana-1164 163 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 163 2 = = PUNCT cana-1164 164 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 164 2 + + PROPN cana-1164 165 1 + + CCONJ cana-1164 165 2 ( ( PUNCT cana-1164 165 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 165 4 + + PROPN cana-1164 165 5 ) ) PUNCT cana-1164 165 6 ๐‘ก ๐‘ก PROPN cana-1164 165 7 iii iii NOUN cana-1164 165 8 . . PUNCT cana-1164 166 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 166 2 = = PUNCT cana-1164 167 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 167 2 โˆ’ โˆ’ PROPN cana-1164 168 1 + + CCONJ cana-1164 168 2 ( ( PUNCT cana-1164 168 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 168 4 โˆ’)๐‘ก โˆ’)๐‘ก NOUN cana-1164 168 5 result result VERB cana-1164 168 6 3.3 3.3 NUM cana-1164 168 7 . . PUNCT cana-1164 169 1 the the DET cana-1164 169 2 sum sum NOUN cana-1164 169 3 of of ADP cana-1164 169 4 entries entry NOUN cana-1164 169 5 in in ADP cana-1164 169 6 the the DET cana-1164 169 7 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 169 8 row row NOUN cana-1164 169 9 of of ADP cana-1164 169 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 169 11 + + CCONJ cana-1164 169 12 is be AUX cana-1164 169 13 1 1 NUM cana-1164 169 14 . . PUNCT cana-1164 170 1 proof proof NOUN cana-1164 170 2 . . PUNCT cana-1164 171 1 let let VERB cana-1164 172 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 172 2 + + NOUN cana-1164 172 3 = = SYM cana-1164 173 1 [ [ X cana-1164 173 2 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— X cana-1164 173 3 ] ] PUNCT cana-1164 173 4 be be VERB cana-1164 173 5 an an DET cana-1164 173 6 out out ADJ cana-1164 173 7 - - PUNCT cana-1164 173 8 adjacency adjacency NOUN cana-1164 173 9 matrix matrix NOUN cana-1164 173 10 of of ADP cana-1164 173 11 the the DET cana-1164 173 12 digraph digraph ADJ cana-1164 173 13 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 173 14 , , PUNCT cana-1164 173 15 ๐‘˜ ๐‘˜ NOUN cana-1164 173 16 ) ) PUNCT cana-1164 173 17 and and CCONJ cana-1164 173 18 let let VERB cana-1164 173 19 ๐‘…๐‘– ๐‘…๐‘– PROPN cana-1164 173 20 = = PUNCT cana-1164 174 1 [ [ X cana-1164 174 2 ๐‘Ž๐‘–1 ๐‘Ž๐‘–1 NOUN cana-1164 174 3 , , PUNCT cana-1164 174 4 ๐‘Ž๐‘–2 ๐‘Ž๐‘–2 NOUN cana-1164 174 5 , , PUNCT cana-1164 174 6 โ‹ฏ โ‹ฏ PROPN cana-1164 174 7 , , PUNCT cana-1164 174 8 ๐‘Ž๐‘–๐‘› ๐‘Ž๐‘–๐‘› PROPN cana-1164 174 9 ] ] PUNCT cana-1164 174 10 be be VERB cana-1164 174 11 the the DET cana-1164 174 12 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 174 13 row row NOUN cana-1164 174 14 of of ADP cana-1164 174 15 ๐ดฮณ ๐ดฮณ PROPN cana-1164 174 16 + + PUNCT cana-1164 174 17 corresponding correspond VERB cana-1164 174 18 to to ADP cana-1164 174 19 the the DET cana-1164 174 20 vertex vertex NOUN cana-1164 174 21 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 174 22 โˆˆ โˆˆ PROPN cana-1164 174 23 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 174 24 ) ) PUNCT cana-1164 174 25 . . PUNCT cana-1164 175 1 as as SCONJ cana-1164 175 2 the the DET cana-1164 175 3 residue residue NOUN cana-1164 175 4 of of ADP cana-1164 175 5 a a DET cana-1164 175 6 number number NOUN cana-1164 175 7 modulo modulo VERB cana-1164 175 8 ๐‘› ๐‘› NOUN cana-1164 175 9 is be AUX cana-1164 175 10 unique unique ADJ cana-1164 175 11 , , PUNCT cana-1164 175 12 the the DET cana-1164 175 13 number number NOUN cana-1164 175 14 of of ADP cana-1164 175 15 directed direct VERB cana-1164 175 16 arcs arc NOUN cana-1164 175 17 leaving leave VERB cana-1164 175 18 the the DET cana-1164 175 19 vertex vertex NOUN cana-1164 175 20 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 175 21 is be AUX cana-1164 175 22 exactly exactly ADV cana-1164 175 23 one one NUM cana-1164 175 24 . . PUNCT cana-1164 176 1 it it PRON cana-1164 176 2 contributes contribute VERB cana-1164 176 3 thereby thereby ADV cana-1164 176 4 1 1 NUM cana-1164 176 5 exactly exactly ADV cana-1164 176 6 in in ADP cana-1164 176 7 one one NUM cana-1164 176 8 of of ADP cana-1164 176 9 the the DET cana-1164 176 10 entries entry NOUN cana-1164 176 11 of of ADP cana-1164 176 12 ๐‘…๐‘– ๐‘…๐‘– PROPN cana-1164 176 13 and and CCONJ cana-1164 176 14 0 0 NUM cana-1164 176 15 in in ADP cana-1164 176 16 the the DET cana-1164 176 17 remaining remain VERB cana-1164 176 18 entries entry NOUN cana-1164 176 19 of of ADP cana-1164 176 20 ๐‘…๐‘–. ๐‘…๐‘–. PROPN cana-1164 176 21 thus thus ADV cana-1164 176 22 , , PUNCT cana-1164 176 23 โˆ‘๐‘›๐‘—=1 โˆ‘๐‘›๐‘—=1 PROPN cana-1164 176 24 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 176 25 = = SYM cana-1164 176 26 1 1 X cana-1164 176 27 . . PUNCT cana-1164 176 28 communications communication NOUN cana-1164 176 29 on on ADP cana-1164 176 30 applied apply VERB cana-1164 176 31 nonlinear nonlinear ADJ cana-1164 176 32 analysis analysis NOUN cana-1164 176 33 issn issn NOUN cana-1164 176 34 : : PUNCT cana-1164 176 35 1074 1074 NUM cana-1164 176 36 - - PUNCT cana-1164 176 37 133x 133x NUM cana-1164 176 38 vol vol NOUN cana-1164 176 39 31 31 NUM cana-1164 176 40 no no NOUN cana-1164 176 41 . . PUNCT cana-1164 177 1 6s 6s NUM cana-1164 177 2 ( ( PUNCT cana-1164 177 3 2024 2024 NUM cana-1164 177 4 ) ) PUNCT cana-1164 177 5 104 104 NUM cana-1164 178 1 https://internationalpubls.com https://internationalpubls.com X cana-1164 178 2 corollary corollary NOUN cana-1164 178 3 3.1 3.1 NUM cana-1164 178 4 . . PUNCT cana-1164 179 1 the the DET cana-1164 179 2 sum sum NOUN cana-1164 179 3 of of ADP cana-1164 179 4 entries entry NOUN cana-1164 179 5 in in ADP cana-1164 179 6 the the DET cana-1164 179 7 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 179 8 row row NOUN cana-1164 179 9 of of ADP cana-1164 179 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 179 11 + + CCONJ cana-1164 179 12 is be AUX cana-1164 179 13 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 179 14 + + NOUN cana-1164 179 15 ( ( PUNCT cana-1164 179 16 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 179 17 ) ) PUNCT cana-1164 179 18 , , PUNCT cana-1164 179 19 where where SCONJ cana-1164 179 20 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 179 21 + + NOUN cana-1164 179 22 ( ( PUNCT cana-1164 179 23 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 179 24 ) ) PUNCT cana-1164 179 25 is be AUX cana-1164 179 26 the the DET cana-1164 179 27 out out ADJ cana-1164 179 28 - - PUNCT cana-1164 179 29 degree degree NOUN cana-1164 179 30 of of ADP cana-1164 179 31 the the DET cana-1164 179 32 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 179 33 vertex vertex NOUN cana-1164 179 34 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADV cana-1164 179 35 . . PUNCT cana-1164 180 1 proof proof NOUN cana-1164 180 2 . . PUNCT cana-1164 181 1 as as SCONJ cana-1164 181 2 the the DET cana-1164 181 3 out out ADJ cana-1164 181 4 - - PUNCT cana-1164 181 5 degree degree NOUN cana-1164 181 6 of of ADP cana-1164 181 7 each each DET cana-1164 181 8 vertex vertex NOUN cana-1164 181 9 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 181 10 โˆˆ โˆˆ PROPN cana-1164 181 11 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 181 12 , , PUNCT cana-1164 181 13 ๐‘˜ ๐‘˜ NOUN cana-1164 181 14 ) ) PUNCT cana-1164 181 15 is be AUX cana-1164 181 16 1 1 NUM cana-1164 181 17 , , PUNCT cana-1164 181 18 so so SCONJ cana-1164 181 19 we we PRON cana-1164 181 20 have have AUX cana-1164 181 21 ๐‘‘ฮณ ๐‘‘ฮณ VERB cana-1164 181 22 + + NOUN cana-1164 181 23 ( ( PUNCT cana-1164 181 24 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 181 25 ) ) PUNCT cana-1164 181 26 = = SYM cana-1164 181 27 1 1 NUM cana-1164 181 28 , , PUNCT cana-1164 181 29 โˆ€ โˆ€ X cana-1164 181 30 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 181 31 โˆˆ โˆˆ PROPN cana-1164 181 32 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 181 33 ) ) PUNCT cana-1164 181 34 . . PUNCT cana-1164 182 1 โ‡’ โ‡’ NOUN cana-1164 182 2 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 182 3 + + PROPN cana-1164 182 4 ( ( PUNCT cana-1164 182 5 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 182 6 ) ) PUNCT cana-1164 182 7 = = SYM cana-1164 182 8 1 1 NUM cana-1164 182 9 = = SYM cana-1164 182 10 โˆ‘ โˆ‘ PUNCT cana-1164 182 11 ๐‘› ๐‘› DET cana-1164 182 12 ๐‘—=1 ๐‘—=1 PROPN cana-1164 182 13 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 182 14 [ [ PUNCT cana-1164 182 15 by by ADP cana-1164 182 16 result result NOUN cana-1164 182 17 3.3 3.3 NUM cana-1164 182 18 ] ] PUNCT cana-1164 182 19 ๐‘–. ๐‘–. ADJ cana-1164 182 20 ๐‘’. ๐‘’. PUNCT cana-1164 183 1 โˆ‘๐‘›๐‘—=1 โˆ‘๐‘›๐‘—=1 NUM cana-1164 183 2 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 183 3 = = SYM cana-1164 183 4 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 183 5 + + PROPN cana-1164 183 6 ( ( PUNCT cana-1164 183 7 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 183 8 ) ) PUNCT cana-1164 183 9 . . PUNCT cana-1164 184 1 result result VERB cana-1164 184 2 3.4 3.4 NUM cana-1164 184 3 . . PUNCT cana-1164 185 1 the the DET cana-1164 185 2 sum sum NOUN cana-1164 185 3 of of ADP cana-1164 185 4 entries entry NOUN cana-1164 185 5 in in ADP cana-1164 185 6 the the DET cana-1164 185 7 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 185 8 row row NOUN cana-1164 185 9 of of ADP cana-1164 185 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 185 11 โˆ’ โˆ’ PROPN cana-1164 185 12 is be AUX cana-1164 185 13 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 185 14 โˆ’(๐‘ฃ๐‘– โˆ’(๐‘ฃ๐‘– PROPN cana-1164 185 15 ) ) PUNCT cana-1164 185 16 , , PUNCT cana-1164 185 17 where where SCONJ cana-1164 185 18 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 185 19 โˆ’(๐‘ฃ๐‘– โˆ’(๐‘ฃ๐‘– PROPN cana-1164 185 20 ) ) PUNCT cana-1164 185 21 is be AUX cana-1164 185 22 the the DET cana-1164 185 23 in in ADP cana-1164 185 24 - - PUNCT cana-1164 185 25 degree degree NOUN cana-1164 185 26 of of ADP cana-1164 185 27 the the DET cana-1164 185 28 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 185 29 vertex vertex NOUN cana-1164 185 30 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADV cana-1164 185 31 . . PUNCT cana-1164 186 1 proof proof NOUN cana-1164 186 2 . . PUNCT cana-1164 187 1 let let VERB cana-1164 188 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 188 2 โˆ’ โˆ’ PROPN cana-1164 189 1 = = PUNCT cana-1164 190 1 [ [ X cana-1164 190 2 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— X cana-1164 190 3 ] ] PUNCT cana-1164 190 4 be be VERB cana-1164 190 5 an an DET cana-1164 190 6 in in ADP cana-1164 190 7 - - PUNCT cana-1164 190 8 adjacency adjacency NOUN cana-1164 190 9 matrix matrix NOUN cana-1164 190 10 of of ADP cana-1164 190 11 the the DET cana-1164 190 12 digraph digraph ADJ cana-1164 190 13 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 190 14 , , PUNCT cana-1164 190 15 ๐‘˜ ๐‘˜ NOUN cana-1164 190 16 ) ) PUNCT cana-1164 190 17 and and CCONJ cana-1164 190 18 let let VERB cana-1164 190 19 ๐‘…๐‘– ๐‘…๐‘– PROPN cana-1164 190 20 = = PUNCT cana-1164 191 1 [ [ X cana-1164 191 2 ๐‘Ž๐‘–1 ๐‘Ž๐‘–1 NOUN cana-1164 191 3 , , PUNCT cana-1164 191 4 ๐‘Ž๐‘–2 ๐‘Ž๐‘–2 NOUN cana-1164 191 5 , , PUNCT cana-1164 191 6 โ‹ฏ โ‹ฏ PROPN cana-1164 191 7 , , PUNCT cana-1164 191 8 ๐‘Ž๐‘–๐‘› ๐‘Ž๐‘–๐‘› PROPN cana-1164 191 9 ] ] PUNCT cana-1164 191 10 be be VERB cana-1164 191 11 the the DET cana-1164 191 12 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 191 13 row row NOUN cana-1164 191 14 of of ADP cana-1164 191 15 the the DET cana-1164 191 16 matrix matrix NOUN cana-1164 192 1 ๐ดฮณ ๐ดฮณ AUX cana-1164 192 2 โˆ’ โˆ’ PROPN cana-1164 192 3 corresponding correspond VERB cana-1164 192 4 to to ADP cana-1164 192 5 the the DET cana-1164 192 6 vertex vertex NOUN cana-1164 192 7 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 192 8 โˆˆ โˆˆ PROPN cana-1164 192 9 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 192 10 ) ) PUNCT cana-1164 192 11 . . PUNCT cana-1164 193 1 we we PRON cana-1164 193 2 now now ADV cana-1164 193 3 consider consider VERB cana-1164 193 4 the the DET cana-1164 193 5 sum sum NOUN cana-1164 193 6 โˆ‘๐‘›๐‘—=1 โˆ‘๐‘›๐‘—=1 PROPN cana-1164 193 7 ๐‘Ž๐‘–๐‘—. ๐‘Ž๐‘–๐‘—. NOUN cana-1164 193 8 clearly clearly ADV cana-1164 193 9 , , PUNCT cana-1164 193 10 1 1 NUM cana-1164 193 11 is be AUX cana-1164 193 12 added add VERB cana-1164 193 13 to to ADP cana-1164 193 14 this this DET cana-1164 193 15 sum sum NOUN cana-1164 193 16 โˆ‘๐‘›๐‘—=1 โˆ‘๐‘›๐‘—=1 PROPN cana-1164 193 17 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 193 18 exactly exactly ADV cana-1164 193 19 once once ADV cana-1164 193 20 for for ADP cana-1164 193 21 each each DET cana-1164 193 22 directed direct VERB cana-1164 193 23 arc arc NOUN cana-1164 193 24 coming come VERB cana-1164 193 25 to to ADP cana-1164 193 26 the the DET cana-1164 193 27 vertex vertex NOUN cana-1164 193 28 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADV cana-1164 193 29 and and CCONJ cana-1164 193 30 thereby thereby ADV cana-1164 193 31 using use VERB cana-1164 193 32 the the DET cana-1164 193 33 definition definition NOUN cana-1164 193 34 of of ADP cana-1164 193 35 the the DET cana-1164 193 36 in in ADP cana-1164 193 37 - - PUNCT cana-1164 193 38 degree degree NOUN cana-1164 193 39 of of ADP cana-1164 193 40 a a DET cana-1164 193 41 vertex vertex NOUN cana-1164 193 42 the the DET cana-1164 193 43 result result NOUN cana-1164 193 44 follows follow VERB cana-1164 193 45 immediately immediately ADV cana-1164 193 46 i.e. i.e. X cana-1164 193 47 โˆ‘๐‘›๐‘—=1 โˆ‘๐‘›๐‘—=1 PROPN cana-1164 193 48 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 193 49 = = SYM cana-1164 193 50 ๐‘–๐‘›๐‘‘๐‘’๐‘”(๐‘ฃ๐‘– ๐‘–๐‘›๐‘‘๐‘’๐‘”(๐‘ฃ๐‘– NOUN cana-1164 193 51 ) ) PUNCT cana-1164 193 52 = = PUNCT cana-1164 193 53 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 193 54 โˆ’(๐‘ฃ๐‘– โˆ’(๐‘ฃ๐‘– PROPN cana-1164 193 55 ) ) PUNCT cana-1164 193 56 . . PUNCT cana-1164 194 1 result result VERB cana-1164 194 2 3.5 3.5 NUM cana-1164 194 3 . . PUNCT cana-1164 195 1 the the DET cana-1164 195 2 sum sum NOUN cana-1164 195 3 of of ADP cana-1164 195 4 entries entry NOUN cana-1164 195 5 in in ADP cana-1164 195 6 the the DET cana-1164 195 7 ๐‘—๐‘กโ„Ž ๐‘—๐‘กโ„Ž ADJ cana-1164 195 8 column column NOUN cana-1164 195 9 of of ADP cana-1164 195 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 195 11 + + CCONJ cana-1164 195 12 is be AUX cana-1164 195 13 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 195 14 โˆ’(๐‘ฃ๐‘— โˆ’(๐‘ฃ๐‘— PROPN cana-1164 195 15 ) ) PUNCT cana-1164 195 16 , , PUNCT cana-1164 195 17 where where SCONJ cana-1164 195 18 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 195 19 โˆ’(๐‘ฃ๐‘— โˆ’(๐‘ฃ๐‘— PROPN cana-1164 195 20 ) ) PUNCT cana-1164 195 21 is be AUX cana-1164 195 22 the the DET cana-1164 195 23 in in ADP cana-1164 195 24 - - PUNCT cana-1164 195 25 degree degree NOUN cana-1164 195 26 of of ADP cana-1164 195 27 the the DET cana-1164 195 28 ๐‘—๐‘กโ„Ž ๐‘—๐‘กโ„Ž ADJ cana-1164 195 29 vertex vertex NOUN cana-1164 195 30 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 195 31 . . PUNCT cana-1164 196 1 proof proof NOUN cana-1164 196 2 . . PUNCT cana-1164 197 1 let let VERB cana-1164 198 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 198 2 + + NOUN cana-1164 198 3 = = SYM cana-1164 199 1 [ [ X cana-1164 199 2 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— X cana-1164 199 3 ] ] PUNCT cana-1164 199 4 be be VERB cana-1164 199 5 an an DET cana-1164 199 6 out out ADJ cana-1164 199 7 - - PUNCT cana-1164 199 8 adjacency adjacency NOUN cana-1164 199 9 matrix matrix NOUN cana-1164 199 10 of of ADP cana-1164 199 11 the the DET cana-1164 199 12 digraph digraph ADJ cana-1164 199 13 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 199 14 , , PUNCT cana-1164 199 15 ๐‘˜ ๐‘˜ NOUN cana-1164 199 16 ) ) PUNCT cana-1164 199 17 and and CCONJ cana-1164 199 18 let let VERB cana-1164 199 19 ๐ถ๐‘— ๐ถ๐‘— NOUN cana-1164 199 20 = = PUNCT cana-1164 199 21 [ [ PUNCT cana-1164 199 22 ๐‘Ž1๐‘— ๐‘Ž1๐‘— X cana-1164 199 23 ๐‘Ž2๐‘— ๐‘Ž2๐‘— ADP cana-1164 199 24 โ‹ฎ โ‹ฎ NOUN cana-1164 199 25 ๐‘Ž๐‘›๐‘— ๐‘Ž๐‘›๐‘— PROPN cana-1164 199 26 ] ] PUNCT cana-1164 199 27 be be AUX cana-1164 199 28 the the DET cana-1164 199 29 ๐‘—๐‘กโ„Žcolumn ๐‘—๐‘กโ„Žcolumn NOUN cana-1164 199 30 of of ADP cana-1164 199 31 ๐ดฮณ ๐ดฮณ PROPN cana-1164 199 32 + + PUNCT cana-1164 199 33 corresponding correspond VERB cana-1164 199 34 to to ADP cana-1164 199 35 the the DET cana-1164 199 36 vertex vertex NOUN cana-1164 199 37 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 199 38 โˆˆ โˆˆ NOUN cana-1164 199 39 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 199 40 ) ) PUNCT cana-1164 199 41 . . PUNCT cana-1164 200 1 we we PRON cana-1164 200 2 now now ADV cana-1164 200 3 consider consider VERB cana-1164 200 4 the the DET cana-1164 200 5 sum sum NOUN cana-1164 200 6 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 200 7 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 200 8 . . PUNCT cana-1164 201 1 clearly clearly ADV cana-1164 201 2 , , PUNCT cana-1164 201 3 1 1 NUM cana-1164 201 4 is be AUX cana-1164 201 5 added add VERB cana-1164 201 6 to to ADP cana-1164 201 7 this this DET cana-1164 201 8 sum sum NOUN cana-1164 201 9 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 201 10 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 201 11 exactly exactly ADV cana-1164 201 12 once once ADV cana-1164 201 13 for for ADP cana-1164 201 14 each each DET cana-1164 201 15 directed direct VERB cana-1164 201 16 arc arc NOUN cana-1164 201 17 coming come VERB cana-1164 201 18 to to ADP cana-1164 201 19 the the DET cana-1164 201 20 vertex vertex NOUN cana-1164 201 21 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 201 22 and and CCONJ cana-1164 201 23 thereby thereby ADV cana-1164 201 24 using use VERB cana-1164 201 25 the the DET cana-1164 201 26 definition definition NOUN cana-1164 201 27 of of ADP cana-1164 201 28 the the DET cana-1164 201 29 in in ADP cana-1164 201 30 - - PUNCT cana-1164 201 31 degree degree NOUN cana-1164 201 32 of of ADP cana-1164 201 33 a a DET cana-1164 201 34 vertex vertex NOUN cana-1164 201 35 the the DET cana-1164 201 36 result result NOUN cana-1164 201 37 follows follow VERB cana-1164 201 38 immediately immediately ADV cana-1164 201 39 i.e. i.e. X cana-1164 201 40 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 201 41 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 201 42 = = SYM cana-1164 201 43 ๐‘–๐‘›๐‘‘๐‘’๐‘”(๐‘ฃ๐‘— ๐‘–๐‘›๐‘‘๐‘’๐‘”(๐‘ฃ๐‘— PROPN cana-1164 201 44 ) ) PUNCT cana-1164 201 45 = = PUNCT cana-1164 201 46 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 201 47 โˆ’(๐‘ฃ๐‘— โˆ’(๐‘ฃ๐‘— PROPN cana-1164 201 48 ) ) PUNCT cana-1164 201 49 . . PUNCT cana-1164 202 1 result result VERB cana-1164 202 2 3.6 3.6 NUM cana-1164 202 3 . . PUNCT cana-1164 203 1 the the DET cana-1164 203 2 sum sum NOUN cana-1164 203 3 of of ADP cana-1164 203 4 entries entry NOUN cana-1164 203 5 in in ADP cana-1164 203 6 the the DET cana-1164 203 7 ๐‘—๐‘กโ„Ž ๐‘—๐‘กโ„Ž ADJ cana-1164 203 8 column column NOUN cana-1164 203 9 of of ADP cana-1164 203 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 203 11 โˆ’ โˆ’ PROPN cana-1164 203 12 is be AUX cana-1164 203 13 1 1 NUM cana-1164 203 14 . . PUNCT cana-1164 204 1 proof proof NOUN cana-1164 204 2 . . PUNCT cana-1164 205 1 let let VERB cana-1164 206 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 206 2 โˆ’ โˆ’ PROPN cana-1164 207 1 = = PUNCT cana-1164 208 1 [ [ X cana-1164 208 2 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— X cana-1164 208 3 ] ] PUNCT cana-1164 208 4 be be VERB cana-1164 208 5 an an DET cana-1164 208 6 in in ADP cana-1164 208 7 - - PUNCT cana-1164 208 8 adjacency adjacency NOUN cana-1164 208 9 matrix matrix NOUN cana-1164 208 10 of of ADP cana-1164 208 11 the the DET cana-1164 208 12 digraph digraph ADJ cana-1164 208 13 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 208 14 , , PUNCT cana-1164 208 15 ๐‘˜ ๐‘˜ NOUN cana-1164 208 16 ) ) PUNCT cana-1164 208 17 and and CCONJ cana-1164 208 18 let let VERB cana-1164 208 19 ๐ถ๐‘— ๐ถ๐‘— NOUN cana-1164 208 20 = = PUNCT cana-1164 208 21 [ [ PUNCT cana-1164 208 22 ๐‘Ž1๐‘— ๐‘Ž1๐‘— X cana-1164 208 23 ๐‘Ž2๐‘— ๐‘Ž2๐‘— ADP cana-1164 208 24 โ‹ฎ โ‹ฎ NOUN cana-1164 208 25 ๐‘Ž๐‘›๐‘— ๐‘Ž๐‘›๐‘— PROPN cana-1164 208 26 ] ] PUNCT cana-1164 208 27 be be AUX cana-1164 208 28 the the DET cana-1164 208 29 ๐‘—๐‘กโ„Ž ๐‘—๐‘กโ„Ž ADJ cana-1164 208 30 column column NOUN cana-1164 208 31 of of ADP cana-1164 208 32 ๐ดฮณ ๐ดฮณ PROPN cana-1164 208 33 โˆ’ โˆ’ PROPN cana-1164 208 34 corresponding correspond VERB cana-1164 208 35 to to ADP cana-1164 208 36 the the DET cana-1164 208 37 vertex vertex NOUN cana-1164 208 38 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 208 39 โˆˆ โˆˆ NOUN cana-1164 208 40 ๐‘‰(ฮณ ๐‘‰(ฮณ PROPN cana-1164 208 41 ) ) PUNCT cana-1164 208 42 . . PUNCT cana-1164 209 1 as as SCONJ cana-1164 209 2 the the DET cana-1164 209 3 residue residue NOUN cana-1164 209 4 of of ADP cana-1164 209 5 a a DET cana-1164 209 6 number number NOUN cana-1164 209 7 modulo modulo VERB cana-1164 209 8 ๐‘› ๐‘› NOUN cana-1164 209 9 is be AUX cana-1164 209 10 unique unique ADJ cana-1164 209 11 , , PUNCT cana-1164 209 12 the the DET cana-1164 209 13 number number NOUN cana-1164 209 14 of of ADP cana-1164 209 15 directed direct VERB cana-1164 209 16 arcs arc NOUN cana-1164 209 17 leaving leave VERB cana-1164 209 18 the the DET cana-1164 209 19 vertex vertex NOUN cana-1164 209 20 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 209 21 is be AUX cana-1164 209 22 exactly exactly ADV cana-1164 209 23 one one NUM cana-1164 209 24 . . PUNCT cana-1164 210 1 it it PRON cana-1164 210 2 contributes contribute VERB cana-1164 210 3 thereby thereby ADV cana-1164 210 4 1 1 NUM cana-1164 210 5 exactly exactly ADV cana-1164 210 6 in in ADP cana-1164 210 7 one one NUM cana-1164 210 8 of of ADP cana-1164 210 9 the the DET cana-1164 210 10 entries entry NOUN cana-1164 210 11 of of ADP cana-1164 210 12 ๐ถ๐‘— ๐ถ๐‘— PROPN cana-1164 210 13 and and CCONJ cana-1164 210 14 0 0 NUM cana-1164 210 15 in in ADP cana-1164 210 16 the the DET cana-1164 210 17 remaining remain VERB cana-1164 210 18 entries entry NOUN cana-1164 210 19 of of ADP cana-1164 210 20 ๐ถ๐‘—. ๐ถ๐‘—. NOUN cana-1164 210 21 thus thus ADV cana-1164 210 22 , , PUNCT cana-1164 210 23 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 210 24 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 210 25 = = SYM cana-1164 210 26 1 1 X cana-1164 210 27 . . PUNCT cana-1164 210 28 corollary corollary ADJ cana-1164 210 29 3.2 3.2 NUM cana-1164 210 30 . . PUNCT cana-1164 211 1 the the DET cana-1164 211 2 sum sum NOUN cana-1164 211 3 of of ADP cana-1164 211 4 entries entry NOUN cana-1164 211 5 in in ADP cana-1164 211 6 the the DET cana-1164 211 7 ๐‘—๐‘กโ„Ž ๐‘—๐‘กโ„Ž ADJ cana-1164 211 8 column column NOUN cana-1164 211 9 of of ADP cana-1164 211 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 211 11 โˆ’ โˆ’ PROPN cana-1164 211 12 is be AUX cana-1164 211 13 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 211 14 โˆ’(๐‘ฃ๐‘— โˆ’(๐‘ฃ๐‘— PROPN cana-1164 211 15 ) ) PUNCT cana-1164 211 16 , , PUNCT cana-1164 211 17 where where SCONJ cana-1164 211 18 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 211 19 โˆ’(๐‘ฃ๐‘— โˆ’(๐‘ฃ๐‘— PROPN cana-1164 211 20 ) ) PUNCT cana-1164 211 21 is be AUX cana-1164 211 22 the the DET cana-1164 211 23 in in ADP cana-1164 211 24 - - PUNCT cana-1164 211 25 degree degree NOUN cana-1164 211 26 of of ADP cana-1164 211 27 the the DET cana-1164 211 28 ๐‘—๐‘กโ„Ž ๐‘—๐‘กโ„Ž ADJ cana-1164 211 29 vertex vertex NOUN cana-1164 211 30 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 211 31 . . PUNCT cana-1164 212 1 result result VERB cana-1164 212 2 3.7 3.7 NUM cana-1164 212 3 . . PUNCT cana-1164 213 1 the the DET cana-1164 213 2 sum sum NOUN cana-1164 213 3 of of ADP cana-1164 213 4 all all DET cana-1164 213 5 entries entry NOUN cana-1164 213 6 in in ADP cana-1164 213 7 the the DET cana-1164 213 8 matrix matrix NOUN cana-1164 214 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 214 2 + + CCONJ cana-1164 214 3 is be AUX cana-1164 214 4 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 214 5 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 214 6 + + PROPN cana-1164 214 7 ( ( PUNCT cana-1164 214 8 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 214 9 ) ) PUNCT cana-1164 214 10 . . PUNCT cana-1164 215 1 communications communication NOUN cana-1164 215 2 on on ADP cana-1164 215 3 applied apply VERB cana-1164 215 4 nonlinear nonlinear ADJ cana-1164 215 5 analysis analysis NOUN cana-1164 215 6 issn issn NOUN cana-1164 215 7 : : PUNCT cana-1164 215 8 1074 1074 NUM cana-1164 215 9 - - PUNCT cana-1164 215 10 133x 133x NUM cana-1164 215 11 vol vol NOUN cana-1164 215 12 31 31 NUM cana-1164 215 13 no no NOUN cana-1164 215 14 . . PUNCT cana-1164 216 1 6s 6s NUM cana-1164 216 2 ( ( PUNCT cana-1164 216 3 2024 2024 NUM cana-1164 216 4 ) ) PUNCT cana-1164 216 5 105 105 NUM cana-1164 216 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 216 7 proof proof NOUN cana-1164 216 8 . . PUNCT cana-1164 217 1 let let VERB cana-1164 218 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 218 2 + + NOUN cana-1164 218 3 = = SYM cana-1164 219 1 [ [ X cana-1164 219 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 219 3 be be AUX cana-1164 219 4 an an DET cana-1164 219 5 out out ADJ cana-1164 219 6 - - PUNCT cana-1164 219 7 adjacency adjacency NOUN cana-1164 219 8 matrix matrix NOUN cana-1164 219 9 of of ADP cana-1164 219 10 the the DET cana-1164 219 11 digraph digraph ADJ cana-1164 219 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 219 13 , , PUNCT cana-1164 219 14 ๐‘˜ ๐‘˜ NOUN cana-1164 219 15 ) ) PUNCT cana-1164 219 16 . . PUNCT cana-1164 220 1 suppose suppose VERB cana-1164 220 2 ๐‘…1 ๐‘…1 PROPN cana-1164 220 3 , , PUNCT cana-1164 220 4 ๐‘…2 ๐‘…2 ADP cana-1164 220 5 , , PUNCT cana-1164 220 6 โ‹ฏ โ‹ฏ PROPN cana-1164 220 7 , , PUNCT cana-1164 220 8 ๐‘…๐‘› ๐‘…๐‘› PROPN cana-1164 220 9 be be VERB cana-1164 220 10 the the DET cana-1164 220 11 ๐‘›-rows ๐‘›-row NOUN cana-1164 220 12 of of ADP cana-1164 220 13 the the DET cana-1164 220 14 matrix matrix NOUN cana-1164 221 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 221 2 + + PROPN cana-1164 221 3 . . PUNCT cana-1164 221 4 by by ADP cana-1164 221 5 corollary corollary ADJ cana-1164 221 6 3.1 3.1 NUM cana-1164 221 7 . . PROPN cana-1164 221 8 , , PUNCT cana-1164 221 9 the the DET cana-1164 221 10 sum sum NOUN cana-1164 221 11 of of ADP cana-1164 221 12 entries entry NOUN cana-1164 221 13 in in ADP cana-1164 221 14 the the DET cana-1164 221 15 ๐‘–๐‘กโ„Ž ๐‘–๐‘กโ„Ž NOUN cana-1164 221 16 row row NOUN cana-1164 221 17 ( ( PUNCT cana-1164 221 18 i.e. i.e. X cana-1164 221 19 ๐‘…๐‘– ๐‘…๐‘– PROPN cana-1164 221 20 ) ) PUNCT cana-1164 221 21 is be AUX cana-1164 221 22 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 221 23 + + NOUN cana-1164 221 24 ( ( PUNCT cana-1164 221 25 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 221 26 ) ) PUNCT cana-1164 221 27 , , PUNCT cana-1164 221 28 for for ADP cana-1164 221 29 all all DET cana-1164 221 30 ๐‘– ๐‘– NOUN cana-1164 221 31 = = SYM cana-1164 221 32 1,2,โ‹ฏ 1,2,โ‹ฏ PROPN cana-1164 221 33 , , PUNCT cana-1164 221 34 ๐‘› ๐‘› PROPN cana-1164 221 35 and and CCONJ cana-1164 221 36 consequently consequently ADV cana-1164 221 37 , , PUNCT cana-1164 221 38 the the DET cana-1164 221 39 sum sum NOUN cana-1164 221 40 of of ADP cana-1164 221 41 entries entry NOUN cana-1164 221 42 in in ADP cana-1164 221 43 all all DET cana-1164 221 44 these these DET cana-1164 221 45 rows row NOUN cana-1164 221 46 is be AUX cana-1164 221 47 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 221 48 + + PROPN cana-1164 221 49 ( ( PUNCT cana-1164 221 50 ๐‘ฃ1 ๐‘ฃ1 NOUN cana-1164 221 51 ) ) PUNCT cana-1164 221 52 + + CCONJ cana-1164 221 53 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 221 54 + + PROPN cana-1164 221 55 ( ( PUNCT cana-1164 221 56 ๐‘ฃ2 ๐‘ฃ2 NOUN cana-1164 221 57 ) ) PUNCT cana-1164 221 58 + + NUM cana-1164 221 59 โ‹ฏ+ โ‹ฏ+ NOUN cana-1164 221 60 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 221 61 + + NOUN cana-1164 221 62 ( ( PUNCT cana-1164 221 63 ๐‘ฃ๐‘› ๐‘ฃ๐‘› NOUN cana-1164 221 64 ) ) PUNCT cana-1164 221 65 = = SYM cana-1164 221 66 โˆ‘ โˆ‘ PUNCT cana-1164 221 67 ๐‘› ๐‘› PRON cana-1164 221 68 ๐‘–=1 ๐‘–=1 PUNCT cana-1164 221 69 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 221 70 + + PROPN cana-1164 221 71 ( ( PUNCT cana-1164 221 72 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 221 73 ) ) PUNCT cana-1164 221 74 i.e. i.e. X cana-1164 221 75 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 221 76 โˆ‘ โˆ‘ PUNCT cana-1164 221 77 ๐‘› ๐‘› DET cana-1164 221 78 ๐‘—=1 ๐‘—=1 PROPN cana-1164 221 79 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 221 80 = = SYM cana-1164 221 81 โˆ‘ โˆ‘ PUNCT cana-1164 221 82 ๐‘› ๐‘› PRON cana-1164 221 83 ๐‘–=1 ๐‘–=1 PUNCT cana-1164 221 84 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 221 85 + + PROPN cana-1164 221 86 ( ( PUNCT cana-1164 221 87 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 221 88 ) ) PUNCT cana-1164 221 89 . . PUNCT cana-1164 222 1 result result VERB cana-1164 222 2 3.8 3.8 NUM cana-1164 222 3 . . PUNCT cana-1164 223 1 the the DET cana-1164 223 2 sum sum NOUN cana-1164 223 3 of of ADP cana-1164 223 4 all all DET cana-1164 223 5 entries entry NOUN cana-1164 223 6 in in ADP cana-1164 223 7 the the DET cana-1164 223 8 matrix matrix NOUN cana-1164 224 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 224 2 + + CCONJ cana-1164 224 3 is be AUX cana-1164 224 4 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 224 5 ๐‘‘ฮณ ๐‘‘ฮณ PROPN cana-1164 224 6 โˆ’(๐‘ฃ๐‘– โˆ’(๐‘ฃ๐‘– PROPN cana-1164 224 7 ) ) PUNCT cana-1164 224 8 . . PUNCT cana-1164 225 1 proof proof NOUN cana-1164 225 2 . . PUNCT cana-1164 226 1 the the DET cana-1164 226 2 result result NOUN cana-1164 226 3 can can AUX cana-1164 226 4 be be AUX cana-1164 226 5 easily easily ADV cana-1164 226 6 established establish VERB cana-1164 226 7 using use VERB cana-1164 226 8 result result NOUN cana-1164 226 9 3.4 3.4 NUM cana-1164 226 10 . . PUNCT cana-1164 226 11 remark remark PROPN cana-1164 226 12 3.2 3.2 NUM cana-1164 226 13 . . PUNCT cana-1164 227 1 if if SCONJ cana-1164 227 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 227 3 + + NOUN cana-1164 227 4 = = SYM cana-1164 228 1 [ [ X cana-1164 228 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 228 3 be be AUX cana-1164 228 4 an an DET cana-1164 228 5 out out ADJ cana-1164 228 6 - - PUNCT cana-1164 228 7 adjacency adjacency NOUN cana-1164 228 8 matrix matrix NOUN cana-1164 228 9 of of ADP cana-1164 228 10 the the DET cana-1164 228 11 digraph digraph ADJ cana-1164 228 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 228 13 , , PUNCT cana-1164 228 14 ๐‘˜ ๐‘˜ NOUN cana-1164 228 15 ) ) PUNCT cana-1164 228 16 then then ADV cana-1164 228 17 i. i. PROPN cana-1164 228 18 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 228 19 โˆ‘ โˆ‘ PUNCT cana-1164 228 20 ๐‘› ๐‘› PROPN cana-1164 228 21 ๐‘—=1 ๐‘—=1 PROPN cana-1164 228 22 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 228 23 = = SYM cana-1164 228 24 โˆ‘ โˆ‘ PUNCT cana-1164 228 25 ๐‘› ๐‘› PRON cana-1164 228 26 ๐‘–=1 ๐‘–=1 PUNCT cana-1164 228 27 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 228 28 + + PROPN cana-1164 228 29 ( ( PUNCT cana-1164 228 30 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 228 31 ) ) PUNCT cana-1164 228 32 = = SYM cana-1164 228 33 โˆ‘ โˆ‘ PUNCT cana-1164 228 34 ๐‘› ๐‘› PRON cana-1164 228 35 ๐‘–=1 ๐‘–=1 PROPN cana-1164 228 36 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 228 37 โˆ’(๐‘ฃ๐‘– โˆ’(๐‘ฃ๐‘– PROPN cana-1164 228 38 ) ) PUNCT cana-1164 228 39 = = SYM cana-1164 228 40 ๐‘› ๐‘› DET cana-1164 228 41 ii ii NOUN cana-1164 228 42 . . PUNCT cana-1164 229 1 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 229 2 โˆ‘ โˆ‘ PUNCT cana-1164 229 3 ๐‘› ๐‘› PRON cana-1164 229 4 ๐‘—=1 ๐‘—=1 PROPN cana-1164 229 5 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— X cana-1164 229 6 = = SYM cana-1164 229 7 |๐ด(ฮณ)| |๐ด(ฮณ)| PROPN cana-1164 229 8 = = SYM cana-1164 229 9 |๐‘‰(ฮณ)| |๐‘‰(ฮณ)| PROPN cana-1164 229 10 = = PUNCT cana-1164 229 11 ๐‘›. ๐‘›. NOUN cana-1164 229 12 result result VERB cana-1164 229 13 3.9 3.9 NUM cana-1164 229 14 . . PUNCT cana-1164 230 1 the the DET cana-1164 230 2 sum sum NOUN cana-1164 230 3 of of ADP cana-1164 230 4 all all DET cana-1164 230 5 entries entry NOUN cana-1164 230 6 in in ADP cana-1164 230 7 the the DET cana-1164 230 8 matrix matrix NOUN cana-1164 230 9 ๐ดฮณ ๐ดฮณ PROPN cana-1164 230 10 โˆ’ โˆ’ PROPN cana-1164 230 11 is be AUX cana-1164 230 12 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 230 13 ๐‘‘ฮณ ๐‘‘ฮณ PROPN cana-1164 230 14 โˆ’(๐‘ฃ๐‘– โˆ’(๐‘ฃ๐‘– PROPN cana-1164 230 15 ) ) PUNCT cana-1164 230 16 . . PUNCT cana-1164 231 1 proof proof NOUN cana-1164 231 2 . . PUNCT cana-1164 232 1 the the DET cana-1164 232 2 proof proof NOUN cana-1164 232 3 is be AUX cana-1164 232 4 left leave VERB cana-1164 232 5 for for ADP cana-1164 232 6 the the DET cana-1164 232 7 reader reader NOUN cana-1164 232 8 . . PUNCT cana-1164 233 1 result result VERB cana-1164 233 2 3.10 3.10 NUM cana-1164 233 3 . . PUNCT cana-1164 234 1 the the DET cana-1164 234 2 sum sum NOUN cana-1164 234 3 of of ADP cana-1164 234 4 all all DET cana-1164 234 5 entries entry NOUN cana-1164 234 6 in in ADP cana-1164 234 7 the the DET cana-1164 234 8 matrix matrix NOUN cana-1164 234 9 ๐ดฮณ ๐ดฮณ PROPN cana-1164 234 10 โˆ’ โˆ’ PROPN cana-1164 234 11 is be AUX cana-1164 234 12 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 234 13 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 234 14 + + PROPN cana-1164 234 15 ( ( PUNCT cana-1164 234 16 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 234 17 ) ) PUNCT cana-1164 234 18 . . PUNCT cana-1164 235 1 proof proof NOUN cana-1164 235 2 . . PUNCT cana-1164 236 1 the the DET cana-1164 236 2 proof proof NOUN cana-1164 236 3 is be AUX cana-1164 236 4 left leave VERB cana-1164 236 5 for for ADP cana-1164 236 6 the the DET cana-1164 236 7 reader reader NOUN cana-1164 236 8 . . PUNCT cana-1164 237 1 remark remark VERB cana-1164 237 2 3.3 3.3 NUM cana-1164 237 3 . . PUNCT cana-1164 238 1 if if SCONJ cana-1164 238 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 238 3 โˆ’ โˆ’ PROPN cana-1164 239 1 = = PUNCT cana-1164 240 1 [ [ X cana-1164 240 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 240 3 be be AUX cana-1164 240 4 an an DET cana-1164 240 5 in in ADP cana-1164 240 6 - - PUNCT cana-1164 240 7 adjacency adjacency NOUN cana-1164 240 8 matrix matrix NOUN cana-1164 240 9 of of ADP cana-1164 240 10 the the DET cana-1164 240 11 digraph digraph ADJ cana-1164 240 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 240 13 , , PUNCT cana-1164 240 14 ๐‘˜ ๐‘˜ NOUN cana-1164 240 15 ) ) PUNCT cana-1164 240 16 then then ADV cana-1164 240 17 i. i. PROPN cana-1164 240 18 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 240 19 โˆ‘ โˆ‘ PUNCT cana-1164 240 20 ๐‘› ๐‘› PROPN cana-1164 240 21 ๐‘—=1 ๐‘—=1 PROPN cana-1164 240 22 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 240 23 = = SYM cana-1164 240 24 โˆ‘ โˆ‘ PUNCT cana-1164 240 25 ๐‘› ๐‘› PRON cana-1164 240 26 ๐‘–=1 ๐‘–=1 PUNCT cana-1164 240 27 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 240 28 + + PROPN cana-1164 240 29 ( ( PUNCT cana-1164 240 30 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 240 31 ) ) PUNCT cana-1164 240 32 = = SYM cana-1164 240 33 โˆ‘ โˆ‘ PUNCT cana-1164 240 34 ๐‘› ๐‘› PRON cana-1164 240 35 ๐‘–=1 ๐‘–=1 PROPN cana-1164 240 36 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 240 37 โˆ’(๐‘ฃ๐‘– โˆ’(๐‘ฃ๐‘– PROPN cana-1164 240 38 ) ) PUNCT cana-1164 240 39 = = SYM cana-1164 240 40 ๐‘› ๐‘› DET cana-1164 240 41 ii ii NOUN cana-1164 240 42 . . PUNCT cana-1164 241 1 โˆ‘๐‘›๐‘–=1 โˆ‘๐‘›๐‘–=1 PROPN cana-1164 241 2 โˆ‘ โˆ‘ PUNCT cana-1164 241 3 ๐‘› ๐‘› PRON cana-1164 241 4 ๐‘—=1 ๐‘—=1 PROPN cana-1164 241 5 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— X cana-1164 241 6 = = SYM cana-1164 241 7 |๐ด(ฮณ)| |๐ด(ฮณ)| PROPN cana-1164 241 8 = = SYM cana-1164 241 9 |๐‘‰(ฮณ)| |๐‘‰(ฮณ)| PROPN cana-1164 241 10 = = PUNCT cana-1164 241 11 ๐‘›. ๐‘›. NOUN cana-1164 241 12 result result VERB cana-1164 241 13 3.11 3.11 NUM cana-1164 241 14 . . PUNCT cana-1164 242 1 let let VERB cana-1164 243 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 243 2 + + NOUN cana-1164 243 3 = = SYM cana-1164 244 1 [ [ X cana-1164 244 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 244 3 be be AUX cana-1164 244 4 an an DET cana-1164 244 5 out out ADJ cana-1164 244 6 - - PUNCT cana-1164 244 7 adjacency adjacency NOUN cana-1164 244 8 matrix matrix NOUN cana-1164 244 9 of of ADP cana-1164 244 10 the the DET cana-1164 244 11 digraph digraph ADJ cana-1164 244 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 244 13 , , PUNCT cana-1164 244 14 ๐‘˜ ๐‘˜ NOUN cana-1164 244 15 ) ) PUNCT cana-1164 244 16 , , PUNCT cana-1164 244 17 then then ADV cana-1164 244 18 the the DET cana-1164 244 19 number number NOUN cana-1164 244 20 of of ADP cana-1164 244 21 directed direct VERB cana-1164 244 22 walks walk NOUN cana-1164 244 23 of of ADP cana-1164 244 24 length length NOUN cana-1164 244 25 ๐‘š ๐‘š NOUN cana-1164 244 26 from from ADP cana-1164 244 27 vertex vertex NOUN cana-1164 244 28 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 244 29 to to AUX cana-1164 244 30 vertex vertex PROPN cana-1164 244 31 ๐‘ฃ๐‘—(๐‘–. ๐‘ฃ๐‘—(๐‘–. PROPN cana-1164 244 32 ๐‘’. ๐‘’. PROPN cana-1164 244 33 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 244 34 โ†’ โ†’ SYM cana-1164 244 35 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 244 36 directed direct VERB cana-1164 244 37 walk walk NOUN cana-1164 244 38 ) ) PUNCT cana-1164 244 39 in in ADP cana-1164 244 40 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 244 41 , , PUNCT cana-1164 244 42 ๐‘˜ ๐‘˜ NOUN cana-1164 244 43 ) ) PUNCT cana-1164 244 44 is be AUX cana-1164 244 45 the the DET cana-1164 244 46 element element NOUN cana-1164 244 47 in in ADP cana-1164 244 48 the the DET cana-1164 244 49 ( ( PUNCT cana-1164 244 50 ๐‘– ๐‘– PROPN cana-1164 244 51 , , PUNCT cana-1164 244 52 ๐‘—)๐‘กโ„Ž ๐‘—)๐‘กโ„Ž PROPN cana-1164 244 53 position position NOUN cana-1164 244 54 of of ADP cana-1164 244 55 the the DET cana-1164 244 56 matrix matrix NOUN cana-1164 244 57 ( ( PUNCT cana-1164 244 58 ๐ดฮณ ๐ดฮณ PROPN cana-1164 244 59 + + PROPN cana-1164 244 60 ) ) PUNCT cana-1164 244 61 ๐‘š ๐‘š NOUN cana-1164 244 62 , , PUNCT cana-1164 244 63 where where SCONJ cana-1164 244 64 ๐‘š ๐‘š PROPN cana-1164 244 65 is be AUX cana-1164 244 66 a a DET cana-1164 244 67 non non ADJ cana-1164 244 68 - - ADJ cana-1164 244 69 negative negative ADJ cana-1164 244 70 integer integer NOUN cana-1164 244 71 . . PUNCT cana-1164 245 1 proof proof NOUN cana-1164 245 2 . . PUNCT cana-1164 246 1 we we PRON cana-1164 246 2 shall shall AUX cana-1164 246 3 try try VERB cana-1164 246 4 to to PART cana-1164 246 5 prove prove VERB cana-1164 246 6 the the DET cana-1164 246 7 result result NOUN cana-1164 246 8 using use VERB cana-1164 246 9 mathematical mathematical ADJ cana-1164 246 10 induction induction NOUN cana-1164 246 11 on on ADP cana-1164 246 12 ๐‘š. ๐‘š. ADJ cana-1164 246 13 if if SCONJ cana-1164 246 14 ๐‘š ๐‘š PROPN cana-1164 246 15 = = SYM cana-1164 246 16 0 0 NUM cana-1164 246 17 , , PUNCT cana-1164 246 18 then then ADV cana-1164 246 19 the the DET cana-1164 246 20 number number NOUN cana-1164 246 21 of of ADP cana-1164 246 22 directed direct VERB cana-1164 246 23 walks walk NOUN cana-1164 246 24 of of ADP cana-1164 246 25 length length NOUN cana-1164 246 26 0 0 NUM cana-1164 246 27 from from ADP cana-1164 246 28 vertex vertex NOUN cana-1164 246 29 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 246 30 to to ADP cana-1164 246 31 vertex vertex NOUN cana-1164 246 32 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 246 33 is be AUX cana-1164 246 34 0 0 NUM cana-1164 246 35 resulting result VERB cana-1164 246 36 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 246 37 = = SYM cana-1164 246 38 0 0 NUM cana-1164 246 39 , , PUNCT cana-1164 246 40 for for ADP cana-1164 246 41 ๐‘– ๐‘– SYM cana-1164 246 42 โ‰  โ‰  PROPN cana-1164 246 43 ๐‘—. ๐‘—. NOUN cana-1164 246 44 also also ADV cana-1164 246 45 the the DET cana-1164 246 46 number number NOUN cana-1164 246 47 of of ADP cana-1164 246 48 directed direct VERB cana-1164 246 49 walks walk NOUN cana-1164 246 50 of of ADP cana-1164 246 51 length length NOUN cana-1164 246 52 0 0 NUM cana-1164 246 53 from from ADP cana-1164 246 54 a a DET cana-1164 246 55 vertex vertex NOUN cana-1164 246 56 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 246 57 to to ADP cana-1164 246 58 itself itself PRON cana-1164 246 59 is be AUX cana-1164 246 60 1 1 NUM cana-1164 246 61 resulting result VERB cana-1164 246 62 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 246 63 = = SYM cana-1164 246 64 1 1 NUM cana-1164 246 65 , , PUNCT cana-1164 246 66 for for ADP cana-1164 246 67 ๐‘– ๐‘– DET cana-1164 246 68 = = PUNCT cana-1164 246 69 ๐‘— ๐‘— NOUN cana-1164 246 70 which which PRON cana-1164 246 71 gives give VERB cana-1164 246 72 us we PRON cana-1164 246 73 the the DET cana-1164 246 74 identity identity NOUN cana-1164 246 75 matrix matrix NOUN cana-1164 246 76 ๐ผ. ๐ผ. NOUN cana-1164 247 1 so so SCONJ cana-1164 247 2 we we PRON cana-1164 247 3 get get VERB cana-1164 247 4 ( ( PUNCT cana-1164 247 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 247 6 + + PROPN cana-1164 247 7 ) ) PUNCT cana-1164 247 8 0 0 PUNCT cana-1164 248 1 = = SYM cana-1164 248 2 ๐ผ. ๐ผ. PROPN cana-1164 248 3 if if SCONJ cana-1164 248 4 ๐‘š ๐‘š X cana-1164 248 5 = = SYM cana-1164 248 6 1 1 NUM cana-1164 248 7 , , PUNCT cana-1164 248 8 then then ADV cana-1164 248 9 the the DET cana-1164 248 10 number number NOUN cana-1164 248 11 of of ADP cana-1164 248 12 directed direct VERB cana-1164 248 13 walks walk NOUN cana-1164 248 14 of of ADP cana-1164 248 15 length length NOUN cana-1164 248 16 1 1 NUM cana-1164 248 17 from from ADP cana-1164 248 18 vertex vertex NOUN cana-1164 248 19 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 248 20 to to ADP cana-1164 248 21 vertex vertex NOUN cana-1164 248 22 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 248 23 is be AUX cana-1164 248 24 the the DET cana-1164 248 25 number number NOUN cana-1164 248 26 of of ADP cana-1164 248 27 directed direct VERB cana-1164 248 28 arcs arc NOUN cana-1164 248 29 from from ADP cana-1164 248 30 the the DET cana-1164 248 31 vertex vertex NOUN cana-1164 248 32 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 248 33 to to ADP cana-1164 248 34 vertex vertex NOUN cana-1164 248 35 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 248 36 which which PRON cana-1164 248 37 is be AUX cana-1164 248 38 equal equal ADJ cana-1164 248 39 to to ADP cana-1164 248 40 ๐‘Ž๐‘–๐‘— ๐‘Ž๐‘–๐‘— NOUN cana-1164 248 41 of of ADP cana-1164 248 42 the the DET cana-1164 248 43 out out ADJ cana-1164 248 44 - - PUNCT cana-1164 248 45 adjacency adjacency NOUN cana-1164 248 46 matrix matrix NOUN cana-1164 248 47 ๐ดฮณ ๐ดฮณ PROPN cana-1164 248 48 + + PROPN cana-1164 248 49 . . PUNCT cana-1164 249 1 so so ADV cana-1164 249 2 we we PRON cana-1164 249 3 get get VERB cana-1164 249 4 ( ( PUNCT cana-1164 249 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 249 6 + + PROPN cana-1164 249 7 ) ) PUNCT cana-1164 249 8 1 1 NUM cana-1164 250 1 = = SYM cana-1164 250 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 250 3 + + PROPN cana-1164 250 4 . . PUNCT cana-1164 251 1 we we PRON cana-1164 251 2 now now ADV cana-1164 251 3 assume assume VERB cana-1164 251 4 that that SCONJ cana-1164 251 5 the the DET cana-1164 251 6 result result NOUN cana-1164 251 7 is be AUX cana-1164 251 8 true true ADJ cana-1164 251 9 for for ADP cana-1164 251 10 ๐‘š ๐‘š PROPN cana-1164 251 11 > > SYM cana-1164 251 12 1 1 NUM cana-1164 251 13 and and CCONJ cana-1164 251 14 try try VERB cana-1164 251 15 to to PART cana-1164 251 16 establish establish VERB cana-1164 251 17 the the DET cana-1164 251 18 result result NOUN cana-1164 251 19 for for ADP cana-1164 251 20 ๐‘š ๐‘š PROPN cana-1164 251 21 + + NOUN cana-1164 251 22 1 1 X cana-1164 251 23 . . PUNCT cana-1164 252 1 let let VERB cana-1164 252 2 us we PRON cana-1164 252 3 denote denote VERB cana-1164 252 4 the the DET cana-1164 252 5 ( ( PUNCT cana-1164 252 6 ๐‘– ๐‘– PROPN cana-1164 252 7 , , PUNCT cana-1164 252 8 ๐‘—)๐‘กโ„Ž ๐‘—)๐‘กโ„Ž PROPN cana-1164 252 9 element element NOUN cana-1164 252 10 of of ADP cana-1164 252 11 ( ( PUNCT cana-1164 252 12 ๐ดฮณ ๐ดฮณ PROPN cana-1164 252 13 + + PROPN cana-1164 252 14 ) ) PUNCT cana-1164 252 15 ๐‘š ๐‘š NOUN cana-1164 252 16 by by ADP cana-1164 252 17 ๐‘๐‘–๐‘— ๐‘๐‘–๐‘— ADV cana-1164 252 18 i.e. i.e. X cana-1164 252 19 ( ( PUNCT cana-1164 252 20 ๐ดฮณ ๐ดฮณ PROPN cana-1164 252 21 + + PROPN cana-1164 252 22 ) ) PUNCT cana-1164 252 23 ๐‘š ๐‘š NOUN cana-1164 253 1 = = PUNCT cana-1164 254 1 [ [ X cana-1164 254 2 ๐‘๐‘–๐‘—]๐‘›ร—๐‘›. ๐‘๐‘–๐‘—]๐‘›ร—๐‘›. NOUN cana-1164 254 3 as as ADP cana-1164 254 4 , , PUNCT cana-1164 254 5 ( ( PUNCT cana-1164 254 6 ๐ดฮณ ๐ดฮณ PROPN cana-1164 254 7 + + PROPN cana-1164 254 8 ) ) PUNCT cana-1164 254 9 ๐‘š+1 ๐‘š+1 NUM cana-1164 254 10 = = SYM cana-1164 254 11 ( ( PUNCT cana-1164 254 12 ๐ดฮณ ๐ดฮณ PROPN cana-1164 254 13 + + PROPN cana-1164 254 14 ) ) PUNCT cana-1164 254 15 ๐‘š ๐‘š X cana-1164 254 16 โ‹… โ‹… PROPN cana-1164 254 17 ( ( PUNCT cana-1164 254 18 ๐ดฮณ ๐ดฮณ PROPN cana-1164 254 19 + + PROPN cana-1164 254 20 ) ) PUNCT cana-1164 254 21 = = NOUN cana-1164 255 1 [ [ X cana-1164 255 2 ๐‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘๐‘–๐‘—]๐‘›ร—๐‘› PROPN cana-1164 255 3 โ‹… โ‹… PROPN cana-1164 256 1 [ [ X cana-1164 256 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 256 3 = = SYM cana-1164 256 4 [ [ X cana-1164 256 5 ๐‘๐‘–๐‘—]๐‘›ร—๐‘› ๐‘๐‘–๐‘—]๐‘›ร—๐‘› PROPN cana-1164 256 6 communications communication NOUN cana-1164 256 7 on on ADP cana-1164 256 8 applied apply VERB cana-1164 256 9 nonlinear nonlinear ADJ cana-1164 256 10 analysis analysis NOUN cana-1164 256 11 issn issn NOUN cana-1164 256 12 : : PUNCT cana-1164 256 13 1074 1074 NUM cana-1164 256 14 - - PUNCT cana-1164 256 15 133x 133x NUM cana-1164 256 16 vol vol NOUN cana-1164 256 17 31 31 NUM cana-1164 256 18 no no NOUN cana-1164 256 19 . . PUNCT cana-1164 257 1 6s 6s NUM cana-1164 257 2 ( ( PUNCT cana-1164 257 3 2024 2024 NUM cana-1164 257 4 ) ) PUNCT cana-1164 257 5 106 106 NUM cana-1164 257 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 257 7 where where SCONJ cana-1164 257 8 , , PUNCT cana-1164 257 9 ๐‘๐‘–๐‘— ๐‘๐‘–๐‘— PROPN cana-1164 257 10 = = PUNCT cana-1164 257 11 โˆ‘ โˆ‘ PUNCT cana-1164 257 12 ๐‘› ๐‘› DET cana-1164 257 13 ๐‘˜=1 ๐‘˜=1 NOUN cana-1164 257 14 ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘—. ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘—. VERB cana-1164 257 15 by by ADP cana-1164 257 16 assumption assumption NOUN cana-1164 257 17 , , PUNCT cana-1164 257 18 ๐‘๐‘–๐‘˜ ๐‘๐‘–๐‘˜ PROPN cana-1164 257 19 is be AUX cana-1164 257 20 the the DET cana-1164 257 21 number number NOUN cana-1164 257 22 of of ADP cana-1164 257 23 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 257 24 โ†’ โ†’ SYM cana-1164 257 25 ๐‘ฃ๐‘˜ ๐‘ฃ๐‘˜ ADV cana-1164 257 26 directed direct VERB cana-1164 257 27 walks walk NOUN cana-1164 257 28 of of ADP cana-1164 257 29 length length NOUN cana-1164 257 30 ๐‘š. ๐‘š. ADV cana-1164 257 31 also also ADV cana-1164 257 32 , , PUNCT cana-1164 257 33 ๐‘Ž๐‘˜๐‘— ๐‘Ž๐‘˜๐‘— NOUN cana-1164 257 34 = = SYM cana-1164 257 35 0 0 NUM cana-1164 257 36 or or CCONJ cana-1164 257 37 1 1 NUM cana-1164 257 38 , , PUNCT cana-1164 257 39 so so ADV cana-1164 257 40 ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘— ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘— ADJ cana-1164 257 41 = = SYM cana-1164 257 42 0 0 NUM cana-1164 257 43 or or CCONJ cana-1164 257 44 ๐‘๐‘–๐‘˜. ๐‘๐‘–๐‘˜. NOUN cana-1164 257 45 then then ADV cana-1164 257 46 ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘— ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘— NOUN cana-1164 257 47 is be AUX cana-1164 257 48 exactly exactly ADV cana-1164 257 49 the the DET cana-1164 257 50 number number NOUN cana-1164 257 51 of of ADP cana-1164 257 52 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 257 53 โ†’ โ†’ SYM cana-1164 257 54 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 257 55 directed direct VERB cana-1164 257 56 walks walk NOUN cana-1164 257 57 of of ADP cana-1164 257 58 length length NOUN cana-1164 257 59 ( ( PUNCT cana-1164 257 60 ๐‘š ๐‘š PROPN cana-1164 257 61 + + NOUN cana-1164 257 62 1 1 NUM cana-1164 257 63 ) ) PUNCT cana-1164 257 64 with with ADP cana-1164 257 65 vertex vertex NOUN cana-1164 257 66 ๐‘ฃ๐‘˜ ๐‘ฃ๐‘˜ ADV cana-1164 257 67 adjacent adjacent ADJ cana-1164 257 68 to to ADP cana-1164 257 69 vertex vertex NOUN cana-1164 257 70 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 257 71 . . PUNCT cana-1164 258 1 as as SCONJ cana-1164 258 2 the the DET cana-1164 258 3 sum sum NOUN cana-1164 258 4 includes include VERB cana-1164 258 5 this this PRON cana-1164 258 6 for for ADP cana-1164 258 7 each each PRON cana-1164 258 8 of of ADP cana-1164 258 9 the the DET cana-1164 258 10 vertices vertex NOUN cana-1164 258 11 , , PUNCT cana-1164 258 12 we we PRON cana-1164 258 13 notice notice VERB cana-1164 258 14 that that SCONJ cana-1164 258 15 ๐‘๐‘–๐‘—(= ๐‘๐‘–๐‘—(= VERB cana-1164 258 16 โˆ‘ โˆ‘ PUNCT cana-1164 258 17 ๐‘› ๐‘› DET cana-1164 258 18 ๐‘˜=1 ๐‘˜=1 NOUN cana-1164 258 19 ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘— ๐‘๐‘–๐‘˜๐‘Ž๐‘˜๐‘— NOUN cana-1164 258 20 ) ) PUNCT cana-1164 258 21 is be AUX cana-1164 258 22 the the DET cana-1164 258 23 number number NOUN cana-1164 258 24 of of ADP cana-1164 258 25 ๐‘ฃ๐‘– ๐‘ฃ๐‘– NOUN cana-1164 258 26 โ†’ โ†’ SYM cana-1164 258 27 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 258 28 directed direct VERB cana-1164 258 29 walks walk NOUN cana-1164 258 30 of of ADP cana-1164 258 31 length length NOUN cana-1164 258 32 ( ( PUNCT cana-1164 258 33 ๐‘š ๐‘š PROPN cana-1164 258 34 + + NOUN cana-1164 258 35 1 1 NUM cana-1164 258 36 ) ) PUNCT cana-1164 258 37 and and CCONJ cana-1164 258 38 hence hence ADV cana-1164 258 39 the the DET cana-1164 258 40 result result NOUN cana-1164 258 41 holds hold VERB cana-1164 258 42 for for ADP cana-1164 258 43 ( ( PUNCT cana-1164 258 44 ๐ดฮณ ๐ดฮณ PROPN cana-1164 258 45 + + PROPN cana-1164 258 46 ) ) PUNCT cana-1164 258 47 ๐‘š+1 ๐‘š+1 NUM cana-1164 258 48 . . PUNCT cana-1164 259 1 so so ADV cana-1164 259 2 by by ADP cana-1164 259 3 induction induction NOUN cana-1164 259 4 , , PUNCT cana-1164 259 5 the the DET cana-1164 259 6 result result NOUN cana-1164 259 7 is be AUX cana-1164 259 8 established establish VERB cana-1164 259 9 . . PUNCT cana-1164 260 1 result result PROPN cana-1164 260 2 3.12 3.12 NUM cana-1164 260 3 . . PUNCT cana-1164 261 1 let let VERB cana-1164 262 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 262 2 โˆ’ โˆ’ PROPN cana-1164 263 1 = = PUNCT cana-1164 264 1 [ [ X cana-1164 264 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 264 3 be be AUX cana-1164 264 4 the the DET cana-1164 264 5 in in ADP cana-1164 264 6 - - PUNCT cana-1164 264 7 adjacency adjacency NOUN cana-1164 264 8 matrix matrix NOUN cana-1164 264 9 of of ADP cana-1164 264 10 the the DET cana-1164 264 11 digraph digraph ADJ cana-1164 264 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 264 13 , , PUNCT cana-1164 264 14 ๐‘˜ ๐‘˜ NOUN cana-1164 264 15 ) ) PUNCT cana-1164 264 16 , , PUNCT cana-1164 264 17 then then ADV cana-1164 264 18 the the DET cana-1164 264 19 number number NOUN cana-1164 264 20 of of ADP cana-1164 264 21 directed direct VERB cana-1164 264 22 walks walk NOUN cana-1164 264 23 of of ADP cana-1164 264 24 length length NOUN cana-1164 264 25 ๐‘š ๐‘š NOUN cana-1164 264 26 from from ADP cana-1164 264 27 vertex vertex NOUN cana-1164 264 28 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 264 29 to to PART cana-1164 264 30 vertex vertex NOUN cana-1164 264 31 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 264 32 ( ( PUNCT cana-1164 264 33 ๐‘–. ๐‘–. ADJ cana-1164 264 34 ๐‘’. ๐‘’. PROPN cana-1164 264 35 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 264 36 โ† โ† PROPN cana-1164 264 37 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 264 38 directed direct VERB cana-1164 264 39 walk walk NOUN cana-1164 264 40 ) ) PUNCT cana-1164 264 41 in in ADP cana-1164 264 42 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 264 43 , , PUNCT cana-1164 264 44 ๐‘˜ ๐‘˜ NOUN cana-1164 264 45 ) ) PUNCT cana-1164 264 46 is be AUX cana-1164 264 47 the the DET cana-1164 264 48 element element NOUN cana-1164 264 49 in in ADP cana-1164 264 50 the the DET cana-1164 264 51 ( ( PUNCT cana-1164 264 52 ๐‘– ๐‘– PROPN cana-1164 264 53 , , PUNCT cana-1164 264 54 ๐‘—)๐‘กโ„Ž ๐‘—)๐‘กโ„Ž PROPN cana-1164 264 55 position position NOUN cana-1164 264 56 of of ADP cana-1164 264 57 the the DET cana-1164 264 58 matrix matrix NOUN cana-1164 264 59 ( ( PUNCT cana-1164 264 60 ๐ดฮณ ๐ดฮณ PROPN cana-1164 264 61 โˆ’)๐‘š โˆ’)๐‘š NOUN cana-1164 264 62 , , PUNCT cana-1164 264 63 where where SCONJ cana-1164 264 64 ๐‘š ๐‘š PROPN cana-1164 264 65 is be AUX cana-1164 264 66 a a DET cana-1164 264 67 non non ADJ cana-1164 264 68 - - ADJ cana-1164 264 69 negative negative ADJ cana-1164 264 70 integer integer NOUN cana-1164 264 71 . . PUNCT cana-1164 265 1 proof proof NOUN cana-1164 265 2 . . PUNCT cana-1164 266 1 it it PRON cana-1164 266 2 can can AUX cana-1164 266 3 be be AUX cana-1164 266 4 proven prove VERB cana-1164 266 5 in in ADP cana-1164 266 6 the the DET cana-1164 266 7 same same ADJ cana-1164 266 8 way way NOUN cana-1164 266 9 as as ADP cana-1164 266 10 result result NOUN cana-1164 266 11 3.11 3.11 NUM cana-1164 266 12 , , PUNCT cana-1164 266 13 using use VERB cana-1164 266 14 the the DET cana-1164 266 15 definition definition NOUN cana-1164 266 16 of of ADP cana-1164 266 17 ๐ดฮณ ๐ดฮณ PROPN cana-1164 266 18 โˆ’. โˆ’. ADV cana-1164 266 19 result result VERB cana-1164 266 20 3.13 3.13 NUM cana-1164 266 21 . . PUNCT cana-1164 267 1 let let VERB cana-1164 268 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 268 2 + + NOUN cana-1164 268 3 = = SYM cana-1164 269 1 [ [ X cana-1164 269 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 269 3 be be AUX cana-1164 269 4 an an DET cana-1164 269 5 out out ADJ cana-1164 269 6 - - PUNCT cana-1164 269 7 adjacency adjacency NOUN cana-1164 269 8 matrix matrix NOUN cana-1164 269 9 of of ADP cana-1164 269 10 the the DET cana-1164 269 11 digraph digraph ADJ cana-1164 269 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 269 13 , , PUNCT cana-1164 269 14 ๐‘˜ ๐‘˜ NOUN cana-1164 269 15 ) ) PUNCT cana-1164 269 16 . . PUNCT cana-1164 270 1 then then ADV cana-1164 270 2 the the DET cana-1164 270 3 matrix matrix NOUN cana-1164 270 4 ๐ตฮณ ๐ตฮณ NOUN cana-1164 270 5 = = PUNCT cana-1164 271 1 [ [ X cana-1164 271 2 ๐‘๐‘–๐‘— ๐‘๐‘–๐‘— NOUN cana-1164 271 3 ] ] PUNCT cana-1164 271 4 has have VERB cana-1164 271 5 at at ADV cana-1164 271 6 least least ADV cana-1164 271 7 two two NUM cana-1164 271 8 entries entry NOUN cana-1164 271 9 which which PRON cana-1164 271 10 is be AUX cana-1164 271 11 zero zero NUM cana-1164 271 12 , , PUNCT cana-1164 271 13 where where SCONJ cana-1164 271 14 ๐ตฮณ ๐ตฮณ NOUN cana-1164 271 15 = = PUNCT cana-1164 271 16 ๐ดฮณ ๐ดฮณ PROPN cana-1164 271 17 + + PUNCT cana-1164 272 1 + + CCONJ cana-1164 272 2 ( ( PUNCT cana-1164 272 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 272 4 + + PROPN cana-1164 272 5 ) ) SYM cana-1164 272 6 2 2 NUM cana-1164 273 1 + + CCONJ cana-1164 273 2 ( ( PUNCT cana-1164 273 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 273 4 + + PROPN cana-1164 273 5 ) ) PUNCT cana-1164 273 6 3 3 NUM cana-1164 273 7 + + NOUN cana-1164 273 8 โ‹ฏ+ โ‹ฏ+ NOUN cana-1164 273 9 ( ( PUNCT cana-1164 273 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 273 11 + + PROPN cana-1164 273 12 ) ) PUNCT cana-1164 273 13 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-1164 273 14 and and CCONJ cana-1164 273 15 ๐‘› ๐‘› ADJ cana-1164 273 16 > > SYM cana-1164 273 17 1 1 X cana-1164 273 18 . . PUNCT cana-1164 274 1 proof proof NOUN cana-1164 274 2 . . PUNCT cana-1164 275 1 by by ADP cana-1164 275 2 definition definition NOUN cana-1164 275 3 of of ADP cana-1164 275 4 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 275 5 , , PUNCT cana-1164 275 6 ๐‘˜ ๐‘˜ NOUN cana-1164 275 7 ) ) PUNCT cana-1164 275 8 , , PUNCT cana-1164 275 9 it it PRON cana-1164 275 10 is be AUX cana-1164 275 11 clear clear ADJ cana-1164 275 12 that that SCONJ cana-1164 275 13 the the DET cana-1164 275 14 digraph digraph ADJ cana-1164 275 15 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 275 16 , , PUNCT cana-1164 275 17 ๐‘˜ ๐‘˜ NOUN cana-1164 275 18 ) ) PUNCT cana-1164 275 19 is be AUX cana-1164 275 20 disconnected disconnect VERB cana-1164 275 21 for for ADP cana-1164 275 22 ๐‘› ๐‘› PROPN cana-1164 275 23 > > SYM cana-1164 275 24 1 1 NUM cana-1164 275 25 . . PUNCT cana-1164 276 1 so so ADV cana-1164 276 2 there there PRON cana-1164 276 3 exists exist VERB cana-1164 276 4 two two NUM cana-1164 276 5 or or CCONJ cana-1164 276 6 more more ADJ cana-1164 276 7 than than ADP cana-1164 276 8 two two NUM cana-1164 276 9 disjoint disjoint ADJ cana-1164 276 10 components component NOUN cana-1164 276 11 of of ADP cana-1164 276 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 276 13 , , PUNCT cana-1164 276 14 ๐‘˜ ๐‘˜ NOUN cana-1164 276 15 ) ) PUNCT cana-1164 276 16 that that PRON cana-1164 276 17 have have VERB cana-1164 276 18 no no DET cana-1164 276 19 directed direct VERB cana-1164 276 20 arcs arcs NOUN cana-1164 276 21 in in ADP cana-1164 276 22 between between ADP cana-1164 276 23 them they PRON cana-1164 276 24 . . PUNCT cana-1164 277 1 let let VERB cana-1164 277 2 there there PRON cana-1164 277 3 be be AUX cana-1164 277 4 such such ADJ cana-1164 277 5 ๐‘  ๐‘  PRON cana-1164 277 6 number number NOUN cana-1164 277 7 of of ADP cana-1164 277 8 components component NOUN cana-1164 277 9 namely namely ADV cana-1164 277 10 ฮณ1 ฮณ1 NOUN cana-1164 277 11 , , PUNCT cana-1164 277 12 ฮณ2 ฮณ2 PROPN cana-1164 277 13 , , PUNCT cana-1164 277 14 โ‹ฏ โ‹ฏ PROPN cana-1164 277 15 , , PUNCT cana-1164 277 16 ฮณ๐‘ . ฮณ๐‘ . NOUN cana-1164 277 17 in in ADP cana-1164 277 18 this this DET cana-1164 277 19 case case NOUN cana-1164 277 20 , , PUNCT cana-1164 277 21 the the DET cana-1164 277 22 out out ADJ cana-1164 277 23 - - PUNCT cana-1164 277 24 adjacency adjacency NOUN cana-1164 277 25 matrix matrix NOUN cana-1164 278 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 278 2 + + X cana-1164 278 3 of of ADP cana-1164 278 4 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 278 5 , , PUNCT cana-1164 278 6 ๐‘˜ ๐‘˜ NOUN cana-1164 278 7 ) ) PUNCT cana-1164 278 8 can can AUX cana-1164 278 9 be be AUX cana-1164 278 10 partitioned partition VERB cana-1164 278 11 into into ADP cana-1164 278 12 block block NOUN cana-1164 278 13 diagonal diagonal ADJ cana-1164 278 14 matrices matrix NOUN cana-1164 278 15 as as ADP cana-1164 278 16 ๐ดฮณ ๐ดฮณ PROPN cana-1164 278 17 + + PROPN cana-1164 278 18 = = SYM cana-1164 278 19 [ [ PUNCT cana-1164 278 20 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 278 21 + + X cana-1164 278 22 0 0 NUM cana-1164 278 23 0 0 NUM cana-1164 278 24 โ‹ฏ โ‹ฏ ADP cana-1164 278 25 0 0 NUM cana-1164 278 26 0 0 NUM cana-1164 278 27 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 279 1 + + X cana-1164 279 2 0 0 NUM cana-1164 279 3 โ‹ฏ โ‹ฏ NOUN cana-1164 279 4 0 0 NUM cana-1164 279 5 โ‹ฎ โ‹ฎ NOUN cana-1164 279 6 โ‹ฎ โ‹ฎ ADJ cana-1164 279 7 โ‹ฎ โ‹ฎ NOUN cana-1164 279 8 โ‹ฑ โ‹ฑ PUNCT cana-1164 279 9 โ‹ฎ โ‹ฎ NOUN cana-1164 279 10 0 0 NUM cana-1164 279 11 0 0 NUM cana-1164 279 12 0 0 NUM cana-1164 279 13 โ‹ฏ โ‹ฏ PROPN cana-1164 279 14 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 279 15 + + X cana-1164 279 16 ] ] PUNCT cana-1164 279 17 where where SCONJ cana-1164 279 18 ๐ดฮณ1 ๐ดฮณ1 NOUN cana-1164 279 19 + + CCONJ cana-1164 279 20 , , PUNCT cana-1164 279 21 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 279 22 + + X cana-1164 279 23 , , PUNCT cana-1164 279 24 โ‹ฏ โ‹ฏ PROPN cana-1164 279 25 , , PUNCT cana-1164 279 26 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 279 27 + + CCONJ cana-1164 279 28 are be AUX cana-1164 279 29 out out ADJ cana-1164 279 30 - - PUNCT cana-1164 279 31 adjacency adjacency NOUN cana-1164 279 32 matrices matrix NOUN cana-1164 279 33 of of ADP cana-1164 279 34 the the DET cana-1164 279 35 components component NOUN cana-1164 279 36 ฮณ1 ฮณ1 PROPN cana-1164 279 37 , , PUNCT cana-1164 279 38 ฮณ2 ฮณ2 PROPN cana-1164 279 39 , , PUNCT cana-1164 279 40 โ‹ฏ โ‹ฏ PROPN cana-1164 279 41 , , PUNCT cana-1164 279 42 ฮณ๐‘  ฮณ๐‘  X cana-1164 279 43 respectively respectively ADV cana-1164 279 44 . . PUNCT cana-1164 280 1 now now ADV cana-1164 280 2 , , PUNCT cana-1164 280 3 let let VERB cana-1164 280 4 us we PRON cana-1164 280 5 consider consider VERB cana-1164 280 6 the the DET cana-1164 280 7 matrix matrix NOUN cana-1164 280 8 , , PUNCT cana-1164 280 9 ๐ตฮณ ๐ตฮณ NOUN cana-1164 281 1 = = PUNCT cana-1164 281 2 ๐ดฮณ ๐ดฮณ PROPN cana-1164 281 3 + + PUNCT cana-1164 282 1 + + CCONJ cana-1164 282 2 ( ( PUNCT cana-1164 282 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 282 4 + + PROPN cana-1164 282 5 ) ) SYM cana-1164 282 6 2 2 NUM cana-1164 283 1 + + CCONJ cana-1164 283 2 ( ( PUNCT cana-1164 283 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 283 4 + + PROPN cana-1164 283 5 ) ) PUNCT cana-1164 283 6 3 3 NUM cana-1164 283 7 + + NOUN cana-1164 283 8 โ‹ฏ+ โ‹ฏ+ NOUN cana-1164 283 9 ( ( PUNCT cana-1164 283 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 283 11 + + PROPN cana-1164 283 12 ) ) PUNCT cana-1164 283 13 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-1164 283 14 . . PUNCT cana-1164 284 1 clearly clearly ADV cana-1164 284 2 , , PUNCT cana-1164 284 3 each each DET cana-1164 284 4 entry entry NOUN cana-1164 284 5 in in ADP cana-1164 284 6 ( ( PUNCT cana-1164 284 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 284 8 + + PROPN cana-1164 284 9 ) ) PUNCT cana-1164 284 10 ๐‘š(1 ๐‘š(1 PROPN cana-1164 284 11 โ‰ค โ‰ค NOUN cana-1164 284 12 ๐‘š ๐‘š ADP cana-1164 284 13 โ‰ค โ‰ค NUM cana-1164 284 14 ๐‘› ๐‘› PRON cana-1164 284 15 โˆ’ โˆ’ PROPN cana-1164 284 16 1 1 NUM cana-1164 284 17 ) ) PUNCT cana-1164 284 18 counts count VERB cana-1164 284 19 the the DET cana-1164 284 20 number number NOUN cana-1164 284 21 of of ADP cana-1164 284 22 directed direct VERB cana-1164 284 23 walks walk NOUN cana-1164 284 24 of of ADP cana-1164 284 25 length length NOUN cana-1164 284 26 ๐‘š ๐‘š NOUN cana-1164 284 27 from from ADP cana-1164 284 28 vertex vertex NOUN cana-1164 284 29 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 284 30 to to ADP cana-1164 284 31 vertex vertex NOUN cana-1164 284 32 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1164 284 33 . . PUNCT cana-1164 285 1 as as SCONJ cana-1164 285 2 the the DET cana-1164 285 3 digraph digraph ADJ cana-1164 285 4 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 285 5 , , PUNCT cana-1164 285 6 ๐‘˜ ๐‘˜ NOUN cana-1164 285 7 ) ) PUNCT cana-1164 285 8 is be AUX cana-1164 285 9 disconnected disconnect VERB cana-1164 285 10 , , PUNCT cana-1164 285 11 so so CCONJ cana-1164 285 12 a a DET cana-1164 285 13 directed direct VERB cana-1164 285 14 walk walk NOUN cana-1164 285 15 from from ADP cana-1164 285 16 one one NUM cana-1164 285 17 component component NOUN cana-1164 285 18 to to ADP cana-1164 285 19 another another DET cana-1164 285 20 component component NOUN cana-1164 285 21 is be AUX cana-1164 285 22 not not PART cana-1164 285 23 possible possible ADJ cana-1164 285 24 , , PUNCT cana-1164 285 25 and and CCONJ cana-1164 285 26 hence hence ADV cana-1164 285 27 the the DET cana-1164 285 28 entry entry NOUN cana-1164 285 29 in in ADP cana-1164 285 30 the the DET cana-1164 285 31 matrix matrix NOUN cana-1164 285 32 ( ( PUNCT cana-1164 285 33 ๐ดฮณ ๐ดฮณ PROPN cana-1164 285 34 + + PROPN cana-1164 285 35 ) ) PUNCT cana-1164 285 36 ๐‘š ๐‘š AUX cana-1164 285 37 corresponding correspond VERB cana-1164 285 38 to to ADP cana-1164 285 39 those those DET cana-1164 285 40 directed direct VERB cana-1164 285 41 walks walk NOUN cana-1164 285 42 will will AUX cana-1164 285 43 be be AUX cana-1164 285 44 zero zero NUM cana-1164 285 45 . . PUNCT cana-1164 286 1 thus thus ADV cana-1164 286 2 the the DET cana-1164 286 3 only only ADJ cana-1164 286 4 non non ADJ cana-1164 286 5 - - ADJ cana-1164 286 6 zero zero NUM cana-1164 286 7 entries entry NOUN cana-1164 286 8 in in ADP cana-1164 286 9 ๐ตฮณ ๐ตฮณ PROPN cana-1164 286 10 will will AUX cana-1164 286 11 come come VERB cana-1164 286 12 from from ADP cana-1164 286 13 the the DET cana-1164 286 14 individual individual ADJ cana-1164 286 15 components component NOUN cana-1164 286 16 ฮณ1 ฮณ1 PROPN cana-1164 286 17 , , PUNCT cana-1164 286 18 ฮณ2 ฮณ2 PROPN cana-1164 286 19 , , PUNCT cana-1164 286 20 โ‹ฏ โ‹ฏ PROPN cana-1164 286 21 , , PUNCT cana-1164 286 22 ฮณ๐‘ . ฮณ๐‘ . NOUN cana-1164 286 23 moreover moreover ADV cana-1164 286 24 , , PUNCT cana-1164 286 25 each each DET cana-1164 286 26 out out ADJ cana-1164 286 27 - - PUNCT cana-1164 286 28 adjacency adjacency NOUN cana-1164 286 29 matrix matrix NOUN cana-1164 287 1 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 287 2 + + PUNCT cana-1164 287 3 corresponding correspond VERB cana-1164 287 4 to to ADP cana-1164 287 5 components component NOUN cana-1164 287 6 ฮณ๐‘–(1 ฮณ๐‘–(1 PROPN cana-1164 287 7 โ‰ค โ‰ค NUM cana-1164 287 8 ๐‘– ๐‘– SYM cana-1164 287 9 โ‰ค โ‰ค NUM cana-1164 287 10 ๐‘  ๐‘  NOUN cana-1164 287 11 ) ) PUNCT cana-1164 287 12 is be AUX cana-1164 287 13 a a DET cana-1164 287 14 non non ADJ cana-1164 287 15 - - ADJ cana-1164 287 16 zero zero NUM cana-1164 287 17 square square ADJ cana-1164 287 18 matrix matrix NOUN cana-1164 287 19 . . PUNCT cana-1164 288 1 so so ADV cana-1164 288 2 , , PUNCT cana-1164 288 3 the the DET cana-1164 288 4 submatrices submatrice NOUN cana-1164 288 5 in in ADP cana-1164 288 6 the the DET cana-1164 288 7 diagonal diagonal ADJ cana-1164 288 8 blocks block NOUN cana-1164 288 9 of of ADP cana-1164 288 10 ๐ตฮณ ๐ตฮณ NOUN cana-1164 288 11 will will AUX cana-1164 288 12 be be AUX cana-1164 288 13 non non ADJ cana-1164 288 14 - - ADJ cana-1164 288 15 zero zero NUM cana-1164 288 16 matrices matrix NOUN cana-1164 288 17 but but CCONJ cana-1164 288 18 nondiagonal nondiagonal ADJ cana-1164 288 19 blocks block NOUN cana-1164 288 20 will will AUX cana-1164 288 21 be be AUX cana-1164 288 22 zero zero NUM cana-1164 288 23 because because SCONJ cana-1164 288 24 there there PRON cana-1164 288 25 are be VERB cana-1164 288 26 no no DET cana-1164 288 27 arcs arcs NOUN cana-1164 288 28 between between ADP cana-1164 288 29 the the DET cana-1164 288 30 components component NOUN cana-1164 288 31 . . PUNCT cana-1164 289 1 hence hence ADV cana-1164 289 2 , , PUNCT cana-1164 289 3 at at ADV cana-1164 289 4 least least ADV cana-1164 289 5 two two NUM cana-1164 289 6 entries entry NOUN cana-1164 289 7 in in ADP cana-1164 289 8 the the DET cana-1164 289 9 matrix matrix NOUN cana-1164 289 10 ๐ตฮณ ๐ตฮณ NOUN cana-1164 289 11 will will AUX cana-1164 289 12 be be AUX cana-1164 289 13 zero zero NUM cana-1164 289 14 . . PUNCT cana-1164 290 1 result result VERB cana-1164 290 2 3.14 3.14 NUM cana-1164 290 3 . . PUNCT cana-1164 291 1 let let VERB cana-1164 292 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 292 2 โˆ’ โˆ’ PROPN cana-1164 293 1 = = PUNCT cana-1164 294 1 [ [ X cana-1164 294 2 ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› ๐‘Ž๐‘–๐‘—]๐‘›ร—๐‘› NOUN cana-1164 294 3 be be AUX cana-1164 294 4 the the DET cana-1164 294 5 in in ADP cana-1164 294 6 - - PUNCT cana-1164 294 7 adjacency adjacency NOUN cana-1164 294 8 matrix matrix NOUN cana-1164 294 9 of of ADP cana-1164 294 10 the the DET cana-1164 294 11 digraph digraph ADJ cana-1164 294 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 294 13 , , PUNCT cana-1164 294 14 ๐‘˜ ๐‘˜ NOUN cana-1164 294 15 ) ) PUNCT cana-1164 294 16 . . PUNCT cana-1164 295 1 then then ADV cana-1164 295 2 the the DET cana-1164 295 3 matrix matrix NOUN cana-1164 295 4 ๐ถฮณ ๐ถฮณ PROPN cana-1164 295 5 = = PUNCT cana-1164 296 1 [ [ X cana-1164 296 2 ๐‘๐‘–๐‘— ๐‘๐‘–๐‘— X cana-1164 296 3 ] ] AUX cana-1164 296 4 has have VERB cana-1164 296 5 at at ADV cana-1164 296 6 least least ADV cana-1164 296 7 two two NUM cana-1164 296 8 entries entry NOUN cana-1164 296 9 which which PRON cana-1164 296 10 is be AUX cana-1164 296 11 zero zero NUM cana-1164 296 12 , , PUNCT cana-1164 296 13 where where SCONJ cana-1164 296 14 ๐ถฮณ ๐ถฮณ PROPN cana-1164 296 15 = = SYM cana-1164 297 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 297 2 โˆ’ โˆ’ PROPN cana-1164 298 1 + + CCONJ cana-1164 298 2 ( ( PUNCT cana-1164 298 3 ๐ดฮณ ๐ดฮณ PROPN cana-1164 298 4 โˆ’)2 โˆ’)2 PROPN cana-1164 298 5 + + CCONJ cana-1164 298 6 ( ( PUNCT cana-1164 298 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 298 8 โˆ’)3 โˆ’)3 PROPN cana-1164 298 9 + + NOUN cana-1164 298 10 โ‹ฏ+ โ‹ฏ+ NOUN cana-1164 298 11 ( ( PUNCT cana-1164 298 12 ๐ดฮณ ๐ดฮณ PROPN cana-1164 298 13 โˆ’)๐‘›โˆ’1 โˆ’)๐‘›โˆ’1 NOUN cana-1164 298 14 and and CCONJ cana-1164 298 15 ๐‘› ๐‘› ADJ cana-1164 298 16 > > SYM cana-1164 298 17 1 1 X cana-1164 298 18 . . PUNCT cana-1164 299 1 proof proof NOUN cana-1164 299 2 . . PUNCT cana-1164 300 1 it it PRON cana-1164 300 2 can can AUX cana-1164 300 3 be be AUX cana-1164 300 4 proven prove VERB cana-1164 300 5 in in ADP cana-1164 300 6 the the DET cana-1164 300 7 same same ADJ cana-1164 300 8 way way NOUN cana-1164 300 9 as as ADP cana-1164 300 10 result result NOUN cana-1164 300 11 3.13 3.13 NUM cana-1164 300 12 . . PUNCT cana-1164 301 1 communications communication NOUN cana-1164 301 2 on on ADP cana-1164 301 3 applied apply VERB cana-1164 301 4 nonlinear nonlinear ADJ cana-1164 301 5 analysis analysis NOUN cana-1164 301 6 issn issn NOUN cana-1164 301 7 : : PUNCT cana-1164 301 8 1074 1074 NUM cana-1164 301 9 - - PUNCT cana-1164 301 10 133x 133x NUM cana-1164 301 11 vol vol NOUN cana-1164 301 12 31 31 NUM cana-1164 301 13 no no NOUN cana-1164 301 14 . . PUNCT cana-1164 302 1 6s 6s NUM cana-1164 302 2 ( ( PUNCT cana-1164 302 3 2024 2024 NUM cana-1164 302 4 ) ) PUNCT cana-1164 302 5 107 107 NUM cana-1164 302 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 302 7 lemma lemma PROPN cana-1164 302 8 3.1 3.1 NUM cana-1164 302 9 . . PUNCT cana-1164 303 1 the the DET cana-1164 303 2 digraph digraph ADJ cana-1164 303 3 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 303 4 , , PUNCT cana-1164 303 5 ๐‘˜ ๐‘˜ NOUN cana-1164 303 6 ) ) PUNCT cana-1164 303 7 has have VERB cana-1164 303 8 at at ADV cana-1164 303 9 least least ADV cana-1164 303 10 one one NUM cana-1164 303 11 vertex vertex NOUN cana-1164 303 12 of of ADP cana-1164 303 13 in in ADP cana-1164 303 14 - - PUNCT cana-1164 303 15 degree degree NOUN cana-1164 303 16 0 0 NUM cana-1164 303 17 iff iff PROPN cana-1164 303 18 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NOUN cana-1164 303 19 ) ) PUNCT cana-1164 303 20 or or CCONJ cana-1164 303 21 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 303 22 , , PUNCT cana-1164 303 23 for for ADP cana-1164 303 24 some some DET cana-1164 303 25 prime prime ADJ cana-1164 303 26 ๐‘. ๐‘. NOUN cana-1164 303 27 proof proof NOUN cana-1164 303 28 . . PUNCT cana-1164 304 1 let let VERB cana-1164 304 2 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 304 3 , , PUNCT cana-1164 304 4 ๐‘˜ ๐‘˜ NOUN cana-1164 304 5 ) ) PUNCT cana-1164 304 6 have have VERB cana-1164 304 7 at at ADV cana-1164 304 8 least least ADV cana-1164 304 9 one one NUM cana-1164 304 10 vertex vertex NOUN cana-1164 304 11 of of ADP cana-1164 304 12 in in ADP cana-1164 304 13 - - PUNCT cana-1164 304 14 degree degree NOUN cana-1164 304 15 0 0 NUM cana-1164 304 16 . . PUNCT cana-1164 304 17 to to PART cana-1164 304 18 show show VERB cana-1164 304 19 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 304 20 ) ) PUNCT cana-1164 304 21 or or CCONJ cana-1164 304 22 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 304 23 , , PUNCT cana-1164 304 24 for for ADP cana-1164 304 25 some some DET cana-1164 304 26 prime prime ADJ cana-1164 304 27 ๐‘. ๐‘. NOUN cana-1164 304 28 let let VERB cana-1164 304 29 ๐‘2 ๐‘2 PROPN cana-1164 304 30 โˆค โˆค SYM cana-1164 304 31 ๐‘› ๐‘› PROPN cana-1164 304 32 , , PUNCT cana-1164 304 33 for for ADP cana-1164 304 34 any any DET cana-1164 304 35 prime prime ADJ cana-1164 304 36 ๐‘. ๐‘. NOUN cana-1164 304 37 in in ADP cana-1164 304 38 this this DET cana-1164 304 39 case case NOUN cana-1164 304 40 , , PUNCT cana-1164 304 41 the the DET cana-1164 304 42 digraph digraph NOUN cana-1164 304 43 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 304 44 , , PUNCT cana-1164 304 45 ๐‘˜ ๐‘˜ PRON cana-1164 304 46 ) ) PUNCT cana-1164 304 47 is be AUX cana-1164 304 48 semi semi ADJ cana-1164 304 49 - - ADJ cana-1164 304 50 regular regular ADJ cana-1164 304 51 and and CCONJ cana-1164 304 52 so so ADV cana-1164 304 53 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 304 54 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 304 55 ) ) PUNCT cana-1164 305 1 = = SYM cana-1164 305 2 0 0 NUM cana-1164 305 3 or or CCONJ cana-1164 305 4 ๐‘˜๐œ”(๐‘› ๐‘˜๐œ”(๐‘› NOUN cana-1164 305 5 ) ) PUNCT cana-1164 305 6 , , PUNCT cana-1164 305 7 for for ADP cana-1164 305 8 ๐‘ฃ ๐‘ฃ DET cana-1164 305 9 โˆˆ โˆˆ PROPN cana-1164 305 10 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 305 11 , , PUNCT cana-1164 305 12 ๐‘˜ ๐‘˜ NOUN cana-1164 305 13 ) ) PUNCT cana-1164 305 14 , , PUNCT cana-1164 305 15 where where SCONJ cana-1164 305 16 ๐œ”(๐‘› ๐œ”(๐‘› X cana-1164 305 17 ) ) PUNCT cana-1164 306 1 = = PRON cana-1164 306 2 { { PUNCT cana-1164 306 3 ๐œ”0(๐‘› ๐œ”0(๐‘› NOUN cana-1164 306 4 ) ) PUNCT cana-1164 306 5 + + NUM cana-1164 306 6 1 1 NUM cana-1164 306 7 , , PUNCT cana-1164 306 8 if if SCONJ cana-1164 306 9 k2|n k2|n PROPN cana-1164 306 10 ๐œ”0(๐‘› ๐œ”0(๐‘› PROPN cana-1164 306 11 ) ) PUNCT cana-1164 306 12 , , PUNCT cana-1164 306 13 if if SCONJ cana-1164 306 14 k2 k2 PROPN cana-1164 306 15 โˆค โˆค PROPN cana-1164 306 16 n n PROPN cana-1164 306 17 and and CCONJ cana-1164 306 18 ๐œ”๐‘œ(๐‘› ๐œ”๐‘œ(๐‘› PUNCT cana-1164 306 19 ) ) PUNCT cana-1164 306 20 is be AUX cana-1164 306 21 the the DET cana-1164 306 22 number number NOUN cana-1164 306 23 of of ADP cana-1164 306 24 distinct distinct ADJ cana-1164 306 25 primes prime NOUN cana-1164 306 26 dividing divide VERB cana-1164 306 27 ๐‘› ๐‘› PRON cana-1164 306 28 which which PRON cana-1164 306 29 are be AUX cana-1164 306 30 congruent congruent ADJ cana-1164 306 31 to to ADP cana-1164 306 32 1(๐‘š๐‘œ๐‘‘ 1(๐‘š๐‘œ๐‘‘ NUM cana-1164 306 33 ๐‘˜ ๐‘˜ NOUN cana-1164 306 34 ) ) PUNCT cana-1164 306 35 . . PUNCT cana-1164 307 1 as as SCONJ cana-1164 307 2 the the DET cana-1164 307 3 set set NOUN cana-1164 307 4 of of ADP cana-1164 307 5 residues residue NOUN cana-1164 307 6 which which PRON cana-1164 307 7 are be AUX cana-1164 307 8 co co ADJ cana-1164 307 9 - - ADJ cana-1164 307 10 prime prime ADJ cana-1164 307 11 to to ADP cana-1164 307 12 ๐‘› ๐‘› PROPN cana-1164 307 13 , , PUNCT cana-1164 307 14 forms form VERB cana-1164 307 15 a a DET cana-1164 307 16 group group NOUN cana-1164 307 17 under under ADP cana-1164 307 18 multiplication multiplication NOUN cana-1164 307 19 modulo modulo NOUN cana-1164 307 20 ๐‘› ๐‘› ADP cana-1164 307 21 of of ADP cana-1164 307 22 order order NOUN cana-1164 307 23 ๐œ™(๐‘› ๐œ™(๐‘› ADJ cana-1164 307 24 ) ) PUNCT cana-1164 307 25 , , PUNCT cana-1164 307 26 so so CCONJ cana-1164 307 27 the the DET cana-1164 307 28 set set NOUN cana-1164 307 29 of of ADP cana-1164 307 30 vertices vertex NOUN cana-1164 307 31 of of ADP cana-1164 307 32 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 307 33 , , PUNCT cana-1164 307 34 ๐‘˜ ๐‘˜ NOUN cana-1164 307 35 ) ) PUNCT cana-1164 307 36 forms form VERB cana-1164 307 37 a a DET cana-1164 307 38 group group NOUN cana-1164 307 39 under under ADP cana-1164 307 40 multiplication multiplication NOUN cana-1164 307 41 modulo modulo NOUN cana-1164 307 42 ๐‘› ๐‘› ADP cana-1164 307 43 of of ADP cana-1164 307 44 order order NOUN cana-1164 307 45 ๐œ™(๐‘› ๐œ™(๐‘› ADJ cana-1164 307 46 ) ) PUNCT cana-1164 307 47 . . PUNCT cana-1164 308 1 let let VERB cana-1164 308 2 ๐‘ฃ ๐‘ฃ PRON cana-1164 308 3 โˆˆ โˆˆ PROPN cana-1164 308 4 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 308 5 , , PUNCT cana-1164 308 6 ๐‘˜ ๐‘˜ NOUN cana-1164 308 7 ) ) PUNCT cana-1164 308 8 such such ADJ cana-1164 308 9 that that SCONJ cana-1164 308 10 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 308 11 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 308 12 ) ) PUNCT cana-1164 308 13 = = SYM cana-1164 308 14 ๐‘˜๐œ”(๐‘› ๐‘˜๐œ”(๐‘› X cana-1164 308 15 ) ) PUNCT cana-1164 308 16 and and CCONJ cana-1164 308 17 let let VERB cana-1164 308 18 h h NOUN cana-1164 308 19 = = PRON cana-1164 308 20 { { PUNCT cana-1164 308 21 0 0 NUM cana-1164 308 22 โ‰ค โ‰ค NUM cana-1164 308 23 ๐‘š ๐‘š NOUN cana-1164 308 24 โ‰ค โ‰ค NUM cana-1164 309 1 ๐‘› ๐‘› PRON cana-1164 309 2 โˆ’ โˆ’ PROPN cana-1164 309 3 1 1 NUM cana-1164 309 4 |(๐‘š |(๐‘š PROPN cana-1164 309 5 , , PUNCT cana-1164 309 6 ๐‘› ๐‘› NOUN cana-1164 309 7 ) ) PUNCT cana-1164 309 8 = = SYM cana-1164 309 9 1,๐‘š๐‘˜ 1,๐‘š๐‘˜ NUM cana-1164 309 10 โ‰ก โ‰ก PROPN cana-1164 309 11 1(๐‘š๐‘œ๐‘‘ 1(๐‘š๐‘œ๐‘‘ PROPN cana-1164 309 12 ๐‘› ๐‘› NOUN cana-1164 309 13 ) ) PUNCT cana-1164 309 14 } } PUNCT cana-1164 309 15 . . PUNCT cana-1164 310 1 then then ADV cana-1164 310 2 h h PROPN cana-1164 310 3 is be AUX cana-1164 310 4 a a DET cana-1164 310 5 subgroup subgroup NOUN cana-1164 310 6 of of ADP cana-1164 310 7 the the DET cana-1164 310 8 group group NOUN cana-1164 310 9 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 310 10 , , PUNCT cana-1164 310 11 ๐‘˜ ๐‘˜ NOUN cana-1164 310 12 ) ) PUNCT cana-1164 310 13 of of ADP cana-1164 310 14 order order NOUN cana-1164 310 15 ๐‘˜๐œ”(๐‘› ๐‘˜๐œ”(๐‘› NOUN cana-1164 310 16 ) ) PUNCT cana-1164 310 17 and and CCONJ cana-1164 310 18 hence hence ADV cana-1164 310 19 ๐‘˜๐œ”(๐‘›)|๐œ™(๐‘› ๐‘˜๐œ”(๐‘›)|๐œ™(๐‘› VERB cana-1164 310 20 ) ) PUNCT cana-1164 310 21 which which PRON cana-1164 310 22 implies imply VERB cana-1164 310 23 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NOUN cana-1164 310 24 ) ) PUNCT cana-1164 310 25 . . PUNCT cana-1164 311 1 now now ADV cana-1164 311 2 , , PUNCT cana-1164 311 3 let let VERB cana-1164 311 4 ๐‘˜ ๐‘˜ PRON cana-1164 311 5 โˆค โˆค NOUN cana-1164 311 6 ๐œ™(๐‘› ๐œ™(๐‘› ADJ cana-1164 311 7 ) ) PUNCT cana-1164 311 8 . . PUNCT cana-1164 312 1 to to PART cana-1164 312 2 show show VERB cana-1164 312 3 ๐‘2|๐‘› ๐‘2|๐‘› PROPN cana-1164 312 4 , , PUNCT cana-1164 312 5 for for ADP cana-1164 312 6 some some DET cana-1164 312 7 prime prime ADJ cana-1164 312 8 ๐‘. ๐‘. NOUN cana-1164 312 9 if if SCONJ cana-1164 312 10 possible possible ADJ cana-1164 312 11 , , PUNCT cana-1164 312 12 let let VERB cana-1164 312 13 ๐‘2 ๐‘2 PROPN cana-1164 312 14 โˆค โˆค PROPN cana-1164 312 15 ๐‘› ๐‘› PROPN cana-1164 312 16 for for ADP cana-1164 312 17 any any DET cana-1164 312 18 prime prime ADJ cana-1164 312 19 ๐‘ ๐‘ NOUN cana-1164 312 20 , , PUNCT cana-1164 312 21 then then ADV cana-1164 312 22 ๐‘› ๐‘› PROPN cana-1164 312 23 is be AUX cana-1164 312 24 a a DET cana-1164 312 25 square square ADJ cana-1164 312 26 - - PUNCT cana-1164 312 27 free free ADJ cana-1164 312 28 integer integer NOUN cana-1164 312 29 . . PUNCT cana-1164 313 1 now now ADV cana-1164 313 2 , , PUNCT cana-1164 313 3 ๐‘› ๐‘› PROPN cana-1164 313 4 is be AUX cana-1164 313 5 square square ADV cana-1164 313 6 - - PUNCT cana-1164 313 7 free free ADJ cana-1164 313 8 and and CCONJ cana-1164 313 9 ๐‘˜ ๐‘˜ PRON cana-1164 313 10 โˆค โˆค NOUN cana-1164 313 11 ๐œ™(๐‘› ๐œ™(๐‘› ADJ cana-1164 313 12 ) ) PUNCT cana-1164 313 13 so so ADV cana-1164 313 14 in in ADP cana-1164 313 15 this this DET cana-1164 313 16 case case NOUN cana-1164 313 17 the the DET cana-1164 313 18 digraph digraph ADJ cana-1164 313 19 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 313 20 , , PUNCT cana-1164 313 21 ๐‘˜ ๐‘˜ NOUN cana-1164 313 22 ) ) PUNCT cana-1164 313 23 is be AUX cana-1164 313 24 cyclic cyclic ADJ cana-1164 313 25 . . PUNCT cana-1164 314 1 by by ADP cana-1164 314 2 definition definition NOUN cana-1164 314 3 , , PUNCT cana-1164 314 4 a a DET cana-1164 314 5 digraph digraph NOUN cana-1164 314 6 is be AUX cana-1164 314 7 cyclic cyclic ADJ cana-1164 314 8 if if SCONJ cana-1164 314 9 all all PRON cana-1164 314 10 of of ADP cana-1164 314 11 its its PRON cana-1164 314 12 components component NOUN cana-1164 314 13 are be AUX cana-1164 314 14 cyclic cyclic ADJ cana-1164 314 15 . . PUNCT cana-1164 315 1 moreover moreover ADV cana-1164 315 2 , , PUNCT cana-1164 315 3 if if SCONJ cana-1164 315 4 all all DET cana-1164 315 5 the the DET cana-1164 315 6 components component NOUN cana-1164 315 7 of of ADP cana-1164 315 8 the the DET cana-1164 315 9 digraph digraph ADJ cana-1164 315 10 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 315 11 , , PUNCT cana-1164 315 12 ๐‘˜ ๐‘˜ NOUN cana-1164 315 13 ) ) PUNCT cana-1164 315 14 are be AUX cana-1164 315 15 cycles cycle NOUN cana-1164 315 16 , , PUNCT cana-1164 315 17 then then ADV cana-1164 315 18 the the DET cana-1164 315 19 digraph digraph ADJ cana-1164 315 20 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 315 21 , , PUNCT cana-1164 315 22 ๐‘˜ ๐‘˜ NOUN cana-1164 315 23 ) ) PUNCT cana-1164 315 24 is be AUX cana-1164 315 25 regular regular ADJ cana-1164 315 26 and and CCONJ cana-1164 315 27 so so ADV cana-1164 315 28 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 315 29 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 315 30 ) ) PUNCT cana-1164 315 31 = = SYM cana-1164 316 1 1 1 NUM cana-1164 316 2 , , PUNCT cana-1164 316 3 โˆ€๐‘ฃ โˆ€๐‘ฃ PROPN cana-1164 316 4 โˆˆ โˆˆ PROPN cana-1164 316 5 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 316 6 , , PUNCT cana-1164 316 7 ๐‘˜ ๐‘˜ NOUN cana-1164 316 8 ) ) PUNCT cana-1164 316 9 , , PUNCT cana-1164 316 10 which which PRON cana-1164 316 11 contradicts contradict VERB cana-1164 316 12 the the DET cana-1164 316 13 fact fact NOUN cana-1164 316 14 that that SCONJ cana-1164 316 15 there there PRON cana-1164 316 16 exists exist VERB cana-1164 316 17 at at ADV cana-1164 316 18 least least ADV cana-1164 316 19 one one NUM cana-1164 316 20 vertex vertex NOUN cana-1164 316 21 of of ADP cana-1164 316 22 in in ADP cana-1164 316 23 - - PUNCT cana-1164 316 24 degree degree NOUN cana-1164 316 25 0 0 NUM cana-1164 316 26 . . PUNCT cana-1164 317 1 this this DET cana-1164 317 2 contradiction contradiction NOUN cana-1164 317 3 implies imply VERB cana-1164 317 4 that that SCONJ cana-1164 317 5 ๐‘2|๐‘› ๐‘2|๐‘› PROPN cana-1164 317 6 , , PUNCT cana-1164 317 7 for for ADP cana-1164 317 8 some some DET cana-1164 317 9 prime prime ADJ cana-1164 317 10 ๐‘. ๐‘. NOUN cana-1164 317 11 conversely conversely ADV cana-1164 317 12 , , PUNCT cana-1164 317 13 let let VERB cana-1164 317 14 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 317 15 ) ) PUNCT cana-1164 317 16 or or CCONJ cana-1164 317 17 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 317 18 , , PUNCT cana-1164 317 19 for for ADP cana-1164 317 20 some some DET cana-1164 317 21 prime prime ADJ cana-1164 317 22 ๐‘. ๐‘. NOUN cana-1164 317 23 to to PART cana-1164 317 24 show show VERB cana-1164 317 25 the the DET cana-1164 317 26 digraph digraph ADJ cana-1164 317 27 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 317 28 , , PUNCT cana-1164 317 29 ๐‘˜ ๐‘˜ NOUN cana-1164 317 30 ) ) PUNCT cana-1164 317 31 has have VERB cana-1164 317 32 at at ADV cana-1164 317 33 least least ADV cana-1164 317 34 one one NUM cana-1164 317 35 vertex vertex NOUN cana-1164 317 36 of of ADP cana-1164 317 37 in in ADP cana-1164 317 38 - - PUNCT cana-1164 317 39 degree degree NOUN cana-1164 317 40 0 0 NUM cana-1164 317 41 . . PUNCT cana-1164 318 1 if if SCONJ cana-1164 318 2 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 318 3 ) ) PUNCT cana-1164 318 4 , , PUNCT cana-1164 318 5 then then ADV cana-1164 318 6 the the DET cana-1164 318 7 digraph digraph NOUN cana-1164 318 8 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 318 9 , , PUNCT cana-1164 318 10 ๐‘˜ ๐‘˜ NOUN cana-1164 318 11 ) ) PUNCT cana-1164 318 12 is be AUX cana-1164 318 13 a a DET cana-1164 318 14 semi semi ADJ cana-1164 318 15 - - ADJ cana-1164 318 16 regular regular ADJ cana-1164 318 17 digraph digraph NOUN cana-1164 318 18 and and CCONJ cana-1164 318 19 hence hence ADV cana-1164 318 20 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 318 21 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 318 22 ) ) PUNCT cana-1164 319 1 = = SYM cana-1164 319 2 0 0 NUM cana-1164 319 3 or or CCONJ cana-1164 319 4 ๐‘˜๐œ”(๐‘› ๐‘˜๐œ”(๐‘› NOUN cana-1164 319 5 ) ) PUNCT cana-1164 319 6 , , PUNCT cana-1164 319 7 for for ADP cana-1164 319 8 ๐‘ฃ ๐‘ฃ DET cana-1164 319 9 โˆˆ โˆˆ PROPN cana-1164 319 10 ฮณ1(๐‘› ฮณ1(๐‘› PROPN cana-1164 319 11 , , PUNCT cana-1164 319 12 ๐‘˜ ๐‘˜ NOUN cana-1164 319 13 ) ) PUNCT cana-1164 319 14 . . PUNCT cana-1164 320 1 thus thus ADV cana-1164 320 2 , , PUNCT cana-1164 320 3 there there PRON cana-1164 320 4 exists exist VERB cana-1164 320 5 at at ADV cana-1164 320 6 least least ADV cana-1164 320 7 one one NUM cana-1164 320 8 vertex vertex NOUN cana-1164 320 9 ๐‘ฃ ๐‘ฃ ADP cana-1164 320 10 such such ADJ cana-1164 320 11 that that SCONJ cana-1164 320 12 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 320 13 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 320 14 ) ) PUNCT cana-1164 320 15 = = PUNCT cana-1164 321 1 0 0 X cana-1164 321 2 . . PUNCT cana-1164 322 1 if if SCONJ cana-1164 322 2 ๐‘2|๐‘› ๐‘2|๐‘› PROPN cana-1164 322 3 , , PUNCT cana-1164 322 4 for for ADP cana-1164 322 5 some some DET cana-1164 322 6 prime prime PROPN cana-1164 322 7 ๐‘ ๐‘ NOUN cana-1164 322 8 , , PUNCT cana-1164 322 9 then then ADV cana-1164 322 10 some some PRON cana-1164 322 11 ( ( PUNCT cana-1164 322 12 or or CCONJ cana-1164 322 13 all all PRON cana-1164 322 14 ) ) PUNCT cana-1164 322 15 vertices vertex NOUN cana-1164 322 16 of of ADP cana-1164 322 17 the the DET cana-1164 322 18 digraph digraph NOUN cana-1164 322 19 ฮณ2(๐‘› ฮณ2(๐‘› PROPN cana-1164 322 20 , , PUNCT cana-1164 322 21 ๐‘˜ ๐‘˜ NOUN cana-1164 322 22 ) ) PUNCT cana-1164 322 23 forms form VERB cana-1164 322 24 a a DET cana-1164 322 25 rooted rooted ADJ cana-1164 322 26 in in ADP cana-1164 322 27 - - PUNCT cana-1164 322 28 tree tree NOUN cana-1164 322 29 with with ADP cana-1164 322 30 root root NOUN cana-1164 322 31 0 0 PUNCT cana-1164 323 1 and and CCONJ cana-1164 323 2 therefore therefore ADV cana-1164 323 3 there there PRON cana-1164 323 4 exists exist VERB cana-1164 323 5 at at ADP cana-1164 323 6 least least ADV cana-1164 323 7 one one NUM cana-1164 323 8 leaf leaf NOUN cana-1164 323 9 ๐‘ฃ ๐‘ฃ NOUN cana-1164 323 10 in in ADP cana-1164 323 11 this this PRON cana-1164 323 12 rooted root VERB cana-1164 323 13 in in ADP cana-1164 323 14 - - PUNCT cana-1164 323 15 tree tree NOUN cana-1164 323 16 such such ADJ cana-1164 323 17 that that SCONJ cana-1164 323 18 ๐‘‘ฮณ ๐‘‘ฮณ ADP cana-1164 323 19 โˆ’(๐‘ฃ โˆ’(๐‘ฃ NOUN cana-1164 323 20 ) ) PUNCT cana-1164 323 21 = = SYM cana-1164 324 1 0 0 X cana-1164 324 2 . . PUNCT cana-1164 325 1 lemma lemma PROPN cana-1164 325 2 3.2 3.2 NUM cana-1164 325 3 . . PUNCT cana-1164 326 1 the the DET cana-1164 326 2 out out ADJ cana-1164 326 3 - - PUNCT cana-1164 326 4 adjacency adjacency NOUN cana-1164 326 5 matrix matrix NOUN cana-1164 326 6 ๐ดฮณ ๐ดฮณ PROPN cana-1164 327 1 + + X cana-1164 327 2 of of ADP cana-1164 327 3 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 327 4 , , PUNCT cana-1164 327 5 ๐‘˜ ๐‘˜ NOUN cana-1164 327 6 ) ) PUNCT cana-1164 327 7 contains contain VERB cana-1164 327 8 at at ADV cana-1164 327 9 least least ADV cana-1164 327 10 one one NUM cana-1164 327 11 block block NOUN cana-1164 327 12 diagonal diagonal ADJ cana-1164 327 13 submatrix submatrix NOUN cana-1164 327 14 whose whose DET cana-1164 327 15 determinant determinant ADJ cana-1164 327 16 is be AUX cana-1164 327 17 ๐‘ง๐‘’๐‘Ÿ๐‘œ ๐‘ง๐‘’๐‘Ÿ๐‘œ ADJ cana-1164 327 18 if if SCONJ cana-1164 327 19 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 327 20 ) ) PUNCT cana-1164 327 21 or or CCONJ cana-1164 327 22 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 327 23 , , PUNCT cana-1164 327 24 for for ADP cana-1164 327 25 some some DET cana-1164 327 26 prime prime ADJ cana-1164 327 27 ๐‘. ๐‘. NOUN cana-1164 327 28 proof proof NOUN cana-1164 327 29 . . PUNCT cana-1164 328 1 let let VERB cana-1164 328 2 us we PRON cana-1164 328 3 consider consider VERB cana-1164 328 4 the the DET cana-1164 328 5 digraph digraph ADJ cana-1164 328 6 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 328 7 , , PUNCT cana-1164 328 8 ๐‘˜ ๐‘˜ NOUN cana-1164 328 9 ) ) PUNCT cana-1164 328 10 with with ADP cana-1164 328 11 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 328 12 ) ) PUNCT cana-1164 328 13 or or CCONJ cana-1164 328 14 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 328 15 , , PUNCT cana-1164 328 16 for for SCONJ cana-1164 328 17 some some DET cana-1164 328 18 prime prime ADJ cana-1164 328 19 ๐‘. ๐‘. NOUN cana-1164 328 20 let let VERB cana-1164 328 21 ฮณ1 ฮณ1 NOUN cana-1164 328 22 , , PUNCT cana-1164 328 23 ฮณ2 ฮณ2 PROPN cana-1164 328 24 , , PUNCT cana-1164 328 25 โ‹ฏ โ‹ฏ PROPN cana-1164 328 26 , , PUNCT cana-1164 328 27 ฮณ๐‘  ฮณ๐‘  PRON cana-1164 328 28 be be AUX cana-1164 328 29 the the DET cana-1164 328 30 ๐‘  ๐‘  PROPN cana-1164 328 31 components component NOUN cana-1164 328 32 of of ADP cana-1164 328 33 the the DET cana-1164 328 34 digraph digraph ADJ cana-1164 328 35 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 328 36 , , PUNCT cana-1164 328 37 ๐‘˜ ๐‘˜ NOUN cana-1164 328 38 ) ) PUNCT cana-1164 328 39 and and CCONJ cana-1164 329 1 ๐‘› ๐‘› ADJ cana-1164 329 2 > > X cana-1164 329 3 2 2 X cana-1164 329 4 . . PUNCT cana-1164 329 5 let let VERB cana-1164 330 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 330 2 + + CCONJ cana-1164 330 3 be be AUX cana-1164 330 4 the the DET cana-1164 330 5 out out ADJ cana-1164 330 6 - - PUNCT cana-1164 330 7 adjacency adjacency NOUN cana-1164 330 8 matrix matrix NOUN cana-1164 330 9 of of ADP cana-1164 330 10 the the DET cana-1164 330 11 digraph digraph ADJ cana-1164 330 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 330 13 , , PUNCT cana-1164 330 14 ๐‘˜ ๐‘˜ NOUN cana-1164 330 15 ) ) PUNCT cana-1164 330 16 and and CCONJ cana-1164 330 17 ๐ดฮณ ๐ดฮณ PROPN cana-1164 330 18 + + NOUN cana-1164 330 19 = = SYM cana-1164 330 20 [ [ PUNCT cana-1164 330 21 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 330 22 + + X cana-1164 330 23 0 0 NUM cana-1164 330 24 0 0 NUM cana-1164 330 25 โ‹ฏ โ‹ฏ ADP cana-1164 330 26 0 0 NUM cana-1164 330 27 0 0 NUM cana-1164 330 28 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 330 29 + + X cana-1164 330 30 0 0 NUM cana-1164 330 31 โ‹ฏ โ‹ฏ NOUN cana-1164 330 32 0 0 NUM cana-1164 330 33 โ‹ฎ โ‹ฎ NOUN cana-1164 330 34 โ‹ฎ โ‹ฎ ADJ cana-1164 330 35 โ‹ฎ โ‹ฎ NOUN cana-1164 330 36 โ‹ฑ โ‹ฑ PUNCT cana-1164 330 37 โ‹ฎ โ‹ฎ NOUN cana-1164 330 38 0 0 NUM cana-1164 330 39 0 0 NUM cana-1164 330 40 0 0 NUM cana-1164 331 1 โ‹ฏ โ‹ฏ PROPN cana-1164 331 2 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 332 1 + + X cana-1164 332 2 ] ] PUNCT cana-1164 332 3 where where SCONJ cana-1164 332 4 ๐ดฮณ1 ๐ดฮณ1 NOUN cana-1164 332 5 + + CCONJ cana-1164 332 6 , , PUNCT cana-1164 332 7 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 332 8 + + X cana-1164 332 9 , , PUNCT cana-1164 332 10 โ‹ฏ โ‹ฏ PROPN cana-1164 332 11 , , PUNCT cana-1164 332 12 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 332 13 + + CCONJ cana-1164 332 14 are be AUX cana-1164 332 15 out out ADJ cana-1164 332 16 - - PUNCT cana-1164 332 17 adjacency adjacency NOUN cana-1164 332 18 matrices matrix NOUN cana-1164 332 19 of of ADP cana-1164 332 20 the the DET cana-1164 332 21 components component NOUN cana-1164 332 22 ฮณ1 ฮณ1 PROPN cana-1164 332 23 , , PUNCT cana-1164 332 24 ฮณ2 ฮณ2 PROPN cana-1164 332 25 , , PUNCT cana-1164 332 26 โ‹ฏ โ‹ฏ PROPN cana-1164 332 27 , , PUNCT cana-1164 332 28 ฮณ๐‘  ฮณ๐‘  X cana-1164 332 29 respectively respectively ADV cana-1164 332 30 . . PUNCT cana-1164 333 1 communications communication NOUN cana-1164 333 2 on on ADP cana-1164 333 3 applied apply VERB cana-1164 333 4 nonlinear nonlinear ADJ cana-1164 333 5 analysis analysis NOUN cana-1164 333 6 issn issn NOUN cana-1164 333 7 : : PUNCT cana-1164 333 8 1074 1074 NUM cana-1164 333 9 - - PUNCT cana-1164 333 10 133x 133x NUM cana-1164 333 11 vol vol NOUN cana-1164 333 12 31 31 NUM cana-1164 333 13 no no NOUN cana-1164 333 14 . . PUNCT cana-1164 334 1 6s 6s NUM cana-1164 334 2 ( ( PUNCT cana-1164 334 3 2024 2024 NUM cana-1164 334 4 ) ) PUNCT cana-1164 334 5 108 108 NUM cana-1164 334 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 334 7 by by ADP cana-1164 334 8 lemma lemma PROPN cana-1164 334 9 3.1 3.1 NUM cana-1164 334 10 . . PROPN cana-1164 334 11 , , PUNCT cana-1164 334 12 the the DET cana-1164 334 13 digraph digraph ADJ cana-1164 334 14 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 334 15 , , PUNCT cana-1164 334 16 ๐‘˜ ๐‘˜ NOUN cana-1164 334 17 ) ) PUNCT cana-1164 334 18 has have VERB cana-1164 334 19 at at ADV cana-1164 334 20 least least ADV cana-1164 334 21 one one NUM cana-1164 334 22 vertex vertex NOUN cana-1164 334 23 of of ADP cana-1164 334 24 in in ADP cana-1164 334 25 - - PUNCT cana-1164 334 26 degree degree NOUN cana-1164 334 27 0 0 NUM cana-1164 334 28 , , PUNCT cana-1164 334 29 so so ADV cana-1164 334 30 let let VERB cana-1164 334 31 ๐‘ฃ๐‘ก ๐‘ฃ๐‘ก PART cana-1164 334 32 be be AUX cana-1164 334 33 such such DET cana-1164 334 34 a a DET cana-1164 334 35 vertex vertex NOUN cana-1164 334 36 of of ADP cana-1164 334 37 the the DET cana-1164 334 38 digraph digraph ADJ cana-1164 334 39 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 334 40 , , PUNCT cana-1164 334 41 ๐‘˜ ๐‘˜ NOUN cana-1164 334 42 ) ) PUNCT cana-1164 334 43 such such ADJ cana-1164 334 44 that that DET cana-1164 334 45 indeg(๐‘ฃ๐‘ก indeg(๐‘ฃ๐‘ก NOUN cana-1164 334 46 ) ) PUNCT cana-1164 334 47 = = SYM cana-1164 335 1 0 0 X cana-1164 335 2 . . PUNCT cana-1164 336 1 therefore therefore ADV cana-1164 336 2 , , PUNCT cana-1164 336 3 each each DET cana-1164 336 4 entry entry NOUN cana-1164 336 5 of of ADP cana-1164 336 6 the the DET cana-1164 336 7 column column NOUN cana-1164 336 8 ๐ถ๐‘ฃ๐‘ก ๐ถ๐‘ฃ๐‘ก PROPN cana-1164 336 9 ( ( PUNCT cana-1164 336 10 say say INTJ cana-1164 336 11 ) ) PUNCT cana-1164 336 12 corresponding correspond VERB cana-1164 336 13 to to ADP cana-1164 336 14 the the DET cana-1164 336 15 vertex vertex NOUN cana-1164 336 16 ๐‘ฃ๐‘ก ๐‘ฃ๐‘ก ADP cana-1164 336 17 in in ADP cana-1164 336 18 ๐ดฮณ ๐ดฮณ PROPN cana-1164 336 19 + + CCONJ cana-1164 336 20 will will AUX cana-1164 336 21 be be AUX cana-1164 336 22 zero zero NUM cana-1164 336 23 . . PUNCT cana-1164 337 1 now now ADV cana-1164 337 2 , , PUNCT cana-1164 337 3 some some DET cana-1164 337 4 element(s element(s PROPN cana-1164 337 5 ) ) PUNCT cana-1164 337 6 of of ADP cana-1164 337 7 ๐ถ๐‘ฃ๐‘ก ๐ถ๐‘ฃ๐‘ก PROPN cana-1164 337 8 is be AUX cana-1164 337 9 ( ( PUNCT cana-1164 337 10 are be AUX cana-1164 337 11 ) ) PUNCT cana-1164 337 12 also also ADV cana-1164 337 13 column column NOUN cana-1164 337 14 element(s element(s PROPN cana-1164 337 15 ) ) PUNCT cana-1164 337 16 of of ADP cana-1164 337 17 one one NUM cana-1164 337 18 of of ADP cana-1164 337 19 the the DET cana-1164 337 20 block block NOUN cana-1164 337 21 diagonal diagonal ADJ cana-1164 337 22 submatrix submatrix NOUN cana-1164 338 1 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 338 2 + + X cana-1164 338 3 ( ( PUNCT cana-1164 338 4 ๐‘ ๐‘Ž๐‘ฆ),1 ๐‘ ๐‘Ž๐‘ฆ),1 PROPN cana-1164 338 5 โ‰ค โ‰ค PROPN cana-1164 338 6 ๐‘– ๐‘– SYM cana-1164 338 7 โ‰ค โ‰ค NOUN cana-1164 338 8 ๐‘  ๐‘  ADP cana-1164 338 9 and and CCONJ cana-1164 338 10 consequently consequently ADV cana-1164 338 11 one one NUM cana-1164 338 12 column column NOUN cana-1164 338 13 of of ADP cana-1164 338 14 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 338 15 + + PROPN cana-1164 338 16 is be AUX cana-1164 338 17 a a DET cana-1164 338 18 zero zero NUM cana-1164 338 19 column column NOUN cana-1164 338 20 resulting result VERB cana-1164 338 21 ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘– ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘– PROPN cana-1164 338 22 + + PUNCT cana-1164 338 23 ) ) PUNCT cana-1164 339 1 = = SYM cana-1164 339 2 0 0 X cana-1164 339 3 . . X cana-1164 339 4 result result VERB cana-1164 339 5 3.15 3.15 NUM cana-1164 339 6 . . PUNCT cana-1164 340 1 if if SCONJ cana-1164 340 2 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 340 3 ) ) PUNCT cana-1164 340 4 or or CCONJ cana-1164 340 5 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 340 6 , , PUNCT cana-1164 340 7 for for ADP cana-1164 340 8 some some DET cana-1164 340 9 prime prime PROPN cana-1164 340 10 ๐‘ ๐‘ PROPN cana-1164 340 11 then then ADV cana-1164 340 12 the the DET cana-1164 340 13 out out ADJ cana-1164 340 14 - - PUNCT cana-1164 340 15 adjacency adjacency NOUN cana-1164 340 16 matrix matrix NOUN cana-1164 341 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 341 2 + + X cana-1164 341 3 of of ADP cana-1164 341 4 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 341 5 , , PUNCT cana-1164 341 6 ๐‘˜ ๐‘˜ NOUN cana-1164 341 7 ) ) PUNCT cana-1164 341 8 is be AUX cana-1164 341 9 a a DET cana-1164 341 10 singular singular ADJ cana-1164 341 11 matrix matrix NOUN cana-1164 341 12 . . PUNCT cana-1164 342 1 proof proof NOUN cana-1164 342 2 . . PUNCT cana-1164 343 1 let let VERB cana-1164 343 2 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 343 3 ) ) PUNCT cana-1164 343 4 or or CCONJ cana-1164 343 5 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 343 6 , , PUNCT cana-1164 343 7 for for ADP cana-1164 343 8 some some DET cana-1164 343 9 prime prime ADJ cana-1164 343 10 ๐‘. ๐‘. NOUN cana-1164 343 11 also also ADV cana-1164 343 12 , , PUNCT cana-1164 343 13 let let VERB cana-1164 343 14 , , PUNCT cana-1164 344 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 344 2 + + NUM cana-1164 344 3 = = SYM cana-1164 344 4 [ [ PUNCT cana-1164 344 5 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 344 6 + + X cana-1164 344 7 0 0 NUM cana-1164 344 8 0 0 NUM cana-1164 344 9 โ‹ฏ โ‹ฏ ADP cana-1164 344 10 0 0 NUM cana-1164 344 11 0 0 NUM cana-1164 344 12 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 344 13 + + X cana-1164 344 14 0 0 NUM cana-1164 344 15 โ‹ฏ โ‹ฏ NOUN cana-1164 344 16 0 0 NUM cana-1164 344 17 โ‹ฎ โ‹ฎ NOUN cana-1164 344 18 โ‹ฎ โ‹ฎ ADJ cana-1164 344 19 โ‹ฎ โ‹ฎ NOUN cana-1164 344 20 โ‹ฑ โ‹ฑ PUNCT cana-1164 344 21 โ‹ฎ โ‹ฎ NOUN cana-1164 344 22 0 0 NUM cana-1164 344 23 0 0 NUM cana-1164 344 24 0 0 NUM cana-1164 344 25 โ‹ฏ โ‹ฏ PROPN cana-1164 344 26 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 344 27 + + X cana-1164 344 28 ] ] PUNCT cana-1164 344 29 where where SCONJ cana-1164 344 30 ๐ดฮณ1 ๐ดฮณ1 NOUN cana-1164 344 31 + + CCONJ cana-1164 344 32 , , PUNCT cana-1164 344 33 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 344 34 + + X cana-1164 344 35 , , PUNCT cana-1164 344 36 โ‹ฏ โ‹ฏ PROPN cana-1164 344 37 , , PUNCT cana-1164 344 38 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 344 39 + + CCONJ cana-1164 344 40 are be AUX cana-1164 344 41 respectively respectively ADV cana-1164 344 42 out out ADJ cana-1164 344 43 - - PUNCT cana-1164 344 44 adjacency adjacency NOUN cana-1164 344 45 matrices matrix NOUN cana-1164 344 46 of of ADP cana-1164 344 47 the the DET cana-1164 344 48 components component NOUN cana-1164 344 49 ฮณ1 ฮณ1 PROPN cana-1164 344 50 , , PUNCT cana-1164 344 51 ฮณ2 ฮณ2 PROPN cana-1164 344 52 , , PUNCT cana-1164 344 53 โ‹ฏ โ‹ฏ PROPN cana-1164 344 54 , , PUNCT cana-1164 344 55 ฮณ๐‘  ฮณ๐‘  NOUN cana-1164 344 56 of of ADP cana-1164 344 57 the the DET cana-1164 344 58 digraph digraph ADJ cana-1164 344 59 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 344 60 , , PUNCT cana-1164 344 61 ๐‘˜ ๐‘˜ NOUN cana-1164 344 62 ) ) PUNCT cana-1164 344 63 . . PUNCT cana-1164 345 1 we we PRON cana-1164 345 2 have have VERB cana-1164 345 3 , , PUNCT cana-1164 345 4 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADV cana-1164 345 5 + + NOUN cana-1164 345 6 ) ) PUNCT cana-1164 345 7 = = SYM cana-1164 345 8 ๐‘‘๐‘’๐‘ก(๐ดฮณ1 ๐‘‘๐‘’๐‘ก(๐ดฮณ1 NOUN cana-1164 345 9 + + NUM cana-1164 345 10 ) ) PUNCT cana-1164 345 11 ร— ร— NOUN cana-1164 345 12 ๐‘‘๐‘’๐‘ก(๐ดฮณ2 ๐‘‘๐‘’๐‘ก(๐ดฮณ2 NOUN cana-1164 345 13 + + CCONJ cana-1164 345 14 ) ) PUNCT cana-1164 345 15 ร— ร— NOUN cana-1164 345 16 โ‹ฏร— โ‹ฏร— NUM cana-1164 345 17 ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘  ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘  PRON cana-1164 345 18 + + PUNCT cana-1164 345 19 ) ) PUNCT cana-1164 345 20 . . PUNCT cana-1164 346 1 ( ( PUNCT cana-1164 346 2 1 1 X cana-1164 346 3 ) ) PUNCT cana-1164 346 4 by by ADP cana-1164 346 5 lemma lemma PROPN cana-1164 346 6 3.2 3.2 NUM cana-1164 346 7 . . PUNCT cana-1164 346 8 , , PUNCT cana-1164 347 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 347 2 + + CCONJ cana-1164 347 3 contains contain VERB cana-1164 347 4 at at ADV cana-1164 347 5 least least ADV cana-1164 347 6 one one NUM cana-1164 347 7 block block NOUN cana-1164 347 8 submatrix submatrix NOUN cana-1164 347 9 ๐ดฮณ๐‘š ๐ดฮณ๐‘š PROPN cana-1164 347 10 + + CCONJ cana-1164 347 11 ( ( PUNCT cana-1164 347 12 say say INTJ cana-1164 347 13 ) ) PUNCT cana-1164 347 14 , , PUNCT cana-1164 348 1 1 1 NUM cana-1164 348 2 โ‰ค โ‰ค NUM cana-1164 348 3 ๐‘š ๐‘š PRON cana-1164 348 4 โ‰ค โ‰ค NUM cana-1164 348 5 ๐‘  ๐‘  ADP cana-1164 348 6 such such ADJ cana-1164 348 7 that that SCONJ cana-1164 348 8 ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘š ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘š PROPN cana-1164 348 9 + + PUNCT cana-1164 348 10 ) ) PUNCT cana-1164 348 11 = = SYM cana-1164 348 12 0 0 PUNCT cana-1164 348 13 and and CCONJ cana-1164 348 14 hence hence ADV cana-1164 348 15 from from ADP cana-1164 348 16 ( ( PUNCT cana-1164 348 17 1 1 X cana-1164 348 18 ) ) PUNCT cana-1164 348 19 we we PRON cana-1164 348 20 get get VERB cana-1164 348 21 , , PUNCT cana-1164 348 22 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADV cana-1164 348 23 + + ADJ cana-1164 348 24 ) ) PUNCT cana-1164 348 25 = = SYM cana-1164 349 1 0 0 X cana-1164 349 2 . . PUNCT cana-1164 350 1 this this PRON cana-1164 350 2 shows show VERB cana-1164 350 3 that that SCONJ cana-1164 350 4 the the DET cana-1164 350 5 matrix matrix NOUN cana-1164 350 6 ๐ดฮณ ๐ดฮณ PROPN cana-1164 350 7 + + CCONJ cana-1164 350 8 is be AUX cana-1164 350 9 a a DET cana-1164 350 10 singular singular ADJ cana-1164 350 11 matrix matrix NOUN cana-1164 350 12 . . PUNCT cana-1164 351 1 result result VERB cana-1164 351 2 3.16 3.16 NUM cana-1164 351 3 . . PUNCT cana-1164 352 1 if if SCONJ cana-1164 352 2 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 352 3 ) ) PUNCT cana-1164 352 4 or or CCONJ cana-1164 352 5 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 352 6 , , PUNCT cana-1164 352 7 for for ADP cana-1164 352 8 some some DET cana-1164 352 9 prime prime PROPN cana-1164 352 10 ๐‘ ๐‘ PROPN cana-1164 352 11 then then ADV cana-1164 352 12 the the DET cana-1164 352 13 in in ADP cana-1164 352 14 - - PUNCT cana-1164 352 15 adjacency adjacency NOUN cana-1164 352 16 matrix matrix NOUN cana-1164 352 17 ๐ดฮณ ๐ดฮณ ADP cana-1164 352 18 โˆ’ โˆ’ PROPN cana-1164 352 19 of of ADP cana-1164 352 20 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 352 21 , , PUNCT cana-1164 352 22 ๐‘˜ ๐‘˜ NOUN cana-1164 352 23 ) ) PUNCT cana-1164 352 24 is be AUX cana-1164 352 25 a a DET cana-1164 352 26 singular singular ADJ cana-1164 352 27 matrix matrix NOUN cana-1164 352 28 . . PUNCT cana-1164 353 1 proof proof NOUN cana-1164 353 2 . . PUNCT cana-1164 354 1 let let VERB cana-1164 354 2 ๐‘˜|๐œ™(๐‘› ๐‘˜|๐œ™(๐‘› NUM cana-1164 354 3 ) ) PUNCT cana-1164 354 4 or or CCONJ cana-1164 354 5 ๐‘2|๐‘› ๐‘2|๐‘› VERB cana-1164 354 6 , , PUNCT cana-1164 354 7 for for ADP cana-1164 354 8 some some DET cana-1164 354 9 prime prime ADJ cana-1164 354 10 ๐‘. ๐‘. NOUN cana-1164 354 11 we we PRON cana-1164 354 12 have have VERB cana-1164 354 13 , , PUNCT cana-1164 354 14 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADV cana-1164 354 15 โˆ’ โˆ’ NOUN cana-1164 354 16 ) ) PUNCT cana-1164 355 1 = = SYM cana-1164 355 2 ๐‘‘๐‘’๐‘ก((๐ดฮณ ๐‘‘๐‘’๐‘ก((๐ดฮณ PROPN cana-1164 355 3 + + NOUN cana-1164 355 4 ) ) PUNCT cana-1164 355 5 ๐‘ก ๐‘ก NOUN cana-1164 355 6 ) ) PUNCT cana-1164 356 1 [ [ X cana-1164 356 2 by by ADP cana-1164 356 3 result result NOUN cana-1164 356 4 3.1 3.1 NUM cana-1164 356 5 . . PUNCT cana-1164 356 6 ] ] PUNCT cana-1164 357 1 = = PUNCT cana-1164 357 2 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADP cana-1164 357 3 + + NOUN cana-1164 357 4 ) ) PUNCT cana-1164 357 5 = = SYM cana-1164 357 6 0 0 PUNCT cana-1164 358 1 [ [ X cana-1164 358 2 by by ADP cana-1164 358 3 result result NOUN cana-1164 358 4 3.15 3.15 NUM cana-1164 358 5 . . PUNCT cana-1164 358 6 ] ] PUNCT cana-1164 359 1 this this PRON cana-1164 359 2 shows show VERB cana-1164 359 3 that that SCONJ cana-1164 359 4 the the DET cana-1164 359 5 matrix matrix NOUN cana-1164 359 6 ๐ดฮณ ๐ดฮณ PROPN cana-1164 359 7 โˆ’ โˆ’ PROPN cana-1164 359 8 is be AUX cana-1164 359 9 a a DET cana-1164 359 10 singular singular ADJ cana-1164 359 11 matrix matrix NOUN cana-1164 359 12 . . PUNCT cana-1164 360 1 4 4 X cana-1164 360 2 . . X cana-1164 360 3 spectrum spectrum NOUN cana-1164 360 4 of of ADP cana-1164 360 5 the the DET cana-1164 360 6 digraph digraph NOUN cana-1164 360 7 ๐šช(๐’ ๐šช(๐’ PROPN cana-1164 360 8 , , PUNCT cana-1164 360 9 ๐’Œ ๐’Œ ADJ cana-1164 360 10 ) ) PUNCT cana-1164 360 11 the the DET cana-1164 360 12 characteristic characteristic ADJ cana-1164 360 13 polynomial polynomial NOUN cana-1164 360 14 of of ADP cana-1164 360 15 a a DET cana-1164 360 16 matrix matrix NOUN cana-1164 361 1 a a PRON cana-1164 361 2 is be AUX cana-1164 361 3 the the DET cana-1164 361 4 polynomial polynomial ADJ cana-1164 361 5 ๐‘‘๐‘’๐‘ก(๐ด ๐‘‘๐‘’๐‘ก(๐ด NUM cana-1164 361 6 โˆ’ โˆ’ PROPN cana-1164 361 7 ๐œ†๐ผ ๐œ†๐ผ NOUN cana-1164 361 8 ) ) PUNCT cana-1164 361 9 . . PUNCT cana-1164 362 1 the the DET cana-1164 362 2 roots root NOUN cana-1164 362 3 of of ADP cana-1164 362 4 the the DET cana-1164 362 5 characteristic characteristic ADJ cana-1164 362 6 polynomial polynomial NOUN cana-1164 362 7 are be AUX cana-1164 362 8 the the DET cana-1164 362 9 eigenvalues eigenvalue NOUN cana-1164 362 10 of of ADP cana-1164 362 11 ๐ด. ๐ด. PROPN cana-1164 362 12 a a DET cana-1164 362 13 non non ADJ cana-1164 362 14 - - ADJ cana-1164 362 15 zero zero NUM cana-1164 362 16 vector vector NOUN cana-1164 362 17 ๐‘ฃ ๐‘ฃ NOUN cana-1164 362 18 is be AUX cana-1164 362 19 an an DET cana-1164 362 20 eigenvector eigenvector NOUN cana-1164 362 21 of of ADP cana-1164 362 22 ๐ด ๐ด PROPN cana-1164 362 23 with with ADP cana-1164 362 24 eigenvalue eigenvalue PROPN cana-1164 362 25 ๐œ† ๐œ† ADP cana-1164 362 26 if if SCONJ cana-1164 362 27 the the DET cana-1164 362 28 equation equation NOUN cana-1164 362 29 ๐ด๐‘ฃ ๐ด๐‘ฃ PROPN cana-1164 362 30 = = PUNCT cana-1164 362 31 ๐œ†๐‘ฃ ๐œ†๐‘ฃ NOUN cana-1164 362 32 is be AUX cana-1164 362 33 satisfied satisfied ADJ cana-1164 362 34 . . PUNCT cana-1164 363 1 the the DET cana-1164 363 2 eigenvalue(s eigenvalue(s NOUN cana-1164 363 3 ) ) PUNCT cana-1164 363 4 of of ADP cana-1164 363 5 a a DET cana-1164 363 6 graph graph NOUN cana-1164 363 7 ๐บ ๐บ NOUN cana-1164 363 8 is be AUX cana-1164 363 9 ( ( PUNCT cana-1164 363 10 are be AUX cana-1164 363 11 ) ) PUNCT cana-1164 363 12 defined define VERB cana-1164 363 13 as as ADP cana-1164 363 14 the the DET cana-1164 363 15 eigenvalue(s eigenvalue(s NOUN cana-1164 363 16 ) ) PUNCT cana-1164 363 17 of of ADP cana-1164 363 18 its its PRON cana-1164 363 19 adjacency adjacency NOUN cana-1164 363 20 matrix matrix NOUN cana-1164 363 21 . . PUNCT cana-1164 364 1 the the DET cana-1164 364 2 spectrum spectrum NOUN cana-1164 364 3 of of ADP cana-1164 364 4 a a DET cana-1164 364 5 graph graph NOUN cana-1164 364 6 ๐บ ๐บ NOUN cana-1164 364 7 is be AUX cana-1164 364 8 the the DET cana-1164 364 9 set set NOUN cana-1164 364 10 of of ADP cana-1164 364 11 eigenvalues eigenvalue NOUN cana-1164 364 12 of of ADP cana-1164 364 13 ๐บ ๐บ PROPN cana-1164 364 14 together together ADV cana-1164 364 15 with with ADP cana-1164 364 16 their their PRON cana-1164 364 17 algebraic algebraic ADJ cana-1164 364 18 multiplicities multiplicity NOUN cana-1164 364 19 . . PUNCT cana-1164 365 1 if if SCONJ cana-1164 365 2 a a DET cana-1164 365 3 communications communication NOUN cana-1164 365 4 on on ADP cana-1164 365 5 applied apply VERB cana-1164 365 6 nonlinear nonlinear ADJ cana-1164 365 7 analysis analysis NOUN cana-1164 365 8 issn issn NOUN cana-1164 365 9 : : PUNCT cana-1164 365 10 1074 1074 NUM cana-1164 365 11 - - PUNCT cana-1164 365 12 133x 133x NUM cana-1164 365 13 vol vol NOUN cana-1164 365 14 31 31 NUM cana-1164 365 15 no no NOUN cana-1164 365 16 . . PUNCT cana-1164 366 1 6s 6s NUM cana-1164 366 2 ( ( PUNCT cana-1164 366 3 2024 2024 NUM cana-1164 366 4 ) ) PUNCT cana-1164 366 5 109 109 NUM cana-1164 366 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 366 7 graph graph NOUN cana-1164 366 8 ๐บ ๐บ PROPN cana-1164 366 9 has have VERB cana-1164 366 10 ๐‘ก ๐‘ก VERB cana-1164 366 11 distinct distinct ADJ cana-1164 366 12 eigenvalues eigenvalue NOUN cana-1164 366 13 ๐œ†1 ๐œ†1 NOUN cana-1164 366 14 > > X cana-1164 366 15 ๐œ†2 ๐œ†2 NOUN cana-1164 366 16 > > X cana-1164 366 17 ๐œ†3 ๐œ†3 PROPN cana-1164 366 18 > > X cana-1164 367 1 โ‹ฏ โ‹ฏ X cana-1164 367 2 > > X cana-1164 367 3 ๐œ†๐‘ก ๐œ†๐‘ก ADP cana-1164 367 4 with with ADP cana-1164 367 5 multiplicities multiplicities PROPN cana-1164 367 6 ๐‘š(๐œ†1),๐‘š(๐œ†2),๐‘š(๐œ†3),โ‹ฏ ๐‘š(๐œ†1),๐‘š(๐œ†2),๐‘š(๐œ†3),โ‹ฏ PROPN cana-1164 367 7 , , PUNCT cana-1164 367 8 ๐‘š(๐œ†๐‘ก ๐‘š(๐œ†๐‘ก PROPN cana-1164 367 9 ) ) PUNCT cana-1164 367 10 then then ADV cana-1164 367 11 the the DET cana-1164 367 12 spectrum spectrum NOUN cana-1164 367 13 of of ADP cana-1164 367 14 ๐บ ๐บ PROPN cana-1164 367 15 is be AUX cana-1164 367 16 ๐‘†๐‘๐‘’๐‘(๐บ)= ๐‘†๐‘๐‘’๐‘(๐บ)= NOUN cana-1164 367 17 ( ( PUNCT cana-1164 367 18 ๐œ†1 ๐œ†1 NOUN cana-1164 367 19 ๐œ†2 ๐œ†2 NOUN cana-1164 367 20 ๐œ†3 ๐œ†3 NOUN cana-1164 367 21 โ‹ฏ โ‹ฏ NOUN cana-1164 367 22 ๐œ†๐‘ก ๐œ†๐‘ก NOUN cana-1164 367 23 ๐‘š(๐œ†1 ๐‘š(๐œ†1 NOUN cana-1164 367 24 ) ) PUNCT cana-1164 367 25 ๐‘š(๐œ†2 ๐‘š(๐œ†2 ADJ cana-1164 367 26 ) ) PUNCT cana-1164 367 27 ๐‘š(๐œ†3 ๐‘š(๐œ†3 NOUN cana-1164 367 28 ) ) PUNCT cana-1164 367 29 โ‹ฏ โ‹ฏ NOUN cana-1164 367 30 ๐‘š(๐œ†๐‘ก ๐‘š(๐œ†๐‘ก PROPN cana-1164 367 31 ) ) PUNCT cana-1164 367 32 ) ) PUNCT cana-1164 368 1 also also ADV cana-1164 368 2 , , PUNCT cana-1164 368 3 we we PRON cana-1164 368 4 have have VERB cana-1164 368 5 , , PUNCT cana-1164 368 6 ๐‘‘๐‘’๐‘ก((๐ดฮณ ๐‘‘๐‘’๐‘ก((๐ดฮณ PROPN cana-1164 368 7 + + NOUN cana-1164 368 8 ) ) PUNCT cana-1164 368 9 ๐‘ก ๐‘ก PROPN cana-1164 368 10 โˆ’ โˆ’ NOUN cana-1164 368 11 ๐œ†๐ผ ๐œ†๐ผ NOUN cana-1164 368 12 ) ) PUNCT cana-1164 369 1 = = SYM cana-1164 370 1 ๐‘‘๐‘’๐‘ก((๐ดฮณ ๐‘‘๐‘’๐‘ก((๐ดฮณ PROPN cana-1164 370 2 + + NOUN cana-1164 370 3 ) ) PUNCT cana-1164 370 4 ๐‘ก ๐‘ก NOUN cana-1164 370 5 โˆ’ โˆ’ NOUN cana-1164 370 6 ๐œ†๐ผ๐‘ก ๐œ†๐ผ๐‘ก ADP cana-1164 370 7 ) ) PUNCT cana-1164 370 8 , , PUNCT cana-1164 370 9 where where SCONJ cana-1164 370 10 i i PRON cana-1164 370 11 is be AUX cana-1164 370 12 an an DET cana-1164 370 13 identity identity NOUN cana-1164 370 14 matrix matrix NOUN cana-1164 370 15 of of ADP cana-1164 370 16 order order NOUN cana-1164 370 17 n. n. NOUN cana-1164 370 18 โ‡’ โ‡’ VERB cana-1164 370 19 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADV cana-1164 370 20 โˆ’ โˆ’ PROPN cana-1164 370 21 โˆ’ โˆ’ PROPN cana-1164 370 22 ๐œ†๐ผ ๐œ†๐ผ NOUN cana-1164 370 23 ) ) PUNCT cana-1164 370 24 = = PUNCT cana-1164 371 1 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADV cana-1164 371 2 + + CCONJ cana-1164 371 3 โˆ’ โˆ’ PROPN cana-1164 371 4 ๐œ†๐ผ)๐‘ก ๐œ†๐ผ)๐‘ก PROPN cana-1164 371 5 [ [ PUNCT cana-1164 371 6 by by ADP cana-1164 371 7 result result NOUN cana-1164 371 8 3.1 3.1 NUM cana-1164 371 9 . . NUM cana-1164 371 10 , , PUNCT cana-1164 371 11 ( ( PUNCT cana-1164 371 12 ๐ดฮณ ๐ดฮณ PROPN cana-1164 371 13 + + PROPN cana-1164 371 14 ) ) PUNCT cana-1164 371 15 ๐‘ก ๐‘ก NOUN cana-1164 371 16 = = PUNCT cana-1164 372 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 372 2 โˆ’ โˆ’ PROPN cana-1164 372 3 ] ] PUNCT cana-1164 372 4 โ‡’ โ‡’ VERB cana-1164 372 5 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADV cana-1164 372 6 โˆ’ โˆ’ PROPN cana-1164 372 7 โˆ’ โˆ’ PROPN cana-1164 372 8 ๐œ†๐ผ ๐œ†๐ผ NOUN cana-1164 372 9 ) ) PUNCT cana-1164 372 10 = = PUNCT cana-1164 373 1 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADP cana-1164 373 2 + + CCONJ cana-1164 373 3 โˆ’ โˆ’ PROPN cana-1164 373 4 ๐œ†๐ผ ๐œ†๐ผ NOUN cana-1164 373 5 ) ) PUNCT cana-1164 374 1 [ [ X cana-1164 374 2 โˆต โˆต NOUN cana-1164 374 3 ๐‘‘๐‘’๐‘ก(๐‘‹๐‘ก ๐‘‘๐‘’๐‘ก(๐‘‹๐‘ก NOUN cana-1164 374 4 ) ) PUNCT cana-1164 374 5 = = VERB cana-1164 374 6 d๐‘’๐‘ก(๐‘‹),where d๐‘’๐‘ก(๐‘‹),where ADJ cana-1164 374 7 x x VERB cana-1164 374 8 is be AUX cana-1164 374 9 a a DET cana-1164 374 10 square square ADJ cana-1164 374 11 matrix matrix NOUN cana-1164 374 12 . . PUNCT cana-1164 374 13 ] ] PUNCT cana-1164 375 1 so so ADV cana-1164 375 2 , , PUNCT cana-1164 375 3 the the DET cana-1164 375 4 characteristic characteristic ADJ cana-1164 375 5 polynomial polynomial NOUN cana-1164 375 6 of of ADP cana-1164 375 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 375 8 + + CCONJ cana-1164 375 9 = = PUNCT cana-1164 375 10 the the DET cana-1164 375 11 characteristic characteristic ADJ cana-1164 375 12 polynomial polynomial NOUN cana-1164 375 13 of of ADP cana-1164 375 14 ๐ดฮณ ๐ดฮณ PROPN cana-1164 375 15 โˆ’. โˆ’. PROPN cana-1164 375 16 in in ADP cana-1164 375 17 this this DET cana-1164 375 18 section section NOUN cana-1164 375 19 , , PUNCT cana-1164 375 20 we we PRON cana-1164 375 21 will will AUX cana-1164 375 22 study study VERB cana-1164 375 23 some some DET cana-1164 375 24 spectral spectral ADJ cana-1164 375 25 properties property NOUN cana-1164 375 26 of of ADP cana-1164 375 27 the the DET cana-1164 375 28 digraph digraph ADJ cana-1164 375 29 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 375 30 , , PUNCT cana-1164 375 31 ๐‘˜ ๐‘˜ NOUN cana-1164 375 32 ) ) PUNCT cana-1164 375 33 using use VERB cana-1164 375 34 the the DET cana-1164 375 35 out out ADJ cana-1164 375 36 - - PUNCT cana-1164 375 37 adjacency adjacency NOUN cana-1164 375 38 matrix matrix NOUN cana-1164 376 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 376 2 + + CCONJ cana-1164 376 3 or or CCONJ cana-1164 376 4 in in ADP cana-1164 376 5 - - PUNCT cana-1164 376 6 adjacency adjacency NOUN cana-1164 376 7 matrix matrix NOUN cana-1164 377 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 377 2 โˆ’. โˆ’. ADV cana-1164 377 3 we we PRON cana-1164 377 4 define define VERB cana-1164 377 5 the the DET cana-1164 377 6 eigenvalues eigenvalue NOUN cana-1164 377 7 of of ADP cana-1164 377 8 the the DET cana-1164 377 9 digraph digraph ADJ cana-1164 377 10 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 377 11 , , PUNCT cana-1164 377 12 ๐‘˜ ๐‘˜ NOUN cana-1164 377 13 ) ) PUNCT cana-1164 377 14 as as ADP cana-1164 377 15 the the DET cana-1164 377 16 eigenvalues eigenvalue NOUN cana-1164 377 17 of of ADP cana-1164 377 18 its its PRON cana-1164 377 19 out out ADJ cana-1164 377 20 - - PUNCT cana-1164 377 21 adjacency adjacency NOUN cana-1164 377 22 matrix matrix NOUN cana-1164 377 23 ( ( PUNCT cana-1164 377 24 or or CCONJ cana-1164 377 25 in in ADP cana-1164 377 26 - - PUNCT cana-1164 377 27 adjacency adjacency NOUN cana-1164 377 28 matrix matrix NOUN cana-1164 377 29 ) ) PUNCT cana-1164 377 30 and and CCONJ cana-1164 377 31 the the DET cana-1164 377 32 spectrum spectrum NOUN cana-1164 377 33 of of ADP cana-1164 377 34 the the DET cana-1164 377 35 digraph digraph ADJ cana-1164 377 36 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 377 37 , , PUNCT cana-1164 377 38 ๐‘˜ ๐‘˜ NOUN cana-1164 377 39 ) ) PUNCT cana-1164 377 40 as as ADP cana-1164 377 41 the the DET cana-1164 377 42 set set NOUN cana-1164 377 43 of of ADP cana-1164 377 44 eigenvalues eigenvalue NOUN cana-1164 377 45 of of ADP cana-1164 377 46 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 377 47 , , PUNCT cana-1164 377 48 ๐‘˜ ๐‘˜ NOUN cana-1164 377 49 ) ) PUNCT cana-1164 377 50 together together ADV cana-1164 377 51 with with ADP cana-1164 377 52 their their PRON cana-1164 377 53 algebraic algebraic ADJ cana-1164 377 54 multiplicities multiplicity NOUN cana-1164 377 55 . . PUNCT cana-1164 377 56 example example NOUN cana-1164 377 57 4.1 4.1 NUM cana-1164 377 58 . . PUNCT cana-1164 378 1 let let VERB cana-1164 378 2 us we PRON cana-1164 378 3 consider consider VERB cana-1164 378 4 the the DET cana-1164 378 5 digraph digraph ADJ cana-1164 378 6 ฮณ(9 ฮณ(9 NOUN cana-1164 378 7 , , PUNCT cana-1164 378 8 11 11 NUM cana-1164 378 9 ) ) PUNCT cana-1164 378 10 . . PUNCT cana-1164 379 1 figure figure VERB cana-1164 379 2 2 2 NUM cana-1164 379 3 : : PUNCT cana-1164 379 4 digraph digraph VERB cana-1164 379 5 ฮณ(9,11 ฮณ(9,11 NOUN cana-1164 379 6 ) ) PUNCT cana-1164 379 7 with with ADP cana-1164 379 8 components component NOUN cana-1164 379 9 ฮณ1 ฮณ1 PROPN cana-1164 379 10 , , PUNCT cana-1164 379 11 ฮณ2 ฮณ2 PROPN cana-1164 379 12 , , PUNCT cana-1164 379 13 ฮณ3 ฮณ3 NOUN cana-1164 379 14 , , PUNCT cana-1164 379 15 ฮณ4 ฮณ4 NOUN cana-1164 379 16 , , PUNCT cana-1164 379 17 ฮณ5 ฮณ5 NOUN cana-1164 379 18 . . PUNCT cana-1164 380 1 we we PRON cana-1164 380 2 have have AUX cana-1164 380 3 , , PUNCT cana-1164 380 4 ๐ดฮณ1 ๐ดฮณ1 VERB cana-1164 380 5 + + X cana-1164 381 1 = = SYM cana-1164 381 2 0 0 NUM cana-1164 381 3 3 3 NUM cana-1164 381 4 6 6 NUM cana-1164 381 5 0 0 NUM cana-1164 381 6 1 1 NUM cana-1164 381 7 0 0 NUM cana-1164 381 8 0 0 NUM cana-1164 381 9 3 3 NUM cana-1164 381 10 1 1 NUM cana-1164 381 11 0 0 NUM cana-1164 381 12 0 0 NUM cana-1164 381 13 6 6 NUM cana-1164 381 14 1 1 NUM cana-1164 381 15 0 0 NUM cana-1164 381 16 0 0 NUM cana-1164 381 17 ๏ƒฉ ๏ƒฉ NUM cana-1164 381 18 ๏ƒน ๏ƒน ADJ cana-1164 381 19 ๏ƒช ๏ƒช NOUN cana-1164 381 20 ๏ƒบ ๏ƒบ PROPN cana-1164 381 21 ๏ƒช ๏ƒช NOUN cana-1164 381 22 ๏ƒบ ๏ƒบ NOUN cana-1164 381 23 ๏ƒช ๏ƒช NOUN cana-1164 381 24 ๏ƒบ๏ƒซ ๏ƒบ๏ƒซ PROPN cana-1164 381 25 ๏ƒป ๏ƒป PROPN cana-1164 381 26 , , PUNCT cana-1164 381 27 ๐ดฮณ2 ๐ดฮณ2 VERB cana-1164 381 28 + + CCONJ cana-1164 382 1 = = SYM cana-1164 382 2 ๏› ๏› ADJ cana-1164 382 3 ๏ ๏ NOUN cana-1164 382 4 1 1 NUM cana-1164 382 5 1 1 NUM cana-1164 382 6 1 1 NUM cana-1164 382 7 , , PUNCT cana-1164 382 8 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 383 1 + + X cana-1164 384 1 = = SYM cana-1164 384 2 2 2 NUM cana-1164 384 3 5 5 NUM cana-1164 384 4 2 2 NUM cana-1164 384 5 0 0 NUM cana-1164 384 6 1 1 NUM cana-1164 384 7 5 5 NUM cana-1164 384 8 1 1 NUM cana-1164 384 9 0 0 NUM cana-1164 384 10 ๏ƒฉ ๏ƒฉ PROPN cana-1164 384 11 ๏ƒน ๏ƒน ADJ cana-1164 384 12 ๏ƒช ๏ƒช NOUN cana-1164 384 13 ๏ƒบ ๏ƒบ NOUN cana-1164 384 14 ๏ƒซ ๏ƒซ NOUN cana-1164 384 15 ๏ƒป ๏ƒป PROPN cana-1164 384 16 , , PUNCT cana-1164 384 17 ๐ดฮณ4 ๐ดฮณ4 VERB cana-1164 384 18 + + X cana-1164 385 1 = = SYM cana-1164 385 2 4 4 NUM cana-1164 385 3 7 7 NUM cana-1164 385 4 4 4 NUM cana-1164 385 5 0 0 NUM cana-1164 385 6 1 1 NUM cana-1164 385 7 7 7 NUM cana-1164 385 8 1 1 NUM cana-1164 385 9 0 0 NUM cana-1164 385 10 ๏ƒฉ ๏ƒฉ PROPN cana-1164 385 11 ๏ƒน ๏ƒน ADJ cana-1164 385 12 ๏ƒช ๏ƒช NOUN cana-1164 385 13 ๏ƒบ ๏ƒบ NOUN cana-1164 385 14 ๏ƒซ ๏ƒซ NOUN cana-1164 385 15 ๏ƒป ๏ƒป PROPN cana-1164 385 16 , , PUNCT cana-1164 385 17 and and CCONJ cana-1164 385 18 ๐ดฮณ5 ๐ดฮณ5 VERB cana-1164 386 1 + + X cana-1164 386 2 = = SYM cana-1164 386 3 ๏› ๏› ADJ cana-1164 386 4 ๏ ๏ NOUN cana-1164 386 5 8 8 NUM cana-1164 386 6 8 8 NUM cana-1164 386 7 1 1 NUM cana-1164 386 8 therefore therefore ADV cana-1164 386 9 , , PUNCT cana-1164 386 10 the the DET cana-1164 386 11 characteristic characteristic ADJ cana-1164 386 12 polynomials polynomial NOUN cana-1164 386 13 of of ADP cana-1164 386 14 ๐ดฮณ1 ๐ดฮณ1 NOUN cana-1164 386 15 + + X cana-1164 386 16 , , PUNCT cana-1164 386 17 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 386 18 + + CCONJ cana-1164 386 19 , , PUNCT cana-1164 386 20 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 386 21 + + CCONJ cana-1164 386 22 , , PUNCT cana-1164 386 23 ๐ดฮณ4 ๐ดฮณ4 NOUN cana-1164 386 24 + + CCONJ cana-1164 386 25 and and CCONJ cana-1164 386 26 ๐ดฮณ5 ๐ดฮณ5 NOUN cana-1164 386 27 + + CCONJ cana-1164 386 28 are be AUX cana-1164 386 29 ๐œ†2(1 ๐œ†2(1 NOUN cana-1164 386 30 โˆ’ โˆ’ NOUN cana-1164 386 31 ๐œ† ๐œ† NOUN cana-1164 386 32 ) ) PUNCT cana-1164 386 33 , , PUNCT cana-1164 386 34 ( ( PUNCT cana-1164 386 35 1 1 NUM cana-1164 386 36 โˆ’ โˆ’ NOUN cana-1164 386 37 ๐œ† ๐œ† NOUN cana-1164 386 38 ) ) PUNCT cana-1164 386 39 , , PUNCT cana-1164 386 40 ( ( PUNCT cana-1164 386 41 ๐œ†2 ๐œ†2 NOUN cana-1164 386 42 โˆ’ โˆ’ PROPN cana-1164 386 43 1 1 NUM cana-1164 386 44 ) ) PUNCT cana-1164 386 45 , , PUNCT cana-1164 386 46 ( ( PUNCT cana-1164 386 47 ๐œ†2 ๐œ†2 NOUN cana-1164 386 48 โˆ’ โˆ’ PROPN cana-1164 386 49 1 1 NUM cana-1164 386 50 ) ) PUNCT cana-1164 386 51 , , PUNCT cana-1164 386 52 and and CCONJ cana-1164 386 53 ( ( PUNCT cana-1164 386 54 1 1 NUM cana-1164 386 55 โˆ’ โˆ’ NOUN cana-1164 386 56 ๐œ† ๐œ† NOUN cana-1164 386 57 ) ) PUNCT cana-1164 386 58 respectively respectively ADV cana-1164 386 59 . . PUNCT cana-1164 387 1 and and CCONJ cana-1164 387 2 , , PUNCT cana-1164 387 3 eigenvalues eigenvalue VERB cana-1164 387 4 of of ADP cana-1164 387 5 ๐ดฮณ1 ๐ดฮณ1 NOUN cana-1164 387 6 + + X cana-1164 387 7 , , PUNCT cana-1164 387 8 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 387 9 + + CCONJ cana-1164 387 10 , , PUNCT cana-1164 387 11 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 387 12 + + CCONJ cana-1164 387 13 , , PUNCT cana-1164 387 14 ๐ดฮณ4 ๐ดฮณ4 NOUN cana-1164 387 15 + + X cana-1164 387 16 , , PUNCT cana-1164 387 17 and and CCONJ cana-1164 387 18 ๐ดฮณ5 ๐ดฮณ5 NOUN cana-1164 387 19 + + CCONJ cana-1164 387 20 are be AUX cana-1164 387 21 0,0,1 0,0,1 NOUN cana-1164 387 22 ; ; PUNCT cana-1164 387 23 1 1 NUM cana-1164 387 24 ; ; PUNCT cana-1164 387 25 โˆ’1,1 โˆ’1,1 X cana-1164 387 26 ; ; PUNCT cana-1164 387 27 โˆ’1,1 โˆ’1,1 NOUN cana-1164 387 28 and and CCONJ cana-1164 387 29 1 1 NUM cana-1164 387 30 respectively respectively ADV cana-1164 387 31 . . PUNCT cana-1164 388 1 communications communication NOUN cana-1164 388 2 on on ADP cana-1164 388 3 applied apply VERB cana-1164 388 4 nonlinear nonlinear ADJ cana-1164 388 5 analysis analysis NOUN cana-1164 388 6 issn issn NOUN cana-1164 388 7 : : PUNCT cana-1164 388 8 1074 1074 NUM cana-1164 388 9 - - PUNCT cana-1164 388 10 133x 133x NUM cana-1164 388 11 vol vol NOUN cana-1164 388 12 31 31 NUM cana-1164 388 13 no no NOUN cana-1164 388 14 . . PUNCT cana-1164 389 1 6s 6s NUM cana-1164 389 2 ( ( PUNCT cana-1164 389 3 2024 2024 NUM cana-1164 389 4 ) ) PUNCT cana-1164 389 5 110 110 NUM cana-1164 389 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 389 7 also also ADV cana-1164 389 8 , , PUNCT cana-1164 389 9 ๐ดฮณ ๐ดฮณ PROPN cana-1164 389 10 + + NUM cana-1164 389 11 = = SYM cana-1164 389 12 [ [ PUNCT cana-1164 389 13 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 389 14 + + X cana-1164 389 15 0 0 NUM cana-1164 389 16 0 0 NUM cana-1164 389 17 0 0 NUM cana-1164 389 18 0 0 NUM cana-1164 389 19 0 0 NUM cana-1164 389 20 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 389 21 + + X cana-1164 389 22 0 0 NUM cana-1164 389 23 0 0 NUM cana-1164 389 24 0 0 NUM cana-1164 389 25 0 0 NUM cana-1164 389 26 0 0 NUM cana-1164 389 27 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 390 1 + + X cana-1164 390 2 0 0 NUM cana-1164 390 3 0 0 NUM cana-1164 390 4 0 0 NUM cana-1164 390 5 0 0 NUM cana-1164 390 6 0 0 NUM cana-1164 390 7 ๐ดฮณ4 ๐ดฮณ4 NOUN cana-1164 390 8 + + X cana-1164 390 9 0 0 NUM cana-1164 390 10 0 0 NUM cana-1164 390 11 0 0 NUM cana-1164 390 12 0 0 NUM cana-1164 390 13 0 0 NUM cana-1164 390 14 ๐ดฮณ5 ๐ดฮณ5 NOUN cana-1164 390 15 + + X cana-1164 390 16 ] ] PUNCT cana-1164 390 17 = = SYM cana-1164 390 18 0 0 NUM cana-1164 390 19 3 3 NUM cana-1164 390 20 6 6 NUM cana-1164 390 21 1 1 NUM cana-1164 390 22 2 2 NUM cana-1164 390 23 5 5 NUM cana-1164 390 24 4 4 NUM cana-1164 390 25 7 7 NUM cana-1164 390 26 8 8 NUM cana-1164 390 27 0 0 NUM cana-1164 390 28 1 1 NUM cana-1164 390 29 0 0 NUM cana-1164 390 30 0 0 NUM cana-1164 390 31 0 0 NUM cana-1164 390 32 0 0 NUM cana-1164 390 33 0 0 NUM cana-1164 390 34 0 0 NUM cana-1164 390 35 0 0 NUM cana-1164 390 36 0 0 NUM cana-1164 390 37 3 3 NUM cana-1164 390 38 1 1 NUM cana-1164 390 39 0 0 NUM cana-1164 390 40 0 0 NUM cana-1164 390 41 0 0 NUM cana-1164 390 42 0 0 NUM cana-1164 390 43 0 0 NUM cana-1164 390 44 0 0 NUM cana-1164 390 45 0 0 NUM cana-1164 390 46 0 0 NUM cana-1164 390 47 6 6 NUM cana-1164 390 48 1 1 NUM cana-1164 390 49 0 0 NUM cana-1164 390 50 0 0 NUM cana-1164 390 51 0 0 NUM cana-1164 390 52 0 0 NUM cana-1164 390 53 0 0 NUM cana-1164 390 54 0 0 NUM cana-1164 390 55 0 0 NUM cana-1164 390 56 0 0 NUM cana-1164 390 57 1 1 NUM cana-1164 390 58 0 0 NUM cana-1164 390 59 0 0 NUM cana-1164 390 60 0 0 NUM cana-1164 390 61 1 1 NUM cana-1164 390 62 0 0 NUM cana-1164 390 63 0 0 NUM cana-1164 390 64 0 0 NUM cana-1164 390 65 0 0 NUM cana-1164 390 66 0 0 NUM cana-1164 390 67 2 2 NUM cana-1164 390 68 0 0 NUM cana-1164 390 69 0 0 NUM cana-1164 390 70 0 0 NUM cana-1164 390 71 0 0 NUM cana-1164 390 72 0 0 NUM cana-1164 390 73 1 1 NUM cana-1164 390 74 0 0 NUM cana-1164 390 75 0 0 NUM cana-1164 390 76 0 0 NUM cana-1164 390 77 5 5 NUM cana-1164 390 78 0 0 NUM cana-1164 390 79 0 0 NUM cana-1164 390 80 0 0 NUM cana-1164 390 81 0 0 NUM cana-1164 390 82 1 1 NUM cana-1164 390 83 0 0 NUM cana-1164 390 84 0 0 NUM cana-1164 390 85 0 0 NUM cana-1164 390 86 0 0 NUM cana-1164 390 87 4 4 NUM cana-1164 390 88 0 0 NUM cana-1164 390 89 0 0 NUM cana-1164 390 90 0 0 NUM cana-1164 390 91 0 0 NUM cana-1164 390 92 0 0 NUM cana-1164 390 93 0 0 NUM cana-1164 390 94 0 0 NUM cana-1164 390 95 1 1 NUM cana-1164 390 96 0 0 NUM cana-1164 390 97 7 7 NUM cana-1164 390 98 0 0 NUM cana-1164 390 99 0 0 NUM cana-1164 390 100 0 0 NUM cana-1164 390 101 0 0 NUM cana-1164 390 102 0 0 NUM cana-1164 390 103 0 0 NUM cana-1164 390 104 1 1 NUM cana-1164 390 105 0 0 NUM cana-1164 390 106 0 0 NUM cana-1164 390 107 8 8 NUM cana-1164 390 108 0 0 NUM cana-1164 390 109 0 0 NUM cana-1164 390 110 0 0 NUM cana-1164 390 111 0 0 NUM cana-1164 390 112 0 0 NUM cana-1164 390 113 0 0 NUM cana-1164 390 114 0 0 NUM cana-1164 390 115 0 0 NUM cana-1164 390 116 1 1 NUM cana-1164 390 117 ๏ƒฉ ๏ƒฉ NUM cana-1164 390 118 ๏ƒน ๏ƒน ADJ cana-1164 390 119 ๏ƒช ๏ƒช NOUN cana-1164 390 120 ๏ƒบ ๏ƒบ PROPN cana-1164 390 121 ๏ƒช ๏ƒช NOUN cana-1164 390 122 ๏ƒบ ๏ƒบ NOUN cana-1164 390 123 ๏ƒช ๏ƒช NOUN cana-1164 390 124 ๏ƒบ ๏ƒบ NOUN cana-1164 390 125 ๏ƒช ๏ƒช NOUN cana-1164 390 126 ๏ƒบ ๏ƒบ NOUN cana-1164 390 127 ๏ƒช ๏ƒช NOUN cana-1164 390 128 ๏ƒบ ๏ƒบ NOUN cana-1164 390 129 ๏ƒช ๏ƒช NOUN cana-1164 390 130 ๏ƒบ ๏ƒบ NOUN cana-1164 390 131 ๏ƒช ๏ƒช NOUN cana-1164 390 132 ๏ƒบ ๏ƒบ NOUN cana-1164 390 133 ๏ƒช ๏ƒช NOUN cana-1164 390 134 ๏ƒบ ๏ƒบ NOUN cana-1164 390 135 ๏ƒช ๏ƒช NOUN cana-1164 390 136 ๏ƒบ ๏ƒบ NOUN cana-1164 390 137 ๏ƒช ๏ƒช NOUN cana-1164 390 138 ๏ƒบ ๏ƒบ NOUN cana-1164 390 139 ๏ƒช ๏ƒช NOUN cana-1164 390 140 ๏ƒบ ๏ƒบ NOUN cana-1164 390 141 ๏ƒช ๏ƒช NOUN cana-1164 390 142 ๏ƒบ ๏ƒบ NOUN cana-1164 390 143 ๏ƒซ ๏ƒซ NOUN cana-1164 390 144 ๏ƒป ๏ƒป PROPN cana-1164 390 145 therefore therefore ADV cana-1164 390 146 , , PUNCT cana-1164 390 147 the the DET cana-1164 390 148 characteristic characteristic ADJ cana-1164 390 149 polynomial polynomial NOUN cana-1164 390 150 of of ADP cana-1164 390 151 ๐ดฮณ ๐ดฮณ PROPN cana-1164 390 152 + + CCONJ cana-1164 390 153 is be AUX cana-1164 390 154 โˆ’๐œ†9 โˆ’๐œ†9 PROPN cana-1164 390 155 + + NOUN cana-1164 390 156 3๐œ†8 3๐œ†8 NUM cana-1164 390 157 โˆ’ โˆ’ NOUN cana-1164 391 1 ๐œ†7 ๐œ†7 NOUN cana-1164 391 2 โˆ’ โˆ’ PROPN cana-1164 391 3 5๐œ†6 5๐œ†6 NUM cana-1164 391 4 + + SYM cana-1164 391 5 5๐œ†5 5๐œ†5 NUM cana-1164 391 6 + + CCONJ cana-1164 391 7 ๐œ†4 ๐œ†4 NOUN cana-1164 391 8 โˆ’ โˆ’ PROPN cana-1164 391 9 3๐œ†3 3๐œ†3 PROPN cana-1164 392 1 + + NUM cana-1164 392 2 ๐œ†2 ๐œ†2 NOUN cana-1164 392 3 = = SYM cana-1164 392 4 โˆ’๐œ†2(๐œ† โˆ’๐œ†2(๐œ† PROPN cana-1164 392 5 โˆ’ โˆ’ PROPN cana-1164 393 1 1)5(๐œ† 1)5(๐œ† NUM cana-1164 393 2 + + CCONJ cana-1164 393 3 1)2 1)2 NUM cana-1164 393 4 and and CCONJ cana-1164 393 5 , , PUNCT cana-1164 393 6 eigenvalues eigenvalue VERB cana-1164 393 7 of of ADP cana-1164 393 8 ๐ดฮณ ๐ดฮณ PROPN cana-1164 393 9 + + CCONJ cana-1164 393 10 are be AUX cana-1164 393 11 0 0 NUM cana-1164 393 12 , , PUNCT cana-1164 393 13 0 0 NUM cana-1164 393 14 , , PUNCT cana-1164 393 15 โˆ’1,โˆ’1 โˆ’1,โˆ’1 NOUN cana-1164 393 16 , , PUNCT cana-1164 393 17 1 1 NUM cana-1164 393 18 , , PUNCT cana-1164 393 19 1 1 NUM cana-1164 393 20 , , PUNCT cana-1164 393 21 1 1 NUM cana-1164 393 22 , , PUNCT cana-1164 393 23 1 1 NUM cana-1164 393 24 , , PUNCT cana-1164 393 25 1 1 NUM cana-1164 393 26 . . PUNCT cana-1164 394 1 so so ADV cana-1164 394 2 , , PUNCT cana-1164 394 3 the the DET cana-1164 394 4 spectrum spectrum NOUN cana-1164 394 5 of of ADP cana-1164 394 6 ฮณ(9 ฮณ(9 PROPN cana-1164 394 7 , , PUNCT cana-1164 394 8 11 11 NUM cana-1164 394 9 ) ) PUNCT cana-1164 394 10 w.r.t w.r.t NOUN cana-1164 394 11 . . PUNCT cana-1164 395 1 the the DET cana-1164 395 2 adjacency adjacency NOUN cana-1164 395 3 matrix matrix NOUN cana-1164 395 4 ๐ดฮณ ๐ดฮณ PROPN cana-1164 395 5 + + CCONJ cana-1164 395 6 is be AUX cana-1164 395 7 ๐‘†๐‘๐‘’๐‘(ฮณ(9 ๐‘†๐‘๐‘’๐‘(ฮณ(9 ADJ cana-1164 395 8 , , PUNCT cana-1164 395 9 11 11 NUM cana-1164 395 10 ) ) PUNCT cana-1164 395 11 ) ) PUNCT cana-1164 396 1 = = PRON cana-1164 396 2 ( ( PUNCT cana-1164 396 3 โˆ’1 โˆ’1 NOUN cana-1164 396 4 0 0 NUM cana-1164 396 5 1 1 NUM cana-1164 396 6 2 2 NUM cana-1164 396 7 2 2 NUM cana-1164 396 8 5 5 NUM cana-1164 396 9 ) ) PUNCT cana-1164 396 10 . . PUNCT cana-1164 397 1 moreover moreover ADV cana-1164 397 2 , , PUNCT cana-1164 397 3 the the DET cana-1164 397 4 characteristic characteristic ADJ cana-1164 397 5 polynomial polynomial NOUN cana-1164 397 6 of of ADP cana-1164 397 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 397 8 โˆ’ โˆ’ PROPN cana-1164 398 1 = = PUNCT cana-1164 398 2 the the DET cana-1164 398 3 characteristic characteristic ADJ cana-1164 398 4 polynomial polynomial NOUN cana-1164 398 5 of of ADP cana-1164 398 6 ๐ดฮณ ๐ดฮณ PROPN cana-1164 398 7 + + PROPN cana-1164 398 8 . . PUNCT cana-1164 399 1 so so ADV cana-1164 399 2 , , PUNCT cana-1164 399 3 the the DET cana-1164 399 4 characteristic characteristic ADJ cana-1164 399 5 polynomial polynomial NOUN cana-1164 399 6 of of ADP cana-1164 399 7 ๐ดฮณ ๐ดฮณ PROPN cana-1164 399 8 โˆ’ โˆ’ PROPN cana-1164 399 9 is be AUX cana-1164 399 10 โˆ’๐œ†9 โˆ’๐œ†9 NOUN cana-1164 399 11 + + CCONJ cana-1164 399 12 3๐œ†8 3๐œ†8 NUM cana-1164 399 13 โˆ’ โˆ’ NOUN cana-1164 400 1 ๐œ†7 ๐œ†7 NOUN cana-1164 400 2 โˆ’ โˆ’ PROPN cana-1164 400 3 5๐œ†6 5๐œ†6 NUM cana-1164 400 4 + + SYM cana-1164 400 5 5๐œ†5 5๐œ†5 NUM cana-1164 400 6 + + CCONJ cana-1164 400 7 ๐œ†4 ๐œ†4 NOUN cana-1164 400 8 โˆ’ โˆ’ PROPN cana-1164 400 9 3๐œ†3 3๐œ†3 PROPN cana-1164 401 1 + + NUM cana-1164 401 2 ๐œ†2 ๐œ†2 NOUN cana-1164 401 3 = = SYM cana-1164 401 4 โˆ’๐œ†2(๐œ† โˆ’๐œ†2(๐œ† PROPN cana-1164 401 5 โˆ’ โˆ’ PROPN cana-1164 402 1 1)5(๐œ† 1)5(๐œ† NUM cana-1164 402 2 + + CCONJ cana-1164 402 3 1)2 1)2 NUM cana-1164 402 4 and and CCONJ cana-1164 402 5 , , PUNCT cana-1164 402 6 eigenvalues eigenvalue VERB cana-1164 402 7 of of ADP cana-1164 402 8 ๐ดฮณ ๐ดฮณ PROPN cana-1164 402 9 โˆ’ โˆ’ PROPN cana-1164 402 10 are be AUX cana-1164 402 11 0 0 NUM cana-1164 402 12 , , PUNCT cana-1164 402 13 0 0 NUM cana-1164 402 14 , , PUNCT cana-1164 402 15 โˆ’1,โˆ’1 โˆ’1,โˆ’1 NOUN cana-1164 402 16 , , PUNCT cana-1164 402 17 1 1 NUM cana-1164 402 18 , , PUNCT cana-1164 402 19 1 1 NUM cana-1164 402 20 , , PUNCT cana-1164 402 21 1 1 NUM cana-1164 402 22 , , PUNCT cana-1164 402 23 1 1 NUM cana-1164 402 24 , , PUNCT cana-1164 402 25 1 1 NUM cana-1164 402 26 . . PUNCT cana-1164 403 1 so so ADV cana-1164 403 2 , , PUNCT cana-1164 403 3 the the DET cana-1164 403 4 spectrum spectrum NOUN cana-1164 403 5 of of ADP cana-1164 403 6 ฮณ(9 ฮณ(9 PROPN cana-1164 403 7 , , PUNCT cana-1164 403 8 11 11 NUM cana-1164 403 9 ) ) PUNCT cana-1164 403 10 w.r.t w.r.t NOUN cana-1164 403 11 . . PUNCT cana-1164 404 1 the the DET cana-1164 404 2 adjacency adjacency NOUN cana-1164 404 3 matrix matrix NOUN cana-1164 404 4 ๐ดฮณ ๐ดฮณ PROPN cana-1164 404 5 โˆ’ โˆ’ PROPN cana-1164 404 6 is be AUX cana-1164 404 7 ๐‘†๐‘๐‘’๐‘(ฮณ(9 ๐‘†๐‘๐‘’๐‘(ฮณ(9 ADJ cana-1164 404 8 , , PUNCT cana-1164 404 9 11 11 NUM cana-1164 404 10 ) ) PUNCT cana-1164 404 11 ) ) PUNCT cana-1164 405 1 = = PRON cana-1164 405 2 ( ( PUNCT cana-1164 405 3 โˆ’1 โˆ’1 NOUN cana-1164 405 4 0 0 NUM cana-1164 405 5 1 1 NUM cana-1164 405 6 2 2 NUM cana-1164 405 7 2 2 NUM cana-1164 405 8 5 5 NUM cana-1164 405 9 ) ) PUNCT cana-1164 405 10 . . PUNCT cana-1164 406 1 result result VERB cana-1164 406 2 4.1 4.1 NUM cana-1164 406 3 . . PUNCT cana-1164 407 1 the the DET cana-1164 407 2 digraph digraph ADJ cana-1164 407 3 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 407 4 , , PUNCT cana-1164 407 5 ๐‘˜ ๐‘˜ NOUN cana-1164 407 6 ) ) PUNCT cana-1164 407 7 has have VERB cana-1164 407 8 ๐‘› ๐‘› DET cana-1164 407 9 eigenvalues eigenvalue NOUN cana-1164 407 10 . . PUNCT cana-1164 408 1 proof proof NOUN cana-1164 408 2 . . PUNCT cana-1164 409 1 let let VERB cana-1164 409 2 us we PRON cana-1164 409 3 consider consider VERB cana-1164 409 4 the the DET cana-1164 409 5 digraph digraph ADJ cana-1164 409 6 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 409 7 , , PUNCT cana-1164 409 8 ๐‘˜ ๐‘˜ NOUN cana-1164 409 9 ) ) PUNCT cana-1164 409 10 . . PUNCT cana-1164 410 1 clearly clearly ADV cana-1164 410 2 |๐‘‰(ฮณ)| |๐‘‰(ฮณ)| NOUN cana-1164 410 3 = = PRON cana-1164 410 4 ๐‘›. ๐‘›. NOUN cana-1164 410 5 the the DET cana-1164 410 6 characteristic characteristic ADJ cana-1164 410 7 polynomial polynomial NOUN cana-1164 410 8 of of ADP cana-1164 410 9 the the DET cana-1164 410 10 digraph digraph ADJ cana-1164 410 11 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 410 12 , , PUNCT cana-1164 410 13 ๐‘˜ ๐‘˜ NOUN cana-1164 410 14 ) ) PUNCT cana-1164 410 15 is be AUX cana-1164 410 16 given give VERB cana-1164 410 17 as as ADP cana-1164 410 18 ๐‘ƒฮณ(๐œ† ๐‘ƒฮณ(๐œ† NOUN cana-1164 410 19 ) ) PUNCT cana-1164 410 20 = = PUNCT cana-1164 411 1 |๐ดฮณ |๐ดฮณ NOUN cana-1164 411 2 + + CCONJ cana-1164 411 3 โˆ’ โˆ’ PROPN cana-1164 411 4 ๐œ†๐ผ๐‘›| ๐œ†๐ผ๐‘›| NOUN cana-1164 411 5 , , PUNCT cana-1164 411 6 which which PRON cana-1164 411 7 is be AUX cana-1164 411 8 a a DET cana-1164 411 9 polynomial polynomial NOUN cana-1164 411 10 of of ADP cana-1164 411 11 degree degree NOUN cana-1164 411 12 ๐‘› ๐‘› PRON cana-1164 411 13 in in ADP cana-1164 411 14 ๐œ†. ๐œ†. NOUN cana-1164 411 15 by by ADP cana-1164 411 16 the the DET cana-1164 411 17 fundamental fundamental ADJ cana-1164 411 18 theorem theorem NOUN cana-1164 411 19 of of ADP cana-1164 411 20 algebra algebra PROPN cana-1164 411 21 , , PUNCT cana-1164 411 22 we we PRON cana-1164 411 23 know know VERB cana-1164 411 24 that that SCONJ cana-1164 411 25 every every DET cana-1164 411 26 polynomial polynomial NOUN cana-1164 411 27 of of ADP cana-1164 411 28 degree degree NOUN cana-1164 411 29 ๐‘› ๐‘› PRON cana-1164 411 30 possesses possess VERB cana-1164 411 31 precisely precisely ADV cana-1164 411 32 ๐‘› ๐‘› PRON cana-1164 411 33 roots root NOUN cana-1164 411 34 , , PUNCT cana-1164 411 35 taking take VERB cana-1164 411 36 into into ADP cana-1164 411 37 account account NOUN cana-1164 411 38 their their PRON cana-1164 411 39 multiplicities multiplicity NOUN cana-1164 411 40 within within ADP cana-1164 411 41 the the DET cana-1164 411 42 complex complex ADJ cana-1164 411 43 number number NOUN cana-1164 411 44 field field NOUN cana-1164 411 45 . . PUNCT cana-1164 412 1 hence hence ADV cana-1164 412 2 , , PUNCT cana-1164 412 3 ๐‘ƒฮณ(๐œ† ๐‘ƒฮณ(๐œ† NOUN cana-1164 412 4 ) ) PUNCT cana-1164 412 5 has have VERB cana-1164 412 6 ๐‘› ๐‘› DET cana-1164 412 7 roots root NOUN cana-1164 412 8 . . PUNCT cana-1164 413 1 this this PRON cana-1164 413 2 shows show VERB cana-1164 413 3 that that SCONJ cana-1164 413 4 the the DET cana-1164 413 5 digraph digraph ADJ cana-1164 413 6 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 413 7 , , PUNCT cana-1164 413 8 ๐‘˜ ๐‘˜ NOUN cana-1164 413 9 ) ) PUNCT cana-1164 413 10 has have VERB cana-1164 413 11 ๐‘›eigenvalues ๐‘›eigenvalue NOUN cana-1164 413 12 . . PUNCT cana-1164 414 1 result result VERB cana-1164 414 2 4.2 4.2 NUM cana-1164 414 3 . . PUNCT cana-1164 415 1 if if SCONJ cana-1164 415 2 ฮณ1 ฮณ1 PROPN cana-1164 415 3 , , PUNCT cana-1164 415 4 ฮณ2 ฮณ2 PROPN cana-1164 415 5 , , PUNCT cana-1164 415 6 ฮณ3 ฮณ3 NOUN cana-1164 415 7 , , PUNCT cana-1164 415 8 โ‹ฏ โ‹ฏ PROPN cana-1164 415 9 , , PUNCT cana-1164 415 10 ฮณ๐‘  ฮณ๐‘  PRON cana-1164 415 11 are be AUX cana-1164 415 12 the the DET cana-1164 415 13 ๐‘ -components ๐‘ -component NOUN cana-1164 415 14 of of ADP cana-1164 415 15 the the DET cana-1164 415 16 digraph digraph ADJ cana-1164 415 17 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 415 18 , , PUNCT cana-1164 415 19 ๐‘˜ ๐‘˜ NOUN cana-1164 415 20 ) ) PUNCT cana-1164 415 21 then then ADV cana-1164 415 22 ๐‘ƒฮณ(๐œ† ๐‘ƒฮณ(๐œ† NOUN cana-1164 415 23 ) ) PUNCT cana-1164 415 24 = = SYM cana-1164 415 25 ๐‘ƒฮณ1(๐œ† ๐‘ƒฮณ1(๐œ† PROPN cana-1164 415 26 ) ) PUNCT cana-1164 415 27 โ‹… โ‹… PROPN cana-1164 415 28 ๐‘ƒฮณ2(๐œ† ๐‘ƒฮณ2(๐œ† NOUN cana-1164 415 29 ) ) PUNCT cana-1164 415 30 โ‹… โ‹… X cana-1164 415 31 ๐‘ƒฮณ3(๐œ†)โ‹ฏ ๐‘ƒฮณ3(๐œ†)โ‹ฏ X cana-1164 415 32 โ‹… โ‹… PROPN cana-1164 415 33 ๐‘ƒฮณ๐‘ (๐œ† ๐‘ƒฮณ๐‘ (๐œ† PROPN cana-1164 415 34 ) ) PUNCT cana-1164 415 35 where where SCONJ cana-1164 415 36 ๐‘ƒฮณ(๐œ† ๐‘ƒฮณ(๐œ† NOUN cana-1164 415 37 ) ) PUNCT cana-1164 415 38 , , PUNCT cana-1164 415 39 ๐‘ƒฮณ1(๐œ† ๐‘ƒฮณ1(๐œ† PROPN cana-1164 415 40 ) ) PUNCT cana-1164 415 41 , , PUNCT cana-1164 415 42 ๐‘ƒฮณ2(๐œ† ๐‘ƒฮณ2(๐œ† PROPN cana-1164 415 43 ) ) PUNCT cana-1164 415 44 , , PUNCT cana-1164 415 45 ๐‘ƒฮณ3(๐œ†),โ‹ฏ ๐‘ƒฮณ3(๐œ†),โ‹ฏ NOUN cana-1164 415 46 , , PUNCT cana-1164 415 47 ๐‘ƒฮณ๐‘ (๐œ† ๐‘ƒฮณ๐‘ (๐œ† PROPN cana-1164 415 48 ) ) PUNCT cana-1164 415 49 are be AUX cana-1164 415 50 the the DET cana-1164 415 51 characteristic characteristic ADJ cana-1164 415 52 polynomials polynomial NOUN cana-1164 415 53 of of ADP cana-1164 415 54 the the DET cana-1164 415 55 digraphs digraphs ADJ cana-1164 415 56 ฮณ ฮณ PROPN cana-1164 415 57 , , PUNCT cana-1164 415 58 ฮณ1 ฮณ1 NOUN cana-1164 415 59 , , PUNCT cana-1164 415 60 ฮณ2 ฮณ2 PROPN cana-1164 415 61 , , PUNCT cana-1164 415 62 ฮณ3,โ‹ฏ ฮณ3,โ‹ฏ INTJ cana-1164 415 63 , , PUNCT cana-1164 415 64 ฮณ๐‘  ฮณ๐‘  X cana-1164 415 65 respectively respectively ADV cana-1164 415 66 . . PUNCT cana-1164 416 1 proof proof NOUN cana-1164 416 2 . . PUNCT cana-1164 417 1 let let VERB cana-1164 417 2 us we PRON cana-1164 417 3 consider consider VERB cana-1164 417 4 the the DET cana-1164 417 5 digraph digraph ADJ cana-1164 417 6 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 417 7 , , PUNCT cana-1164 417 8 ๐‘˜ ๐‘˜ NOUN cana-1164 417 9 ) ) PUNCT cana-1164 417 10 , , PUNCT cana-1164 417 11 where where SCONJ cana-1164 417 12 |๐‘‰(ฮณ)| |๐‘‰(ฮณ)| PROPN cana-1164 417 13 = = PRON cana-1164 417 14 ๐‘›. ๐‘›. NOUN cana-1164 417 15 the the DET cana-1164 417 16 characteristic characteristic ADJ cana-1164 417 17 polynomial polynomial NOUN cana-1164 417 18 of of ADP cana-1164 417 19 the the DET cana-1164 417 20 digraph digraph ADJ cana-1164 417 21 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 417 22 , , PUNCT cana-1164 417 23 ๐‘˜ ๐‘˜ NOUN cana-1164 417 24 ) ) PUNCT cana-1164 417 25 is be AUX cana-1164 417 26 given give VERB cana-1164 417 27 as as ADP cana-1164 417 28 ๐‘ƒฮณ(๐œ† ๐‘ƒฮณ(๐œ† NOUN cana-1164 417 29 ) ) PUNCT cana-1164 417 30 = = PUNCT cana-1164 418 1 |๐ดฮณ |๐ดฮณ NOUN cana-1164 418 2 + + CCONJ cana-1164 418 3 โˆ’ โˆ’ PROPN cana-1164 418 4 ๐œ†๐ผ๐‘›| ๐œ†๐ผ๐‘›| NOUN cana-1164 418 5 . . PUNCT cana-1164 419 1 let let VERB cana-1164 419 2 ๐ดฮณ1 ๐ดฮณ1 NOUN cana-1164 419 3 + + CCONJ cana-1164 419 4 , , PUNCT cana-1164 419 5 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 419 6 + + CCONJ cana-1164 419 7 , , PUNCT cana-1164 419 8 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 419 9 + + X cana-1164 419 10 , , PUNCT cana-1164 419 11 โ‹ฏ โ‹ฏ PROPN cana-1164 419 12 , , PUNCT cana-1164 419 13 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 419 14 + + CCONJ cana-1164 419 15 be be AUX cana-1164 419 16 the the DET cana-1164 419 17 out out ADJ cana-1164 419 18 - - PUNCT cana-1164 419 19 adjacency adjacency NOUN cana-1164 419 20 matrices matrix NOUN cana-1164 419 21 of of ADP cana-1164 419 22 the the DET cana-1164 419 23 component component NOUN cana-1164 419 24 digraphs digraph NOUN cana-1164 419 25 ฮณ1 ฮณ1 NOUN cana-1164 419 26 , , PUNCT cana-1164 419 27 ฮณ2 ฮณ2 PROPN cana-1164 419 28 , , PUNCT cana-1164 419 29 ฮณ3 ฮณ3 NOUN cana-1164 419 30 , , PUNCT cana-1164 419 31 โ‹ฏ โ‹ฏ PROPN cana-1164 419 32 , , PUNCT cana-1164 419 33 ฮณ๐‘  ฮณ๐‘  X cana-1164 419 34 respectively respectively ADV cana-1164 419 35 . . PUNCT cana-1164 420 1 also also ADV cana-1164 420 2 , , PUNCT cana-1164 420 3 let let VERB cana-1164 420 4 |๐‘‰(ฮณ๐‘–(๐‘› |๐‘‰(ฮณ๐‘–(๐‘› ADV cana-1164 420 5 , , PUNCT cana-1164 420 6 ๐‘˜))| ๐‘˜))| PROPN cana-1164 420 7 = = SYM cana-1164 420 8 ๐‘›๐‘– ๐‘›๐‘– PROPN cana-1164 420 9 , , PUNCT cana-1164 420 10 1 1 NUM cana-1164 420 11 โ‰ค โ‰ค NUM cana-1164 420 12 ๐‘– ๐‘– SYM cana-1164 420 13 โ‰ค โ‰ค NUM cana-1164 420 14 ๐‘  ๐‘  ADP cana-1164 420 15 such such ADJ cana-1164 420 16 that that DET cana-1164 420 17 โˆ‘๐‘ ๐‘–=1 โˆ‘๐‘ ๐‘–=1 PROPN cana-1164 420 18 ๐‘›๐‘– ๐‘›๐‘– NOUN cana-1164 420 19 = = PUNCT cana-1164 421 1 ๐‘›. ๐‘›. NOUN cana-1164 421 2 then then ADV cana-1164 421 3 we we PRON cana-1164 421 4 have have VERB cana-1164 421 5 communications communication NOUN cana-1164 421 6 on on ADP cana-1164 421 7 applied apply VERB cana-1164 421 8 nonlinear nonlinear ADJ cana-1164 421 9 analysis analysis NOUN cana-1164 421 10 issn issn NOUN cana-1164 421 11 : : PUNCT cana-1164 421 12 1074 1074 NUM cana-1164 421 13 - - PUNCT cana-1164 421 14 133x 133x NUM cana-1164 421 15 vol vol NOUN cana-1164 421 16 31 31 NUM cana-1164 421 17 no no NOUN cana-1164 421 18 . . PUNCT cana-1164 422 1 6s 6s NUM cana-1164 422 2 ( ( PUNCT cana-1164 422 3 2024 2024 NUM cana-1164 422 4 ) ) PUNCT cana-1164 422 5 111 111 NUM cana-1164 422 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 423 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 423 2 + + NOUN cana-1164 423 3 = = SYM cana-1164 423 4 [ [ PUNCT cana-1164 423 5 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 423 6 + + X cana-1164 423 7 0 0 NUM cana-1164 423 8 0 0 NUM cana-1164 423 9 โ‹ฏ โ‹ฏ ADP cana-1164 423 10 0 0 NUM cana-1164 423 11 0 0 NUM cana-1164 423 12 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 424 1 + + X cana-1164 424 2 0 0 NUM cana-1164 424 3 โ‹ฏ โ‹ฏ NOUN cana-1164 424 4 0 0 NUM cana-1164 424 5 โ‹ฎ โ‹ฎ NOUN cana-1164 424 6 โ‹ฎ โ‹ฎ ADJ cana-1164 424 7 โ‹ฎ โ‹ฎ NOUN cana-1164 424 8 โ‹ฑ โ‹ฑ PUNCT cana-1164 424 9 โ‹ฎ โ‹ฎ NOUN cana-1164 424 10 0 0 NUM cana-1164 424 11 0 0 NUM cana-1164 424 12 0 0 NUM cana-1164 424 13 โ‹ฏ โ‹ฏ PROPN cana-1164 424 14 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 424 15 + + CCONJ cana-1164 424 16 ] ] X cana-1164 424 17 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-1164 424 18 , , PUNCT cana-1164 424 19 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ ADV cana-1164 424 20 + + CCONJ cana-1164 424 21 โˆ’ โˆ’ PROPN cana-1164 424 22 ๐œ†๐ผ๐‘› ๐œ†๐ผ๐‘› NOUN cana-1164 424 23 ) ) PUNCT cana-1164 424 24 = = PUNCT cana-1164 424 25 ๐‘‘๐‘’๐‘ก(๐ดฮณ1 ๐‘‘๐‘’๐‘ก(๐ดฮณ1 NOUN cana-1164 424 26 + + CCONJ cana-1164 424 27 โˆ’ โˆ’ PROPN cana-1164 424 28 ๐œ†๐ผ๐‘›1 ๐œ†๐ผ๐‘›1 PROPN cana-1164 424 29 ) ) PUNCT cana-1164 424 30 โ‹… โ‹… X cana-1164 424 31 ๐‘‘๐‘’๐‘ก(๐ดฮณ2 ๐‘‘๐‘’๐‘ก(๐ดฮณ2 ADJ cana-1164 424 32 + + CCONJ cana-1164 424 33 โˆ’ โˆ’ PROPN cana-1164 424 34 ๐œ†๐ผ๐‘›2 ๐œ†๐ผ๐‘›2 PROPN cana-1164 424 35 ) ) PUNCT cana-1164 424 36 โ‹… โ‹… PROPN cana-1164 424 37 ๐‘‘๐‘’๐‘ก(๐ดฮณ3 ๐‘‘๐‘’๐‘ก(๐ดฮณ3 NOUN cana-1164 425 1 + + CCONJ cana-1164 426 1 โˆ’ โˆ’ PROPN cana-1164 426 2 ๐œ†๐ผ๐‘›3)โ‹ฏ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘  ๐œ†๐ผ๐‘›3)โ‹ฏ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘  PROPN cana-1164 427 1 + + CCONJ cana-1164 427 2 โˆ’ โˆ’ PROPN cana-1164 427 3 ๐œ†๐ผ๐‘›๐‘  ๐œ†๐ผ๐‘›๐‘  PROPN cana-1164 427 4 ) ) PUNCT cana-1164 427 5 ๐‘–. ๐‘–. ADJ cana-1164 427 6 ๐‘’. ๐‘’. PROPN cana-1164 428 1 ๐‘ƒฮณ(๐œ† ๐‘ƒฮณ(๐œ† NOUN cana-1164 428 2 ) ) PUNCT cana-1164 428 3 = = SYM cana-1164 428 4 ๐‘ƒฮณ1(๐œ† ๐‘ƒฮณ1(๐œ† PROPN cana-1164 428 5 ) ) PUNCT cana-1164 428 6 โ‹… โ‹… PROPN cana-1164 428 7 ๐‘ƒฮณ2(๐œ† ๐‘ƒฮณ2(๐œ† NOUN cana-1164 428 8 ) ) PUNCT cana-1164 428 9 โ‹… โ‹… X cana-1164 428 10 ๐‘ƒฮณ3(๐œ†)โ‹ฏ ๐‘ƒฮณ3(๐œ†)โ‹ฏ X cana-1164 428 11 โ‹… โ‹… PROPN cana-1164 428 12 ๐‘ƒฮณ๐‘ (๐œ† ๐‘ƒฮณ๐‘ (๐œ† PROPN cana-1164 428 13 ) ) PUNCT cana-1164 428 14 . . PUNCT cana-1164 429 1 result result VERB cana-1164 429 2 4.3 4.3 NUM cana-1164 429 3 . . PUNCT cana-1164 430 1 let let VERB cana-1164 430 2 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 430 3 , , PUNCT cana-1164 430 4 ๐‘˜ ๐‘˜ NOUN cana-1164 430 5 ) ) PUNCT cana-1164 430 6 be be VERB cana-1164 430 7 a a DET cana-1164 430 8 digraph digraph NOUN cana-1164 430 9 with with ADP cana-1164 430 10 ๐‘ -components ๐‘ -component NOUN cana-1164 430 11 ฮณ1 ฮณ1 PROPN cana-1164 430 12 , , PUNCT cana-1164 430 13 ฮณ2 ฮณ2 PROPN cana-1164 430 14 , , PUNCT cana-1164 430 15 ฮณ3 ฮณ3 NOUN cana-1164 430 16 , , PUNCT cana-1164 430 17 โ‹ฏ โ‹ฏ PROPN cana-1164 430 18 , , PUNCT cana-1164 431 1 ฮณ๐‘  ฮณ๐‘  PRON cana-1164 431 2 then then ADV cana-1164 431 3 the the DET cana-1164 431 4 spectrum spectrum NOUN cana-1164 431 5 of of ADP cana-1164 431 6 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 431 7 , , PUNCT cana-1164 431 8 ๐‘˜ ๐‘˜ NOUN cana-1164 431 9 ) ) PUNCT cana-1164 431 10 is be AUX cana-1164 431 11 the the DET cana-1164 431 12 union union NOUN cana-1164 431 13 of of ADP cana-1164 431 14 the the DET cana-1164 431 15 spectra spectra NOUN cana-1164 431 16 of of ADP cana-1164 431 17 ฮณ1 ฮณ1 PROPN cana-1164 431 18 , , PUNCT cana-1164 431 19 ฮณ2 ฮณ2 PROPN cana-1164 431 20 , , PUNCT cana-1164 431 21 ฮณ3 ฮณ3 NOUN cana-1164 431 22 , , PUNCT cana-1164 431 23 โ‹ฏ โ‹ฏ NOUN cana-1164 431 24 , , PUNCT cana-1164 431 25 ฮณ๐‘ . ฮณ๐‘ . NOUN cana-1164 431 26 proof proof NOUN cana-1164 431 27 . . PUNCT cana-1164 432 1 to to PART cana-1164 432 2 prove prove VERB cana-1164 432 3 this this PRON cana-1164 432 4 , , PUNCT cana-1164 432 5 we we PRON cana-1164 432 6 try try VERB cana-1164 432 7 to to PART cana-1164 432 8 show show VERB cana-1164 432 9 that that SCONJ cana-1164 432 10 each each DET cana-1164 432 11 eigenvalue eigenvalue NOUN cana-1164 432 12 of of ADP cana-1164 432 13 ฮณ ฮณ PROPN cana-1164 432 14 is be AUX cana-1164 432 15 also also ADV cana-1164 432 16 an an DET cana-1164 432 17 eigenvalue eigenvalue NOUN cana-1164 432 18 of of ADP cana-1164 432 19 at at ADV cana-1164 432 20 least least ADJ cana-1164 432 21 one one NUM cana-1164 432 22 of of ADP cana-1164 432 23 the the DET cana-1164 432 24 components component NOUN cana-1164 432 25 ฮณ๐‘– ฮณ๐‘– PROPN cana-1164 432 26 and and CCONJ cana-1164 432 27 conversely conversely ADV cana-1164 432 28 , , PUNCT cana-1164 432 29 each each DET cana-1164 432 30 eigenvalue eigenvalue NOUN cana-1164 432 31 of of ADP cana-1164 432 32 ฮณ๐‘– ฮณ๐‘– PRON cana-1164 432 33 is be AUX cana-1164 432 34 an an DET cana-1164 432 35 eigenvalue eigenvalue NOUN cana-1164 432 36 of of ADP cana-1164 432 37 ฮณ ฮณ NOUN cana-1164 432 38 ; ; PUNCT cana-1164 432 39 1 1 NUM cana-1164 432 40 โ‰ค โ‰ค NUM cana-1164 432 41 ๐‘– ๐‘– PUNCT cana-1164 432 42 โ‰ค โ‰ค NOUN cana-1164 432 43 ๐‘  ๐‘  AUX cana-1164 432 44 let let VERB cana-1164 432 45 ๐ดฮณ ๐ดฮณ PROPN cana-1164 432 46 + + PROPN cana-1164 432 47 , , PUNCT cana-1164 432 48 ๐ดฮณ1 ๐ดฮณ1 VERB cana-1164 432 49 + + X cana-1164 432 50 , , PUNCT cana-1164 432 51 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 432 52 + + CCONJ cana-1164 432 53 , , PUNCT cana-1164 432 54 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 432 55 + + X cana-1164 432 56 , , PUNCT cana-1164 432 57 โ‹ฏ โ‹ฏ PROPN cana-1164 432 58 , , PUNCT cana-1164 432 59 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 432 60 + + CCONJ cana-1164 432 61 be be AUX cana-1164 432 62 the the DET cana-1164 432 63 out out ADJ cana-1164 432 64 - - PUNCT cana-1164 432 65 adjacency adjacency NOUN cana-1164 432 66 matrices matrix NOUN cana-1164 432 67 of of ADP cana-1164 432 68 the the DET cana-1164 432 69 digraphs digraphs ADJ cana-1164 432 70 ฮณ ฮณ PROPN cana-1164 432 71 , , PUNCT cana-1164 432 72 ฮณ1 ฮณ1 NOUN cana-1164 432 73 , , PUNCT cana-1164 432 74 ฮณ2 ฮณ2 PROPN cana-1164 432 75 , , PUNCT cana-1164 432 76 ฮณ3 ฮณ3 NOUN cana-1164 432 77 , , PUNCT cana-1164 432 78 โ‹ฏ โ‹ฏ PROPN cana-1164 432 79 , , PUNCT cana-1164 432 80 ฮณ๐‘  ฮณ๐‘  X cana-1164 432 81 respectively respectively ADV cana-1164 432 82 . . PUNCT cana-1164 433 1 then then ADV cana-1164 433 2 we we PRON cana-1164 433 3 have have VERB cana-1164 433 4 ๐ดฮณ ๐ดฮณ PROPN cana-1164 433 5 + + NOUN cana-1164 433 6 = = SYM cana-1164 433 7 [ [ PUNCT cana-1164 433 8 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 433 9 + + X cana-1164 433 10 0 0 NUM cana-1164 433 11 0 0 NUM cana-1164 433 12 โ‹ฏ โ‹ฏ ADP cana-1164 433 13 0 0 NUM cana-1164 433 14 0 0 NUM cana-1164 433 15 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 434 1 + + X cana-1164 434 2 0 0 NUM cana-1164 434 3 โ‹ฏ โ‹ฏ NOUN cana-1164 434 4 0 0 NUM cana-1164 434 5 โ‹ฎ โ‹ฎ NOUN cana-1164 434 6 โ‹ฎ โ‹ฎ ADJ cana-1164 434 7 โ‹ฎ โ‹ฎ NOUN cana-1164 434 8 โ‹ฑ โ‹ฑ PUNCT cana-1164 434 9 โ‹ฎ โ‹ฎ NOUN cana-1164 434 10 0 0 NUM cana-1164 434 11 0 0 NUM cana-1164 434 12 0 0 NUM cana-1164 434 13 โ‹ฏ โ‹ฏ PROPN cana-1164 434 14 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 434 15 + + CCONJ cana-1164 434 16 ] ] X cana-1164 434 17 let let VERB cana-1164 434 18 ๐œ† ๐œ† NOUN cana-1164 434 19 be be AUX cana-1164 434 20 an an DET cana-1164 434 21 eigenvalue eigenvalue NOUN cana-1164 434 22 of of ADP cana-1164 434 23 the the DET cana-1164 434 24 digraph digraph ADJ cana-1164 434 25 ฮณ ฮณ NOUN cana-1164 434 26 and and CCONJ cana-1164 434 27 let let VERB cana-1164 434 28 ๐‘ฃ ๐‘ฃ PART cana-1164 434 29 be be AUX cana-1164 434 30 the the DET cana-1164 434 31 corresponding corresponding ADJ cana-1164 434 32 eigenvector eigenvector NOUN cana-1164 434 33 , , PUNCT cana-1164 434 34 then then ADV cana-1164 434 35 ๐ดฮณ ๐ดฮณ PROPN cana-1164 434 36 + + CCONJ cana-1164 434 37 โ‹… โ‹… PROPN cana-1164 434 38 ๐‘ฃ ๐‘ฃ X cana-1164 434 39 = = PUNCT cana-1164 434 40 ๐œ† ๐œ† DET cana-1164 434 41 โ‹… โ‹… PROPN cana-1164 434 42 ๐‘ฃ ๐‘ฃ ADP cana-1164 434 43 โ‡’ โ‡’ NOUN cana-1164 434 44 [ [ PUNCT cana-1164 434 45 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 434 46 + + X cana-1164 434 47 0 0 NUM cana-1164 434 48 0 0 NUM cana-1164 434 49 โ‹ฏ โ‹ฏ ADP cana-1164 434 50 0 0 NUM cana-1164 434 51 0 0 NUM cana-1164 434 52 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 434 53 + + X cana-1164 434 54 0 0 NUM cana-1164 434 55 โ‹ฏ โ‹ฏ NOUN cana-1164 434 56 0 0 NUM cana-1164 434 57 โ‹ฎ โ‹ฎ NOUN cana-1164 434 58 โ‹ฎ โ‹ฎ ADJ cana-1164 434 59 โ‹ฎ โ‹ฎ NOUN cana-1164 434 60 โ‹ฑ โ‹ฑ PUNCT cana-1164 434 61 โ‹ฎ โ‹ฎ NOUN cana-1164 434 62 0 0 NUM cana-1164 434 63 0 0 NUM cana-1164 434 64 0 0 NUM cana-1164 434 65 โ‹ฏ โ‹ฏ VERB cana-1164 434 66 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 434 67 + + CCONJ cana-1164 434 68 ] ] PUNCT cana-1164 434 69 โ‹… โ‹… PROPN cana-1164 434 70 [ [ PUNCT cana-1164 434 71 ๐‘ฃ1 ๐‘ฃ1 NOUN cana-1164 434 72 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 434 73 โ‹ฎ โ‹ฎ NOUN cana-1164 434 74 ๐‘ฃ๐‘  ๐‘ฃ๐‘  X cana-1164 434 75 ] ] PUNCT cana-1164 434 76 = = PUNCT cana-1164 434 77 ๐œ† ๐œ† DET cana-1164 434 78 โ‹… โ‹… PROPN cana-1164 434 79 [ [ PUNCT cana-1164 434 80 ๐‘ฃ1 ๐‘ฃ1 PROPN cana-1164 434 81 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 434 82 โ‹ฎ โ‹ฎ NOUN cana-1164 434 83 ๐‘ฃ๐‘  ๐‘ฃ๐‘  X cana-1164 434 84 ] ] X cana-1164 434 85 , , PUNCT cana-1164 434 86 where where SCONJ cana-1164 434 87 ๐‘ฃ ๐‘ฃ ADP cana-1164 434 88 = = PRON cana-1164 434 89 [ [ PUNCT cana-1164 434 90 ๐‘ฃ1 ๐‘ฃ1 PROPN cana-1164 434 91 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 434 92 โ‹ฎ โ‹ฎ NOUN cana-1164 434 93 ๐‘ฃ๐‘  ๐‘ฃ๐‘  X cana-1164 434 94 ] ] PUNCT cana-1164 434 95 is be AUX cana-1164 434 96 an an DET cana-1164 434 97 eigenvector eigenvector NOUN cana-1164 434 98 of of ADP cana-1164 434 99 ฮณ ฮณ PROPN cana-1164 434 100 . . PROPN cana-1164 434 101 โ‡’ โ‡’ PROPN cana-1164 434 102 ๐ดฮณ1 ๐ดฮณ1 VERB cana-1164 434 103 + + CCONJ cana-1164 434 104 โ‹… โ‹… PROPN cana-1164 434 105 ๐‘ฃ1 ๐‘ฃ1 NOUN cana-1164 434 106 = = PROPN cana-1164 434 107 ๐œ† ๐œ† DET cana-1164 434 108 โ‹… โ‹… PROPN cana-1164 434 109 ๐‘ฃ1 ๐‘ฃ1 NOUN cana-1164 434 110 , , PUNCT cana-1164 434 111 ๐ดฮณ2 ๐ดฮณ2 VERB cana-1164 434 112 + + CCONJ cana-1164 434 113 โ‹… โ‹… PROPN cana-1164 434 114 ๐‘ฃ2 ๐‘ฃ2 NOUN cana-1164 434 115 = = PUNCT cana-1164 434 116 ๐œ† ๐œ† PROPN cana-1164 434 117 โ‹… โ‹… PROPN cana-1164 434 118 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1164 434 119 , , PUNCT cana-1164 434 120 โ‹ฏ โ‹ฏ PROPN cana-1164 434 121 , , PUNCT cana-1164 434 122 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 434 123 + + CCONJ cana-1164 434 124 โ‹… โ‹… PROPN cana-1164 434 125 ๐‘ฃ๐‘  ๐‘ฃ๐‘  X cana-1164 434 126 = = PUNCT cana-1164 434 127 ๐œ† ๐œ† PROPN cana-1164 434 128 โ‹… โ‹… PROPN cana-1164 434 129 ๐‘ฃ๐‘  ๐‘ฃ๐‘  INTJ cana-1164 434 130 โ‡’ โ‡’ PROPN cana-1164 434 131 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 434 132 + + CCONJ cana-1164 434 133 โ‹… โ‹… PROPN cana-1164 434 134 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 434 135 = = SYM cana-1164 434 136 ๐œ† ๐œ† DET cana-1164 434 137 โ‹… โ‹… PROPN cana-1164 434 138 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADV cana-1164 434 139 ; ; PUNCT cana-1164 434 140 ๐‘– ๐‘– SYM cana-1164 434 141 = = SYM cana-1164 434 142 1 1 NUM cana-1164 434 143 , , PUNCT cana-1164 434 144 2,โ‹ฏ 2,โ‹ฏ NUM cana-1164 434 145 , , PUNCT cana-1164 434 146 ๐‘ . ๐‘ . VERB cana-1164 434 147 this this PRON cana-1164 434 148 shows show VERB cana-1164 434 149 that that SCONJ cana-1164 434 150 ๐œ† ๐œ† PRON cana-1164 434 151 is be AUX cana-1164 434 152 an an DET cana-1164 434 153 eigenvalue eigenvalue NOUN cana-1164 434 154 of of ADP cana-1164 434 155 the the DET cana-1164 434 156 component component NOUN cana-1164 434 157 digraphs digraph VERB cana-1164 434 158 ฮณ๐‘– ฮณ๐‘– PRON cana-1164 434 159 with with ADP cana-1164 434 160 eigenvalue eigenvalue PROPN cana-1164 434 161 ๐‘ฃ๐‘–. ๐‘ฃ๐‘–. NOUN cana-1164 434 162 since since SCONJ cana-1164 434 163 ๐œ† ๐œ† PRON cana-1164 434 164 is be AUX cana-1164 434 165 an an DET cana-1164 434 166 eigenvalue eigenvalue NOUN cana-1164 434 167 of of ADP cana-1164 434 168 at at ADV cana-1164 434 169 least least ADJ cana-1164 434 170 one one NUM cana-1164 434 171 of of ADP cana-1164 434 172 the the DET cana-1164 434 173 components component NOUN cana-1164 434 174 ฮณ๐‘– ฮณ๐‘– NUM cana-1164 434 175 , , PUNCT cana-1164 434 176 it it PRON cana-1164 434 177 is be AUX cana-1164 434 178 included include VERB cana-1164 434 179 in in ADP cana-1164 434 180 the the DET cana-1164 434 181 spectrum spectrum NOUN cana-1164 434 182 of of ADP cana-1164 434 183 ฮณ ฮณ PROPN cana-1164 434 184 . . PUNCT cana-1164 435 1 conversely conversely ADV cana-1164 435 2 , , PUNCT cana-1164 435 3 let let VERB cana-1164 435 4 ๐œ† ๐œ† NOUN cana-1164 435 5 be be AUX cana-1164 435 6 an an DET cana-1164 435 7 eigenvalue eigenvalue NOUN cana-1164 435 8 of of ADP cana-1164 435 9 a a DET cana-1164 435 10 component component NOUN cana-1164 435 11 ฮณ๐‘– ฮณ๐‘– NOUN cana-1164 435 12 , , PUNCT cana-1164 435 13 then then ADV cana-1164 435 14 there there PRON cana-1164 435 15 exists exist VERB cana-1164 435 16 a a DET cana-1164 435 17 non non ADJ cana-1164 435 18 - - ADJ cana-1164 435 19 zero zero NUM cana-1164 435 20 vector vector NOUN cana-1164 435 21 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 435 22 such such ADJ cana-1164 435 23 that that SCONJ cana-1164 435 24 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 435 25 + + CCONJ cana-1164 435 26 โ‹… โ‹… PROPN cana-1164 435 27 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 435 28 = = SYM cana-1164 435 29 ๐œ† ๐œ† DET cana-1164 435 30 โ‹… โ‹… PROPN cana-1164 435 31 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADV cana-1164 435 32 ; ; PUNCT cana-1164 435 33 1 1 NUM cana-1164 435 34 โ‰ค โ‰ค NUM cana-1164 435 35 ๐‘– ๐‘– SYM cana-1164 435 36 โ‰ค โ‰ค NUM cana-1164 435 37 ๐‘  ๐‘  ADP cana-1164 435 38 โ‡’ โ‡’ NOUN cana-1164 435 39 [ [ PUNCT cana-1164 435 40 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 435 41 + + X cana-1164 435 42 0 0 NUM cana-1164 435 43 0 0 NUM cana-1164 435 44 โ‹ฏ โ‹ฏ ADP cana-1164 435 45 0 0 NUM cana-1164 435 46 0 0 NUM cana-1164 435 47 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 435 48 + + X cana-1164 435 49 0 0 NUM cana-1164 435 50 โ‹ฏ โ‹ฏ NOUN cana-1164 435 51 0 0 NUM cana-1164 435 52 โ‹ฎ โ‹ฎ NOUN cana-1164 435 53 โ‹ฎ โ‹ฎ ADJ cana-1164 435 54 โ‹ฎ โ‹ฎ NOUN cana-1164 435 55 โ‹ฑ โ‹ฑ PUNCT cana-1164 435 56 โ‹ฎ โ‹ฎ NOUN cana-1164 435 57 0 0 NUM cana-1164 435 58 0 0 NUM cana-1164 435 59 0 0 NUM cana-1164 435 60 โ‹ฏ โ‹ฏ VERB cana-1164 435 61 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 435 62 + + CCONJ cana-1164 435 63 ] ] PUNCT cana-1164 435 64 โ‹… โ‹… X cana-1164 435 65 [ [ PUNCT cana-1164 435 66 0 0 NUM cana-1164 435 67 โ‹ฎ โ‹ฎ NOUN cana-1164 435 68 0 0 NUM cana-1164 435 69 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 435 70 0 0 NUM cana-1164 435 71 โ‹ฎ โ‹ฎ NOUN cana-1164 435 72 0 0 NUM cana-1164 435 73 ] ] PUNCT cana-1164 436 1 = = PUNCT cana-1164 436 2 ๐œ† ๐œ† DET cana-1164 436 3 โ‹… โ‹… PROPN cana-1164 436 4 [ [ PUNCT cana-1164 436 5 0 0 NUM cana-1164 436 6 โ‹ฎ โ‹ฎ NOUN cana-1164 436 7 0 0 NUM cana-1164 436 8 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 436 9 0 0 NUM cana-1164 436 10 โ‹ฎ โ‹ฎ NOUN cana-1164 436 11 0 0 PUNCT cana-1164 436 12 ] ] PUNCT cana-1164 436 13 โ‡’ โ‡’ VERB cana-1164 437 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 437 2 + + CCONJ cana-1164 437 3 โ‹… โ‹… NUM cana-1164 437 4 ๐‘ฃ/ ๐‘ฃ/ NOUN cana-1164 437 5 = = NOUN cana-1164 437 6 ๐œ† ๐œ† DET cana-1164 437 7 โ‹… โ‹… PROPN cana-1164 437 8 ๐‘ฃ/ ๐‘ฃ/ NOUN cana-1164 437 9 ; ; PUNCT cana-1164 437 10 where where SCONJ cana-1164 437 11 ๐‘ฃ/ ๐‘ฃ/ NOUN cana-1164 437 12 = = PUNCT cana-1164 438 1 [ [ X cana-1164 438 2 0 0 NUM cana-1164 438 3 โ‹ฏ โ‹ฏ SYM cana-1164 438 4 0 0 NUM cana-1164 438 5 ๐‘ฃ๐‘– ๐‘ฃ๐‘– ADP cana-1164 438 6 0 0 NUM cana-1164 438 7 โ‹ฏ โ‹ฏ NOUN cana-1164 438 8 0]๐‘ก. 0]๐‘ก. VERB cana-1164 438 9 communications communication NOUN cana-1164 438 10 on on ADP cana-1164 438 11 applied apply VERB cana-1164 438 12 nonlinear nonlinear ADJ cana-1164 438 13 analysis analysis NOUN cana-1164 438 14 issn issn NOUN cana-1164 438 15 : : PUNCT cana-1164 438 16 1074 1074 NUM cana-1164 438 17 - - PUNCT cana-1164 438 18 133x 133x NUM cana-1164 438 19 vol vol NOUN cana-1164 438 20 31 31 NUM cana-1164 438 21 no no NOUN cana-1164 438 22 . . PUNCT cana-1164 439 1 6s 6s NUM cana-1164 439 2 ( ( PUNCT cana-1164 439 3 2024 2024 NUM cana-1164 439 4 ) ) PUNCT cana-1164 439 5 112 112 NUM cana-1164 439 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 439 7 this this PRON cana-1164 439 8 shows show VERB cana-1164 439 9 that that SCONJ cana-1164 439 10 ๐œ† ๐œ† PRON cana-1164 439 11 is be AUX cana-1164 439 12 an an DET cana-1164 439 13 eigenvalue eigenvalue NOUN cana-1164 439 14 of of ADP cana-1164 439 15 the the DET cana-1164 439 16 digraph digraph ADJ cana-1164 439 17 ฮณ ฮณ PROPN cana-1164 439 18 . . PUNCT cana-1164 440 1 thus thus ADV cana-1164 440 2 we we PRON cana-1164 440 3 have have AUX cana-1164 440 4 shown show VERB cana-1164 440 5 that that SCONJ cana-1164 440 6 every every DET cana-1164 440 7 eigenvalue eigenvalue NOUN cana-1164 440 8 of of ADP cana-1164 440 9 ฮณ ฮณ PROPN cana-1164 440 10 is be AUX cana-1164 440 11 also also ADV cana-1164 440 12 an an DET cana-1164 440 13 eigenvalue eigenvalue NOUN cana-1164 440 14 of of ADP cana-1164 440 15 at at ADV cana-1164 440 16 least least ADJ cana-1164 440 17 one one NUM cana-1164 440 18 of of ADP cana-1164 440 19 the the DET cana-1164 440 20 components component NOUN cana-1164 440 21 ฮณ๐‘– ฮณ๐‘– PROPN cana-1164 440 22 and and CCONJ cana-1164 440 23 conversely conversely ADV cana-1164 440 24 , , PUNCT cana-1164 440 25 every every DET cana-1164 440 26 eigenvalue eigenvalue NOUN cana-1164 440 27 of of ADP cana-1164 440 28 ฮณ๐‘– ฮณ๐‘– PRON cana-1164 440 29 is be AUX cana-1164 440 30 an an DET cana-1164 440 31 eigenvalue eigenvalue NOUN cana-1164 440 32 of of ADP cana-1164 440 33 ฮณ ฮณ PROPN cana-1164 440 34 . . PUNCT cana-1164 441 1 this this PRON cana-1164 441 2 proves prove VERB cana-1164 441 3 that that SCONJ cana-1164 441 4 the the DET cana-1164 441 5 spectrum spectrum NOUN cana-1164 441 6 of of ADP cana-1164 441 7 ฮณ ฮณ PROPN cana-1164 441 8 is be AUX cana-1164 441 9 the the DET cana-1164 441 10 union union NOUN cana-1164 441 11 of of ADP cana-1164 441 12 the the DET cana-1164 441 13 spectra spectra NOUN cana-1164 441 14 of of ADP cana-1164 441 15 ฮณ๐‘–. ฮณ๐‘–. PROPN cana-1164 441 16 result result VERB cana-1164 441 17 4.4 4.4 NUM cana-1164 441 18 . . PUNCT cana-1164 442 1 let let VERB cana-1164 442 2 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 442 3 , , PUNCT cana-1164 442 4 ๐‘˜ ๐‘˜ NOUN cana-1164 442 5 ) ) PUNCT cana-1164 442 6 be be VERB cana-1164 442 7 a a DET cana-1164 442 8 digraph digraph NOUN cana-1164 442 9 with with ADP cana-1164 442 10 ๐‘ -components ๐‘ -component NOUN cana-1164 442 11 ฮณ1 ฮณ1 PROPN cana-1164 442 12 , , PUNCT cana-1164 442 13 ฮณ2 ฮณ2 PROPN cana-1164 442 14 , , PUNCT cana-1164 442 15 ฮณ3 ฮณ3 NOUN cana-1164 442 16 , , PUNCT cana-1164 442 17 โ‹ฏ โ‹ฏ PROPN cana-1164 442 18 , , PUNCT cana-1164 442 19 ฮณ๐‘ . ฮณ๐‘ . NUM cana-1164 442 20 then then ADV cana-1164 442 21 1 1 NUM cana-1164 442 22 is be AUX cana-1164 442 23 an an DET cana-1164 442 24 eigenvalue eigenvalue NOUN cana-1164 442 25 of of ADP cana-1164 442 26 each each PRON cana-1164 442 27 of of ADP cana-1164 442 28 the the DET cana-1164 442 29 out out ADJ cana-1164 442 30 - - PUNCT cana-1164 442 31 adjacency adjacency NOUN cana-1164 442 32 matrix matrix NOUN cana-1164 443 1 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 443 2 + + X cana-1164 443 3 ( ( PUNCT cana-1164 443 4 1 1 NUM cana-1164 443 5 โ‰ค โ‰ค NUM cana-1164 443 6 ๐‘– ๐‘– SYM cana-1164 443 7 โ‰ค โ‰ค NUM cana-1164 443 8 ๐‘  ๐‘  NUM cana-1164 443 9 ) ) PUNCT cana-1164 443 10 with with ADP cana-1164 443 11 algebraic algebraic ADJ cana-1164 443 12 multiplicity multiplicity NOUN cana-1164 443 13 one one NUM cana-1164 443 14 . . PUNCT cana-1164 444 1 proof proof NOUN cana-1164 444 2 . . PUNCT cana-1164 445 1 let let VERB cana-1164 445 2 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 445 3 + + CCONJ cana-1164 445 4 be be AUX cana-1164 445 5 the the DET cana-1164 445 6 out out ADJ cana-1164 445 7 - - PUNCT cana-1164 445 8 adjacency adjacency NOUN cana-1164 445 9 matrix matrix NOUN cana-1164 445 10 of of ADP cana-1164 445 11 the the DET cana-1164 445 12 component component NOUN cana-1164 445 13 digraph digraph NOUN cana-1164 445 14 ฮณ๐‘– ฮณ๐‘– PRON cana-1164 445 15 with with ADP cana-1164 445 16 ๐‘›๐‘– ๐‘›๐‘– PROPN cana-1164 445 17 vertices vertex NOUN cana-1164 445 18 , , PUNCT cana-1164 445 19 where where SCONJ cana-1164 445 20 ๐‘›๐‘– ๐‘›๐‘– PROPN cana-1164 445 21 โ‰ค โ‰ค PROPN cana-1164 445 22 ๐‘› ๐‘› PRON cana-1164 445 23 and and CCONJ cana-1164 445 24 1 1 NUM cana-1164 445 25 โ‰ค โ‰ค NUM cana-1164 445 26 ๐‘– ๐‘– PRON cana-1164 445 27 โ‰ค โ‰ค NOUN cana-1164 445 28 ๐‘ . ๐‘ . NOUN cana-1164 445 29 clearly clearly ADV cana-1164 446 1 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 446 2 + + PROPN cana-1164 446 3 is be AUX cana-1164 446 4 an an DET cana-1164 446 5 ๐‘›๐‘– ๐‘›๐‘– PROPN cana-1164 446 6 ร— ร— NOUN cana-1164 446 7 ๐‘›๐‘– ๐‘›๐‘– PROPN cana-1164 446 8 matrix matrix NOUN cana-1164 446 9 . . PUNCT cana-1164 447 1 as as SCONJ cana-1164 447 2 the the DET cana-1164 447 3 out out ADJ cana-1164 447 4 - - PUNCT cana-1164 447 5 degree degree NOUN cana-1164 447 6 of of ADP cana-1164 447 7 each each DET cana-1164 447 8 vertex vertex NOUN cana-1164 447 9 in in ADP cana-1164 447 10 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 447 11 , , PUNCT cana-1164 447 12 ๐‘˜ ๐‘˜ NOUN cana-1164 447 13 ) ) PUNCT cana-1164 447 14 is be AUX cana-1164 447 15 1 1 NUM cana-1164 447 16 , , PUNCT cana-1164 447 17 so so ADV cana-1164 447 18 the the DET cana-1164 447 19 out out ADJ cana-1164 447 20 - - PUNCT cana-1164 447 21 degree degree NOUN cana-1164 447 22 of of ADP cana-1164 447 23 each each DET cana-1164 447 24 vertex vertex NOUN cana-1164 447 25 in in ADP cana-1164 447 26 ฮณ๐‘– ฮณ๐‘– PROPN cana-1164 447 27 is be AUX cana-1164 447 28 also also ADV cana-1164 447 29 1 1 NUM cana-1164 447 30 and and CCONJ cana-1164 447 31 hence hence ADV cana-1164 447 32 1 1 NUM cana-1164 447 33 appears appear VERB cana-1164 447 34 exactly exactly ADV cana-1164 447 35 once once ADV cana-1164 447 36 in in ADP cana-1164 447 37 each each DET cana-1164 447 38 row row NOUN cana-1164 447 39 of of ADP cana-1164 447 40 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 447 41 + + PROPN cana-1164 447 42 with with ADP cana-1164 447 43 other other ADJ cana-1164 447 44 entries entry NOUN cana-1164 447 45 as as ADP cana-1164 447 46 0 0 NUM cana-1164 447 47 . . PUNCT cana-1164 448 1 we we PRON cana-1164 448 2 now now ADV cana-1164 448 3 consider consider VERB cana-1164 448 4 the the DET cana-1164 448 5 matrix matrix NOUN cana-1164 448 6 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 448 7 + + CCONJ cana-1164 448 8 โˆ’ โˆ’ PROPN cana-1164 448 9 ๐œ†๐ผ๐‘›๐‘– ๐œ†๐ผ๐‘›๐‘– PROPN cana-1164 448 10 and and CCONJ cana-1164 448 11 we we PRON cana-1164 448 12 apply apply VERB cana-1164 448 13 the the DET cana-1164 448 14 column column NOUN cana-1164 448 15 operation operation NOUN cana-1164 448 16 ๐ถ1 ๐ถ1 PRON cana-1164 448 17 โ†’ โ†’ SYM cana-1164 448 18 ๐ถ1 ๐ถ1 X cana-1164 448 19 + + CCONJ cana-1164 448 20 ๐ถ2 ๐ถ2 ADJ cana-1164 449 1 + + NOUN cana-1164 449 2 โ‹ฏ+ โ‹ฏ+ NOUN cana-1164 449 3 ๐ถ๐‘›๐‘– ๐ถ๐‘›๐‘– ADJ cana-1164 449 4 in in ADP cana-1164 449 5 the the DET cana-1164 449 6 matrix matrix NOUN cana-1164 450 1 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 450 2 + + CCONJ cana-1164 450 3 โˆ’ โˆ’ PROPN cana-1164 450 4 ๐œ†๐ผ๐‘›๐‘– ๐œ†๐ผ๐‘›๐‘– PROPN cana-1164 450 5 , , PUNCT cana-1164 450 6 then then ADV cana-1164 450 7 it it PRON cana-1164 450 8 can can AUX cana-1164 450 9 be be AUX cana-1164 450 10 easily easily ADV cana-1164 450 11 seen see VERB cana-1164 450 12 that that SCONJ cana-1164 450 13 each each DET cana-1164 450 14 element element NOUN cana-1164 450 15 of of ADP cana-1164 450 16 ๐ถ1 ๐ถ1 PROPN cana-1164 450 17 is be AUX cana-1164 450 18 ( ( PUNCT cana-1164 450 19 1 1 NUM cana-1164 450 20 โˆ’ โˆ’ NOUN cana-1164 450 21 ๐œ† ๐œ† NOUN cana-1164 450 22 ) ) PUNCT cana-1164 450 23 and and CCONJ cana-1164 450 24 hence hence ADV cana-1164 450 25 ( ( PUNCT cana-1164 450 26 1 1 NUM cana-1164 450 27 โˆ’ โˆ’ NOUN cana-1164 450 28 ๐œ† ๐œ† X cana-1164 450 29 ) ) PUNCT cana-1164 450 30 will will AUX cana-1164 450 31 be be AUX cana-1164 450 32 a a DET cana-1164 450 33 factor factor NOUN cana-1164 450 34 of of ADP cana-1164 450 35 ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘– ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘– PROPN cana-1164 450 36 + + CCONJ cana-1164 450 37 โˆ’ โˆ’ PROPN cana-1164 450 38 ๐œ†๐ผ๐‘›๐‘– ๐œ†๐ผ๐‘›๐‘– PROPN cana-1164 450 39 ) ) PUNCT cana-1164 450 40 . . PUNCT cana-1164 451 1 this this PRON cana-1164 451 2 shows show VERB cana-1164 451 3 that that SCONJ cana-1164 451 4 1 1 NUM cana-1164 451 5 is be AUX cana-1164 451 6 an an DET cana-1164 451 7 eigenvalue eigenvalue NOUN cana-1164 451 8 of of ADP cana-1164 451 9 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 451 10 + + X cana-1164 451 11 ( ( PUNCT cana-1164 451 12 1 1 NUM cana-1164 451 13 โ‰ค โ‰ค NUM cana-1164 451 14 ๐‘– ๐‘– SYM cana-1164 451 15 โ‰ค โ‰ค NUM cana-1164 451 16 ๐‘  ๐‘  NUM cana-1164 451 17 ) ) PUNCT cana-1164 451 18 . . PUNCT cana-1164 452 1 next next ADV cana-1164 452 2 , , PUNCT cana-1164 452 3 to to PART cana-1164 452 4 show show VERB cana-1164 452 5 that that SCONJ cana-1164 452 6 the the DET cana-1164 452 7 algebraic algebraic ADJ cana-1164 452 8 multiplicity multiplicity NOUN cana-1164 452 9 of of ADP cana-1164 452 10 1 1 NUM cana-1164 452 11 is be AUX cana-1164 452 12 one one NUM cana-1164 452 13 . . PUNCT cana-1164 453 1 if if SCONJ cana-1164 453 2 possible possible ADJ cana-1164 453 3 , , PUNCT cana-1164 453 4 let let VERB cana-1164 453 5 the the DET cana-1164 453 6 algebraic algebraic ADJ cana-1164 453 7 multiplicity multiplicity NOUN cana-1164 453 8 of of ADP cana-1164 453 9 1 1 NUM cana-1164 453 10 be be AUX cana-1164 453 11 greater great ADJ cana-1164 453 12 than than ADP cana-1164 453 13 one one NUM cana-1164 453 14 . . PUNCT cana-1164 454 1 then then ADV cana-1164 454 2 there there PRON cana-1164 454 3 exists exist VERB cana-1164 454 4 at at ADP cana-1164 454 5 least least ADV cana-1164 454 6 two two NUM cana-1164 454 7 linearly linearly ADV cana-1164 454 8 independent independent ADJ cana-1164 454 9 vectors vector NOUN cana-1164 454 10 ๐‘ข ๐‘ข NOUN cana-1164 454 11 and and CCONJ cana-1164 454 12 ๐‘ฃ ๐‘ฃ X cana-1164 454 13 with with ADP cana-1164 454 14 eigenvalue eigenvalue PROPN cana-1164 454 15 1 1 NUM cana-1164 454 16 such such ADJ cana-1164 454 17 that that SCONJ cana-1164 454 18 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 454 19 + + CCONJ cana-1164 454 20 โ‹… โ‹… PROPN cana-1164 454 21 ๐‘ข ๐‘ข X cana-1164 454 22 = = SYM cana-1164 454 23 1 1 NUM cana-1164 454 24 โ‹… โ‹… PROPN cana-1164 454 25 ๐‘ข ๐‘ข X cana-1164 454 26 and and CCONJ cana-1164 454 27 ๐ดฮณ๐‘– ๐ดฮณ๐‘– PROPN cana-1164 454 28 + + CCONJ cana-1164 454 29 โ‹… โ‹… PROPN cana-1164 454 30 ๐‘ฃ ๐‘ฃ X cana-1164 454 31 = = SYM cana-1164 454 32 1 1 NUM cana-1164 454 33 โ‹… โ‹… PROPN cana-1164 454 34 ๐‘ฃ ๐‘ฃ ADP cana-1164 454 35 which which PRON cana-1164 454 36 is be AUX cana-1164 454 37 possible possible ADJ cana-1164 454 38 if if SCONJ cana-1164 454 39 ๐‘ข ๐‘ข NOUN cana-1164 454 40 and and CCONJ cana-1164 454 41 ๐‘ฃ ๐‘ฃ PROPN cana-1164 454 42 are be AUX cana-1164 454 43 scalar scalar ADJ cana-1164 454 44 multiples multiple NOUN cana-1164 454 45 of of ADP cana-1164 454 46 each each DET cana-1164 454 47 other other ADJ cana-1164 454 48 and and CCONJ cana-1164 454 49 in in ADP cana-1164 454 50 this this DET cana-1164 454 51 case case NOUN cana-1164 454 52 , , PUNCT cana-1164 454 53 ๐‘ข ๐‘ข NOUN cana-1164 454 54 and and CCONJ cana-1164 454 55 ๐‘ฃ ๐‘ฃ PROPN cana-1164 454 56 are be AUX cana-1164 454 57 linearly linearly ADV cana-1164 454 58 dependent dependent ADJ cana-1164 454 59 , , PUNCT cana-1164 454 60 which which PRON cana-1164 454 61 is be AUX cana-1164 454 62 a a DET cana-1164 454 63 contradiction contradiction NOUN cana-1164 454 64 . . PUNCT cana-1164 455 1 hence hence ADV cana-1164 455 2 , , PUNCT cana-1164 455 3 the the DET cana-1164 455 4 algebraic algebraic ADJ cana-1164 455 5 multiplicity multiplicity NOUN cana-1164 455 6 of of ADP cana-1164 455 7 1 1 NUM cana-1164 455 8 is be AUX cana-1164 455 9 one one NUM cana-1164 455 10 . . PUNCT cana-1164 456 1 result result VERB cana-1164 456 2 4.5 4.5 NUM cana-1164 456 3 . . PUNCT cana-1164 457 1 the the DET cana-1164 457 2 algebraic algebraic ADJ cana-1164 457 3 multiplicity multiplicity NOUN cana-1164 457 4 of of ADP cana-1164 457 5 1 1 NUM cana-1164 457 6 as as ADP cana-1164 457 7 an an DET cana-1164 457 8 eigenvalue eigenvalue NOUN cana-1164 457 9 of of ADP cana-1164 457 10 ๐ดฮณ ๐ดฮณ PROPN cana-1164 457 11 + + CCONJ cana-1164 457 12 is be AUX cana-1164 457 13 the the DET cana-1164 457 14 number number NOUN cana-1164 457 15 of of ADP cana-1164 457 16 components component NOUN cana-1164 457 17 of of ADP cana-1164 457 18 the the DET cana-1164 457 19 digraph digraph ADJ cana-1164 457 20 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 457 21 , , PUNCT cana-1164 457 22 ๐‘˜ ๐‘˜ NOUN cana-1164 457 23 ) ) PUNCT cana-1164 457 24 . . PUNCT cana-1164 458 1 proof proof NOUN cana-1164 458 2 . . PUNCT cana-1164 459 1 let let VERB cana-1164 460 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 460 2 + + CCONJ cana-1164 460 3 be be AUX cana-1164 460 4 the the DET cana-1164 460 5 out out ADJ cana-1164 460 6 - - PUNCT cana-1164 460 7 adjacency adjacency NOUN cana-1164 460 8 matrix matrix NOUN cana-1164 460 9 of of ADP cana-1164 460 10 the the DET cana-1164 460 11 digraph digraph ADJ cana-1164 460 12 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 460 13 , , PUNCT cana-1164 460 14 ๐‘˜ ๐‘˜ NOUN cana-1164 460 15 ) ) PUNCT cana-1164 460 16 where where SCONJ cana-1164 460 17 |๐‘‰(ฮณ(๐‘› |๐‘‰(ฮณ(๐‘› PROPN cana-1164 460 18 , , PUNCT cana-1164 460 19 ๐‘˜))| ๐‘˜))| PROPN cana-1164 460 20 = = PUNCT cana-1164 460 21 ๐‘›. ๐‘›. NOUN cana-1164 460 22 suppose suppose VERB cana-1164 460 23 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 460 24 , , PUNCT cana-1164 460 25 ๐‘˜ ๐‘˜ NOUN cana-1164 460 26 ) ) PUNCT cana-1164 460 27 has have VERB cana-1164 460 28 ๐‘ -components ๐‘ -component NOUN cana-1164 460 29 ฮณ1 ฮณ1 PROPN cana-1164 460 30 , , PUNCT cana-1164 460 31 ฮณ2 ฮณ2 PROPN cana-1164 460 32 , , PUNCT cana-1164 460 33 โ‹ฏ โ‹ฏ PROPN cana-1164 460 34 , , PUNCT cana-1164 460 35 ฮณ๐‘  ฮณ๐‘  NOUN cana-1164 460 36 with with SCONJ cana-1164 460 37 their their PRON cana-1164 460 38 out out ADJ cana-1164 460 39 - - PUNCT cana-1164 460 40 adjacency adjacency NOUN cana-1164 460 41 matrices matrix NOUN cana-1164 460 42 ๐ดฮณ1 ๐ดฮณ1 VERB cana-1164 460 43 + + X cana-1164 460 44 , , PUNCT cana-1164 460 45 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 460 46 + + CCONJ cana-1164 460 47 , , PUNCT cana-1164 460 48 ๐ดฮณ3 ๐ดฮณ3 NOUN cana-1164 460 49 + + X cana-1164 460 50 , , PUNCT cana-1164 460 51 โ‹ฏ โ‹ฏ PROPN cana-1164 460 52 , , PUNCT cana-1164 460 53 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 460 54 + + CCONJ cana-1164 460 55 respectively respectively ADV cana-1164 460 56 . . PUNCT cana-1164 461 1 also also ADV cana-1164 461 2 , , PUNCT cana-1164 461 3 let let VERB cana-1164 461 4 |๐‘‰(ฮณ๐‘–(๐‘› |๐‘‰(ฮณ๐‘–(๐‘› ADV cana-1164 461 5 , , PUNCT cana-1164 461 6 ๐‘˜))| ๐‘˜))| PROPN cana-1164 461 7 = = SYM cana-1164 461 8 ๐‘›๐‘– ๐‘›๐‘– PROPN cana-1164 461 9 , , PUNCT cana-1164 461 10 1 1 NUM cana-1164 461 11 โ‰ค โ‰ค NUM cana-1164 461 12 ๐‘– ๐‘– SYM cana-1164 461 13 โ‰ค โ‰ค NUM cana-1164 461 14 ๐‘  ๐‘  ADP cana-1164 461 15 such such ADJ cana-1164 461 16 that that DET cana-1164 461 17 โˆ‘๐‘ ๐‘–=1 โˆ‘๐‘ ๐‘–=1 PROPN cana-1164 461 18 ๐‘›๐‘– ๐‘›๐‘– NOUN cana-1164 461 19 = = PUNCT cana-1164 462 1 ๐‘›. ๐‘›. NOUN cana-1164 462 2 then then ADV cana-1164 462 3 we we PRON cana-1164 462 4 have have AUX cana-1164 462 5 ๐ดฮณ ๐ดฮณ PROPN cana-1164 462 6 + + NOUN cana-1164 462 7 = = SYM cana-1164 462 8 [ [ PUNCT cana-1164 462 9 ๐ดฮณ1 ๐ดฮณ1 X cana-1164 462 10 + + X cana-1164 462 11 0 0 NUM cana-1164 462 12 0 0 NUM cana-1164 462 13 โ‹ฏ โ‹ฏ ADP cana-1164 462 14 0 0 NUM cana-1164 462 15 0 0 NUM cana-1164 462 16 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 462 17 + + X cana-1164 462 18 0 0 NUM cana-1164 462 19 โ‹ฏ โ‹ฏ NOUN cana-1164 462 20 0 0 NUM cana-1164 462 21 โ‹ฎ โ‹ฎ NOUN cana-1164 462 22 โ‹ฎ โ‹ฎ ADJ cana-1164 462 23 โ‹ฎ โ‹ฎ NOUN cana-1164 462 24 โ‹ฑ โ‹ฑ PUNCT cana-1164 462 25 โ‹ฎ โ‹ฎ NOUN cana-1164 462 26 0 0 NUM cana-1164 462 27 0 0 NUM cana-1164 462 28 0 0 NUM cana-1164 462 29 โ‹ฏ โ‹ฏ VERB cana-1164 462 30 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 462 31 + + CCONJ cana-1164 462 32 ] ] PUNCT cana-1164 462 33 by by ADP cana-1164 462 34 result result NOUN cana-1164 462 35 4.4 4.4 NUM cana-1164 462 36 . . PUNCT cana-1164 462 37 , , PUNCT cana-1164 462 38 each each DET cana-1164 462 39 block block NOUN cana-1164 462 40 matrices matrix NOUN cana-1164 462 41 ๐ดฮณ1 ๐ดฮณ1 VERB cana-1164 462 42 + + X cana-1164 462 43 , , PUNCT cana-1164 462 44 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 462 45 + + X cana-1164 462 46 , , PUNCT cana-1164 462 47 โ‹ฏ โ‹ฏ PROPN cana-1164 462 48 , , PUNCT cana-1164 462 49 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 462 50 + + PROPN cana-1164 462 51 has have AUX cana-1164 462 52 eigenvalue eigenvalue PROPN cana-1164 462 53 1 1 NUM cana-1164 462 54 with with ADP cana-1164 462 55 algebraic algebraic ADJ cana-1164 462 56 multiplicity multiplicity NOUN cana-1164 462 57 1 1 NUM cana-1164 462 58 . . PUNCT cana-1164 463 1 also also ADV cana-1164 463 2 , , PUNCT cana-1164 463 3 we we PRON cana-1164 463 4 have have VERB cana-1164 463 5 ๐‘‘๐‘’๐‘ก(๐ดฮณ ๐‘‘๐‘’๐‘ก(๐ดฮณ NOUN cana-1164 463 6 + + CCONJ cana-1164 463 7 โˆ’ โˆ’ PROPN cana-1164 463 8 ๐œ†๐ผ๐‘› ๐œ†๐ผ๐‘› NOUN cana-1164 463 9 ) ) PUNCT cana-1164 463 10 = = PUNCT cana-1164 463 11 ๐‘‘๐‘’๐‘ก(๐ดฮณ1 ๐‘‘๐‘’๐‘ก(๐ดฮณ1 NOUN cana-1164 464 1 + + CCONJ cana-1164 464 2 โˆ’ โˆ’ PROPN cana-1164 464 3 ๐œ†๐ผ๐‘›1 ๐œ†๐ผ๐‘›1 PROPN cana-1164 464 4 ) ) PUNCT cana-1164 464 5 โ‹… โ‹… X cana-1164 464 6 ๐‘‘๐‘’๐‘ก(๐ดฮณ2 ๐‘‘๐‘’๐‘ก(๐ดฮณ2 ADJ cana-1164 465 1 + + CCONJ cana-1164 465 2 โˆ’ โˆ’ PROPN cana-1164 465 3 ๐œ†๐ผ๐‘›2 ๐œ†๐ผ๐‘›2 PROPN cana-1164 465 4 ) ) PUNCT cana-1164 465 5 โ‹… โ‹… PROPN cana-1164 465 6 ๐‘‘๐‘’๐‘ก(๐ดฮณ3 ๐‘‘๐‘’๐‘ก(๐ดฮณ3 NOUN cana-1164 465 7 + + CCONJ cana-1164 465 8 โˆ’ โˆ’ PROPN cana-1164 465 9 ๐œ†๐ผ๐‘›3)โ‹ฏ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘  ๐œ†๐ผ๐‘›3)โ‹ฏ๐‘‘๐‘’๐‘ก(๐ดฮณ๐‘  PROPN cana-1164 466 1 + + CCONJ cana-1164 466 2 โˆ’ โˆ’ PROPN cana-1164 466 3 ๐œ†๐ผ๐‘›๐‘  ๐œ†๐ผ๐‘›๐‘  NOUN cana-1164 466 4 ) ) PUNCT cana-1164 466 5 so so ADV cana-1164 466 6 , , PUNCT cana-1164 466 7 the the DET cana-1164 466 8 algebraic algebraic ADJ cana-1164 466 9 multiplicity multiplicity NOUN cana-1164 466 10 of of ADP cana-1164 466 11 1 1 NUM cana-1164 466 12 for for ADP cana-1164 466 13 the the DET cana-1164 466 14 out out ADJ cana-1164 466 15 - - PUNCT cana-1164 466 16 adjacency adjacency NOUN cana-1164 466 17 matrix matrix NOUN cana-1164 467 1 ๐ดฮณ ๐ดฮณ PROPN cana-1164 467 2 + + CCONJ cana-1164 467 3 is be AUX cana-1164 467 4 the the DET cana-1164 467 5 sum sum NOUN cana-1164 467 6 of of ADP cana-1164 467 7 the the DET cana-1164 467 8 algebraic algebraic ADJ cana-1164 467 9 multiplicities multiplicity NOUN cana-1164 467 10 of of ADP cana-1164 467 11 1 1 NUM cana-1164 467 12 for for ADP cana-1164 467 13 each each DET cana-1164 467 14 ๐ดฮณ1 ๐ดฮณ1 NOUN cana-1164 467 15 + + X cana-1164 467 16 , , PUNCT cana-1164 467 17 ๐ดฮณ2 ๐ดฮณ2 NOUN cana-1164 467 18 + + X cana-1164 467 19 , , PUNCT cana-1164 467 20 โ‹ฏ โ‹ฏ PROPN cana-1164 467 21 , , PUNCT cana-1164 467 22 ๐ดฮณ๐‘  ๐ดฮณ๐‘  PROPN cana-1164 467 23 + + PUNCT cana-1164 467 24 . . PUNCT cana-1164 468 1 then then ADV cana-1164 468 2 this this DET cana-1164 468 3 sum sum NOUN cana-1164 468 4 is be AUX cana-1164 468 5 1 1 NUM cana-1164 468 6 + + CCONJ cana-1164 468 7 1 1 NUM cana-1164 468 8 + + SYM cana-1164 468 9 1 1 NUM cana-1164 468 10 + + ADJ cana-1164 468 11 โ‹ฏ+ โ‹ฏ+ NOUN cana-1164 468 12 1โŸ 1โŸ NUM cana-1164 468 13 ๐‘ โˆ’๐‘ก๐‘’๐‘Ÿ๐‘š๐‘  ๐‘ โˆ’๐‘ก๐‘’๐‘Ÿ๐‘š๐‘  NOUN cana-1164 468 14 = = SYM cana-1164 468 15 ๐‘  ๐‘  PROPN cana-1164 468 16 ( ( PUNCT cana-1164 468 17 as as ADP cana-1164 468 18 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 468 19 , , PUNCT cana-1164 468 20 ๐‘˜ ๐‘˜ NOUN cana-1164 468 21 ) ) PUNCT cana-1164 468 22 has have VERB cana-1164 468 23 ๐‘  ๐‘  DET cana-1164 468 24 -components -component NOUN cana-1164 468 25 ) ) PUNCT cana-1164 468 26 . . PUNCT cana-1164 469 1 this this PRON cana-1164 469 2 shows show VERB cana-1164 469 3 that that SCONJ cana-1164 469 4 the the DET cana-1164 469 5 algebraic algebraic ADJ cana-1164 469 6 multiplicity multiplicity NOUN cana-1164 469 7 of of ADP cana-1164 469 8 1 1 NUM cana-1164 469 9 as as ADP cana-1164 469 10 an an DET cana-1164 469 11 eigenvalue eigenvalue NOUN cana-1164 469 12 of of ADP cana-1164 469 13 ๐ดฮณ ๐ดฮณ PROPN cana-1164 469 14 + + CCONJ cana-1164 469 15 is be AUX cana-1164 469 16 ๐‘  ๐‘  PROPN cana-1164 469 17 , , PUNCT cana-1164 469 18 which which PRON cana-1164 469 19 is be AUX cana-1164 469 20 the the DET cana-1164 469 21 number number NOUN cana-1164 469 22 of of ADP cana-1164 469 23 components component NOUN cana-1164 469 24 of of ADP cana-1164 469 25 the the DET cana-1164 469 26 digraph digraph ADJ cana-1164 469 27 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 469 28 , , PUNCT cana-1164 469 29 ๐‘˜ ๐‘˜ NOUN cana-1164 469 30 ) ) PUNCT cana-1164 469 31 . . PUNCT cana-1164 470 1 communications communication NOUN cana-1164 470 2 on on ADP cana-1164 470 3 applied apply VERB cana-1164 470 4 nonlinear nonlinear ADJ cana-1164 470 5 analysis analysis NOUN cana-1164 470 6 issn issn NOUN cana-1164 470 7 : : PUNCT cana-1164 470 8 1074 1074 NUM cana-1164 470 9 - - PUNCT cana-1164 470 10 133x 133x NUM cana-1164 470 11 vol vol NOUN cana-1164 470 12 31 31 NUM cana-1164 470 13 no no NOUN cana-1164 470 14 . . PUNCT cana-1164 471 1 6s 6s NUM cana-1164 471 2 ( ( PUNCT cana-1164 471 3 2024 2024 NUM cana-1164 471 4 ) ) PUNCT cana-1164 471 5 113 113 NUM cana-1164 471 6 https://internationalpubls.com https://internationalpubls.com X cana-1164 471 7 5 5 NUM cana-1164 471 8 . . PUNCT cana-1164 472 1 conclusion conclusion NOUN cana-1164 472 2 we we PRON cana-1164 472 3 introduced introduce VERB cana-1164 472 4 the the DET cana-1164 472 5 adjacency adjacency NOUN cana-1164 472 6 matrix matrix NOUN cana-1164 472 7 of of ADP cana-1164 472 8 the the DET cana-1164 472 9 power power NOUN cana-1164 472 10 digraph digraph NOUN cana-1164 473 1 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 473 2 , , PUNCT cana-1164 473 3 ๐‘˜ ๐‘˜ NOUN cana-1164 473 4 ) ) PUNCT cana-1164 473 5 , , PUNCT cana-1164 473 6 defining define VERB cana-1164 473 7 the the DET cana-1164 473 8 out out ADJ cana-1164 473 9 - - PUNCT cana-1164 473 10 adjacency adjacency NOUN cana-1164 473 11 matrix matrix NOUN cana-1164 473 12 ( ( PUNCT cana-1164 473 13 ๐ดฮณ ๐ดฮณ PROPN cana-1164 473 14 + + PROPN cana-1164 473 15 ) ) PUNCT cana-1164 473 16 and and CCONJ cana-1164 473 17 the the DET cana-1164 473 18 in in ADP cana-1164 473 19 - - PUNCT cana-1164 473 20 adjacency adjacency NOUN cana-1164 473 21 matrix matrix NOUN cana-1164 473 22 ( ( PUNCT cana-1164 473 23 ๐ดฮณ ๐ดฮณ PROPN cana-1164 473 24 โˆ’ โˆ’ PROPN cana-1164 473 25 ) ) PUNCT cana-1164 473 26 . . PUNCT cana-1164 474 1 we we PRON cana-1164 474 2 demonstrated demonstrate VERB cana-1164 474 3 that that SCONJ cana-1164 474 4 these these DET cana-1164 474 5 matrices matrix NOUN cana-1164 474 6 are be AUX cana-1164 474 7 singular singular ADJ cana-1164 474 8 if if SCONJ cana-1164 474 9 certain certain ADJ cana-1164 474 10 conditions condition NOUN cana-1164 474 11 are be AUX cana-1164 474 12 met meet VERB cana-1164 474 13 and and CCONJ cana-1164 474 14 discussed discuss VERB cana-1164 474 15 the the DET cana-1164 474 16 spectral spectral ADJ cana-1164 474 17 properties property NOUN cana-1164 474 18 of of ADP cana-1164 474 19 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1164 474 20 , , PUNCT cana-1164 474 21 ๐‘˜ ๐‘˜ NOUN cana-1164 474 22 ) ) PUNCT cana-1164 474 23 . . PUNCT cana-1164 475 1 additionally additionally ADV cana-1164 475 2 , , PUNCT cana-1164 475 3 we we PRON cana-1164 475 4 proved prove VERB cana-1164 475 5 that that SCONJ cana-1164 475 6 the the DET cana-1164 475 7 algebraic algebraic ADJ cana-1164 475 8 multiplicity multiplicity NOUN cana-1164 475 9 of of ADP cana-1164 475 10 1 1 NUM cana-1164 475 11 as as ADP cana-1164 475 12 an an DET cana-1164 475 13 eigenvalue eigenvalue NOUN cana-1164 475 14 of of ADP cana-1164 475 15 ( ( PUNCT cana-1164 475 16 ๐ดฮณ ๐ดฮณ PROPN cana-1164 475 17 + + PROPN cana-1164 475 18 ) ) PUNCT cana-1164 475 19 corresponds correspond VERB cana-1164 475 20 to to ADP cana-1164 475 21 the the DET cana-1164 475 22 number number NOUN cana-1164 475 23 of of ADP cana-1164 475 24 components component NOUN cana-1164 475 25 in in ADP cana-1164 475 26 the the DET cana-1164 475 27 digraph digraph NOUN cana-1164 475 28 . . PUNCT cana-1164 476 1 conflicts conflict NOUN cana-1164 476 2 of of ADP cana-1164 476 3 interest interest NOUN cana-1164 476 4 the the DET cana-1164 476 5 authors author NOUN cana-1164 476 6 declare declare VERB cana-1164 476 7 no no DET cana-1164 476 8 conflict conflict NOUN cana-1164 476 9 of of ADP cana-1164 476 10 interest interest NOUN cana-1164 476 11 . . PUNCT cana-1164 477 1 references reference NOUN cana-1164 477 2 [ [ X cana-1164 477 3 1 1 NUM cana-1164 477 4 ] ] PUNCT cana-1164 477 5 a. a. NOUN cana-1164 477 6 adbollahi adbollahi NOUN cana-1164 477 7 , , PUNCT cana-1164 477 8 determinants determinant NOUN cana-1164 477 9 of of ADP cana-1164 477 10 adjacency adjacency NOUN cana-1164 477 11 matrices matrix NOUN cana-1164 477 12 of of ADP cana-1164 477 13 graph graph NOUN cana-1164 477 14 , , PUNCT cana-1164 477 15 transactions transaction NOUN cana-1164 477 16 on on ADP cana-1164 477 17 combinatorics combinatoric NOUN cana-1164 477 18 , , PUNCT cana-1164 477 19 1 1 NUM cana-1164 477 20 no no NOUN cana-1164 477 21 . . NOUN cana-1164 477 22 4 4 NUM cana-1164 477 23 ( ( PUNCT cana-1164 477 24 2012 2012 NUM cana-1164 477 25 ) ) PUNCT cana-1164 477 26 9 9 NUM cana-1164 477 27 - - SYM cana-1164 477 28 16 16 NUM cana-1164 477 29 . . PUNCT cana-1164 478 1 [ [ X cana-1164 478 2 2 2 NUM cana-1164 478 3 ] ] PUNCT cana-1164 478 4 r. r. PROPN cana-1164 478 5 b. b. PROPN cana-1164 478 6 bapat bapat PROPN cana-1164 478 7 , , PUNCT cana-1164 478 8 graphs graph NOUN cana-1164 478 9 and and CCONJ cana-1164 478 10 matrices matrix NOUN cana-1164 478 11 , , PUNCT cana-1164 478 12 springer springer NOUN cana-1164 478 13 2010 2010 NUM cana-1164 478 14 . . PUNCT cana-1164 479 1 [ [ X cana-1164 479 2 3 3 X cana-1164 479 3 ] ] X cana-1164 479 4 e. e. PROPN cana-1164 479 5 blanton blanton PROPN cana-1164 479 6 jr jr PROPN cana-1164 479 7 . . PROPN cana-1164 479 8 , , PUNCT cana-1164 479 9 s. s. PROPN cana-1164 479 10 hurd hurd PROPN cana-1164 479 11 , , PUNCT cana-1164 479 12 and and CCONJ cana-1164 479 13 j. j. PROPN cana-1164 479 14 mccranie mccranie PROPN cana-1164 479 15 , , PUNCT cana-1164 479 16 on on ADP cana-1164 479 17 a a DET cana-1164 479 18 digraph digraph NOUN cana-1164 479 19 defined define VERB cana-1164 479 20 by by ADP cana-1164 479 21 square square ADJ cana-1164 479 22 modulo modulo PROPN cana-1164 479 23 n n CCONJ cana-1164 479 24 , , PUNCT cana-1164 479 25 fibonacci fibonacci PROPN cana-1164 479 26 quarterly quarterly ADV cana-1164 479 27 , , PUNCT cana-1164 479 28 34 34 NUM cana-1164 479 29 ( ( PUNCT cana-1164 479 30 1992 1992 NUM cana-1164 479 31 ) ) PUNCT cana-1164 479 32 322โ€“334 322โ€“334 NUM cana-1164 479 33 . . PUNCT cana-1164 480 1 [ [ X cana-1164 480 2 4 4 X cana-1164 480 3 ] ] X cana-1164 480 4 s. s. PROPN cana-1164 480 5 bryant bryant PROPN cana-1164 480 6 , , PUNCT cana-1164 480 7 groups group NOUN cana-1164 480 8 , , PUNCT cana-1164 480 9 graphs graph NOUN cana-1164 480 10 and and CCONJ cana-1164 480 11 fermat fermat PROPN cana-1164 480 12 โ€™s โ€™s PART cana-1164 480 13 last last ADJ cana-1164 480 14 theorem theorem NOUN cana-1164 480 15 , , PUNCT cana-1164 480 16 amer amer PROPN cana-1164 480 17 . . PROPN cana-1164 480 18 math math PROPN cana-1164 480 19 . . PUNCT cana-1164 481 1 monthly monthly ADV cana-1164 481 2 . . PUNCT cana-1164 482 1 , , PUNCT cana-1164 482 2 74 74 NUM cana-1164 482 3 ( ( PUNCT cana-1164 482 4 1967 1967 NUM cana-1164 482 5 ) ) PUNCT cana-1164 482 6 152โ€“156 152โ€“156 NUM cana-1164 482 7 . . PUNCT cana-1164 483 1 [ [ X cana-1164 483 2 5 5 X cana-1164 483 3 ] ] PUNCT cana-1164 483 4 t. t. PROPN cana-1164 483 5 ju ju PROPN cana-1164 483 6 , , PUNCT cana-1164 483 7 m. m. PROPN cana-1164 483 8 wu wu PROPN cana-1164 483 9 , , PUNCT cana-1164 483 10 on on ADP cana-1164 483 11 iteration iteration NOUN cana-1164 483 12 digraph digraph NOUN cana-1164 483 13 and and CCONJ cana-1164 483 14 zero zero NUM cana-1164 483 15 - - PUNCT cana-1164 483 16 divisor divisor NOUN cana-1164 483 17 graph graph NOUN cana-1164 483 18 of of ADP cana-1164 483 19 the the DET cana-1164 483 20 ring ring NOUN cana-1164 483 21 โ„ค๐‘› โ„ค๐‘› PROPN cana-1164 483 22 , , PUNCT cana-1164 483 23 czechoslovak czechoslovak ADJ cana-1164 483 24 mathematical mathematical ADJ cana-1164 483 25 journal journal NOUN cana-1164 483 26 , , PUNCT cana-1164 483 27 64 64 NUM cana-1164 483 28 ( ( PUNCT cana-1164 483 29 2008 2008 NUM cana-1164 483 30 ) ) PUNCT cana-1164 483 31 611โ€“628 611โ€“628 NUM cana-1164 483 32 . . PUNCT cana-1164 484 1 [ [ X cana-1164 484 2 6 6 NUM cana-1164 484 3 ] ] PUNCT cana-1164 484 4 w. w. NOUN cana-1164 484 5 carlip carlip PROPN cana-1164 484 6 and and CCONJ cana-1164 484 7 m. m. NOUN cana-1164 484 8 mincheva mincheva PROPN cana-1164 484 9 , , PUNCT cana-1164 484 10 symmetry symmetry NOUN cana-1164 484 11 of of ADP cana-1164 484 12 iteration iteration NOUN cana-1164 484 13 graphs graph NOUN cana-1164 484 14 , , PUNCT cana-1164 484 15 czechoslovak czechoslovak ADJ cana-1164 484 16 mathematical mathematical ADJ cana-1164 484 17 journal journal NOUN cana-1164 484 18 , , PUNCT cana-1164 484 19 58 58 NUM cana-1164 484 20 ( ( PUNCT cana-1164 484 21 2008 2008 NUM cana-1164 484 22 ) ) PUNCT cana-1164 484 23 131 131 NUM cana-1164 484 24 - - SYM cana-1164 484 25 145 145 NUM cana-1164 484 26 . . PUNCT cana-1164 485 1 [ [ X cana-1164 485 2 7 7 X cana-1164 485 3 ] ] X cana-1164 485 4 p. p. NOUN cana-1164 485 5 m. m. NOUN cana-1164 485 6 cvetkovic cvetkovic PROPN cana-1164 485 7 , , PUNCT cana-1164 485 8 m. m. NOUN cana-1164 485 9 doob doob PROPN cana-1164 485 10 , , PUNCT cana-1164 485 11 h. h. PROPN cana-1164 485 12 sachs sachs PROPN cana-1164 485 13 a a DET cana-1164 485 14 , , PUNCT cana-1164 485 15 spectra spectra NOUN cana-1164 485 16 of of ADP cana-1164 485 17 graphs graph NOUN cana-1164 485 18 : : PUNCT cana-1164 485 19 theory theory NOUN cana-1164 485 20 and and CCONJ cana-1164 485 21 application application NOUN cana-1164 485 22 , , PUNCT cana-1164 485 23 academic academic ADJ cana-1164 485 24 press press NOUN cana-1164 485 25 , , PUNCT cana-1164 485 26 1980 1980 NUM cana-1164 485 27 . . PUNCT cana-1164 486 1 [ [ X cana-1164 486 2 8 8 NUM cana-1164 486 3 ] ] PUNCT cana-1164 486 4 p. p. NOUN cana-1164 486 5 goswami goswami PROPN cana-1164 486 6 , , PUNCT cana-1164 486 7 s. s. PROPN cana-1164 486 8 k. k. PROPN cana-1164 486 9 thakur thakur PROPN cana-1164 486 10 , , PUNCT cana-1164 486 11 and and CCONJ cana-1164 486 12 g. g. PROPN cana-1164 486 13 c. c. PROPN cana-1164 486 14 ray ray PROPN cana-1164 486 15 , , PUNCT cana-1164 486 16 the the DET cana-1164 486 17 structure structure NOUN cana-1164 486 18 of of ADP cana-1164 486 19 the the DET cana-1164 486 20 power power NOUN cana-1164 486 21 digraph digraph NOUN cana-1164 486 22 connected connect VERB cana-1164 486 23 with with ADP cana-1164 486 24 the the DET cana-1164 486 25 congruence congruence PROPN cana-1164 486 26 ๐‘Ž11 ๐‘Ž11 VERB cana-1164 486 27 โ‰ก โ‰ก PROPN cana-1164 486 28 b b PROPN cana-1164 486 29 ( ( PUNCT cana-1164 486 30 mod mod PROPN cana-1164 486 31 n n CCONJ cana-1164 486 32 ) ) PUNCT cana-1164 486 33 , , PUNCT cana-1164 486 34 proyecciones proyecciones PROPN cana-1164 486 35 journal journal NOUN cana-1164 486 36 of of ADP cana-1164 486 37 mathematics mathematic NOUN cana-1164 486 38 , , PUNCT cana-1164 486 39 42 42 NUM cana-1164 486 40 no no NOUN cana-1164 486 41 . . NOUN cana-1164 486 42 2 2 NUM cana-1164 486 43 ( ( PUNCT cana-1164 486 44 2023 2023 NUM cana-1164 486 45 ) ) PUNCT cana-1164 486 46 457 457 NUM cana-1164 486 47 - - SYM cana-1164 486 48 477 477 NUM cana-1164 486 49 . . PUNCT cana-1164 487 1 [ [ X cana-1164 487 2 9 9 NUM cana-1164 487 3 ] ] PUNCT cana-1164 487 4 w. w. PROPN cana-1164 487 5 y. y. PROPN cana-1164 487 6 jiang jiang PROPN cana-1164 487 7 , , PUNCT cana-1164 487 8 t. t. PROPN cana-1164 487 9 g. g. PROPN cana-1164 487 10 hua hua PROPN cana-1164 487 11 , , PUNCT cana-1164 487 12 the the DET cana-1164 487 13 square square ADJ cana-1164 487 14 mapping mapping NOUN cana-1164 487 15 graphs graph NOUN cana-1164 487 16 of of ADP cana-1164 487 17 the the DET cana-1164 487 18 ring ring NOUN cana-1164 488 1 โ„ค๐‘› โ„ค๐‘› PROPN cana-1164 489 1 [ [ X cana-1164 489 2 i i X cana-1164 489 3 ] ] X cana-1164 489 4 , , PUNCT cana-1164 489 5 journal journal NOUN cana-1164 489 6 of of ADP cana-1164 489 7 math math NOUN cana-1164 489 8 ( ( PUNCT cana-1164 489 9 prc prc PROPN cana-1164 489 10 ) ) PUNCT cana-1164 489 11 , , PUNCT cana-1164 489 12 36 36 NUM cana-1164 489 13 no no NOUN cana-1164 489 14 . . NOUN cana-1164 489 15 4 4 NUM cana-1164 489 16 ( ( PUNCT cana-1164 489 17 2016 2016 NUM cana-1164 489 18 ) ) PUNCT cana-1164 489 19 676682 676682 NUM cana-1164 489 20 . . PUNCT cana-1164 490 1 [ [ X cana-1164 490 2 10 10 NUM cana-1164 490 3 ] ] X cana-1164 490 4 c. c. PROPN cana-1164 490 5 lucheta lucheta PROPN cana-1164 490 6 , , PUNCT cana-1164 490 7 e. e. PROPN cana-1164 490 8 miller miller PROPN cana-1164 490 9 , , PUNCT cana-1164 490 10 and and CCONJ cana-1164 490 11 c. c. PROPN cana-1164 490 12 reiter reiter PROPN cana-1164 490 13 , , PUNCT cana-1164 490 14 digraphs digraph VERB cana-1164 490 15 from from ADP cana-1164 490 16 powers power NOUN cana-1164 490 17 modulo modulo VERB cana-1164 490 18 p p X cana-1164 490 19 , , PUNCT cana-1164 490 20 fibonacci fibonacci NOUN cana-1164 490 21 quart quart NOUN cana-1164 490 22 . . PUNCT cana-1164 491 1 , , PUNCT cana-1164 491 2 34 34 NUM cana-1164 491 3 ( ( PUNCT cana-1164 491 4 1996 1996 NUM cana-1164 491 5 ) ) PUNCT cana-1164 491 6 226 226 NUM cana-1164 491 7 - - SYM cana-1164 491 8 239 239 NUM cana-1164 491 9 . . PUNCT cana-1164 492 1 [ [ X cana-1164 492 2 11 11 NUM cana-1164 492 3 ] ] PUNCT cana-1164 492 4 m. m. NOUN cana-1164 492 5 haris haris PROPN cana-1164 492 6 mateen mateen PROPN cana-1164 492 7 , , PUNCT cana-1164 492 8 and and CCONJ cana-1164 492 9 m. m. PROPN cana-1164 492 10 khalid khalid PROPN cana-1164 492 11 mahmood mahmood PROPN cana-1164 492 12 , , PUNCT cana-1164 492 13 power power NOUN cana-1164 492 14 digraphs digraph NOUN cana-1164 492 15 associated associate VERB cana-1164 492 16 with with ADP cana-1164 492 17 the the DET cana-1164 492 18 congruence congruence PROPN cana-1164 492 19 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1164 492 20 โ‰ก โ‰ก PROPN cana-1164 492 21 y y PROPN cana-1164 492 22 ( ( PUNCT cana-1164 492 23 mod mod PROPN cana-1164 492 24 n n CCONJ cana-1164 492 25 ) ) PUNCT cana-1164 492 26 , , PUNCT cana-1164 492 27 punjab punjab PROPN cana-1164 492 28 univ univ PROPN cana-1164 492 29 . . PUNCT cana-1164 493 1 j. j. PROPN cana-1164 493 2 math math PROPN cana-1164 493 3 . . PROPN cana-1164 493 4 , , PUNCT cana-1164 493 5 51 51 NUM cana-1164 493 6 ( ( PUNCT cana-1164 493 7 2019 2019 NUM cana-1164 493 8 ) ) PUNCT cana-1164 493 9 93 93 NUM cana-1164 493 10 - - SYM cana-1164 493 11 102 102 NUM cana-1164 493 12 . . PUNCT cana-1164 494 1 [ [ X cana-1164 494 2 12 12 NUM cana-1164 494 3 ] ] X cana-1164 494 4 e. e. PROPN cana-1164 494 5 a. a. PROPN cana-1164 494 6 osba osba PROPN cana-1164 494 7 , , PUNCT cana-1164 494 8 s. s. PROPN cana-1164 494 9 a. a. PROPN cana-1164 494 10 addasi addasi PROPN cana-1164 494 11 and and CCONJ cana-1164 494 12 n.a n.a PROPN cana-1164 494 13 . . PROPN cana-1164 494 14 jaradeh jaradeh PROPN cana-1164 494 15 , , PUNCT cana-1164 494 16 zero zero NUM cana-1164 494 17 divisor divisor NOUN cana-1164 494 18 graph graph NOUN cana-1164 494 19 for for ADP cana-1164 494 20 the the DET cana-1164 494 21 ring ring NOUN cana-1164 494 22 of of ADP cana-1164 494 23 gaussian gaussian ADJ cana-1164 494 24 integers integer NOUN cana-1164 494 25 modulo modulo PROPN cana-1164 494 26 n n CCONJ cana-1164 494 27 , , PUNCT cana-1164 494 28 taylor taylor PROPN cana-1164 494 29 & & CCONJ cana-1164 494 30 francis francis PROPN cana-1164 494 31 , , PUNCT cana-1164 494 32 communication communication NOUN cana-1164 494 33 in in ADP cana-1164 494 34 algebra algebra NOUN cana-1164 494 35 , , PUNCT cana-1164 494 36 36 36 NUM cana-1164 494 37 ( ( PUNCT cana-1164 494 38 2008 2008 NUM cana-1164 494 39 ) ) PUNCT cana-1164 494 40 3865 3865 NUM cana-1164 494 41 - - SYM cana-1164 494 42 3877 3877 NUM cana-1164 494 43 . . PUNCT cana-1164 495 1 [ [ X cana-1164 495 2 13 13 NUM cana-1164 495 3 ] ] PUNCT cana-1164 495 4 m. m. NOUN cana-1164 495 5 rahmati rahmati PROPN cana-1164 495 6 , , PUNCT cana-1164 495 7 some some DET cana-1164 495 8 digraphs digraph NOUN cana-1164 495 9 attached attach VERB cana-1164 495 10 with with ADP cana-1164 495 11 congruence congruence PROPN cana-1164 495 12 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1164 495 13 โ‰ก โ‰ก PROPN cana-1164 495 14 y y PROPN cana-1164 495 15 ( ( PUNCT cana-1164 495 16 mod mod PROPN cana-1164 495 17 n n CCONJ cana-1164 495 18 ) ) PUNCT cana-1164 495 19 , , PUNCT cana-1164 495 20 journal journal NOUN cana-1164 495 21 of of ADP cana-1164 495 22 mathematical mathematical ADJ cana-1164 495 23 extension extension NOUN cana-1164 495 24 , , PUNCT cana-1164 495 25 11 11 NUM cana-1164 495 26 no no NOUN cana-1164 495 27 . . NOUN cana-1164 495 28 1 1 NUM cana-1164 495 29 ( ( PUNCT cana-1164 495 30 2017 2017 NUM cana-1164 495 31 ) ) PUNCT cana-1164 495 32 47 47 NUM cana-1164 495 33 - - SYM cana-1164 495 34 56 56 NUM cana-1164 495 35 . . PUNCT cana-1164 496 1 [ [ X cana-1164 496 2 14 14 NUM cana-1164 496 3 ] ] PUNCT cana-1164 496 4 t. t. PROPN cana-1164 496 5 d. d. PROPN cana-1164 496 6 rogers rogers PROPN cana-1164 496 7 , , PUNCT cana-1164 496 8 the the DET cana-1164 496 9 graph graph NOUN cana-1164 496 10 of of ADP cana-1164 496 11 the the DET cana-1164 496 12 square square ADJ cana-1164 496 13 mapping mapping NOUN cana-1164 496 14 on on ADP cana-1164 496 15 the the DET cana-1164 496 16 prime prime ADJ cana-1164 496 17 fields field NOUN cana-1164 496 18 , , PUNCT cana-1164 496 19 discrete discrete ADJ cana-1164 496 20 math math NOUN cana-1164 496 21 . . PUNCT cana-1164 496 22 , , PUNCT cana-1164 496 23 148 148 NUM cana-1164 496 24 ( ( PUNCT cana-1164 496 25 1996 1996 NUM cana-1164 496 26 ) ) PUNCT cana-1164 496 27 317 317 NUM cana-1164 496 28 - - SYM cana-1164 496 29 324 324 NUM cana-1164 496 30 . . PUNCT cana-1164 497 1 [ [ X cana-1164 497 2 15 15 NUM cana-1164 497 3 ] ] X cana-1164 497 4 j. j. PROPN cana-1164 497 5 skowronek skowronek PROPN cana-1164 497 6 - - PUNCT cana-1164 497 7 kaziow kaziow NOUN cana-1164 497 8 , , PUNCT cana-1164 497 9 some some DET cana-1164 497 10 digraphs digraph VERB cana-1164 497 11 arising arise VERB cana-1164 497 12 from from ADP cana-1164 497 13 number number NOUN cana-1164 497 14 theory theory NOUN cana-1164 497 15 and and CCONJ cana-1164 497 16 remarks remark NOUN cana-1164 497 17 on on ADP cana-1164 497 18 the the DET cana-1164 497 19 zero zero NUM cana-1164 497 20 - - PUNCT cana-1164 497 21 divisor divisor NOUN cana-1164 497 22 graph graph NOUN cana-1164 497 23 of of ADP cana-1164 497 24 the the DET cana-1164 497 25 ring ring NOUN cana-1164 497 26 โ„ค๐‘› โ„ค๐‘› PROPN cana-1164 497 27 , , PUNCT cana-1164 497 28 information information NOUN cana-1164 497 29 processing processing NOUN cana-1164 497 30 letters letter NOUN cana-1164 497 31 , , PUNCT cana-1164 497 32 108 108 NUM cana-1164 497 33 ( ( PUNCT cana-1164 497 34 2008 2008 NUM cana-1164 497 35 ) ) PUNCT cana-1164 497 36 165 165 NUM cana-1164 497 37 - - SYM cana-1164 497 38 169 169 NUM cana-1164 497 39 . . PUNCT cana-1164 498 1 [ [ X cana-1164 498 2 16 16 NUM cana-1164 498 3 ] ] X cana-1164 498 4 j. j. PROPN cana-1164 498 5 skowronek skowronek PROPN cana-1164 498 6 - - PUNCT cana-1164 498 7 kaziow kaziow PROPN cana-1164 498 8 , , PUNCT cana-1164 498 9 z. z. PROPN cana-1164 498 10 gora gora PROPN cana-1164 498 11 , , PUNCT cana-1164 498 12 properties property NOUN cana-1164 498 13 of of ADP cana-1164 498 14 digraphs digraph NOUN cana-1164 498 15 connected connect VERB cana-1164 498 16 with with ADP cana-1164 498 17 some some DET cana-1164 498 18 congruence congruence PROPN cana-1164 498 19 relations relation NOUN cana-1164 498 20 , , PUNCT cana-1164 498 21 czechoslovak czechoslovak ADJ cana-1164 498 22 mathematical mathematical ADJ cana-1164 498 23 journal journal NOUN cana-1164 498 24 , , PUNCT cana-1164 498 25 59 59 NUM cana-1164 498 26 no no NOUN cana-1164 498 27 . . PROPN cana-1164 499 1 134 134 NUM cana-1164 499 2 ( ( PUNCT cana-1164 499 3 2009 2009 NUM cana-1164 499 4 ) ) PUNCT cana-1164 499 5 39 39 NUM cana-1164 499 6 - - SYM cana-1164 499 7 49 49 NUM cana-1164 499 8 . . PUNCT cana-1164 500 1 [ [ X cana-1164 500 2 17 17 NUM cana-1164 500 3 ] ] X cana-1164 500 4 l. l. PROPN cana-1164 500 5 somer somer PROPN cana-1164 500 6 , , PUNCT cana-1164 500 7 and and CCONJ cana-1164 500 8 m. m. NOUN cana-1164 500 9 krizek krizek PROPN cana-1164 500 10 , , PUNCT cana-1164 500 11 on on ADP cana-1164 500 12 a a DET cana-1164 500 13 connection connection NOUN cana-1164 500 14 of of ADP cana-1164 500 15 number number NOUN cana-1164 500 16 theory theory NOUN cana-1164 500 17 with with ADP cana-1164 500 18 graph graph NOUN cana-1164 500 19 theory theory NOUN cana-1164 500 20 , , PUNCT cana-1164 500 21 czechoslovak czechoslovak ADJ cana-1164 500 22 mathematical mathematical ADJ cana-1164 500 23 journal journal NOUN cana-1164 500 24 , , PUNCT cana-1164 500 25 54 54 NUM cana-1164 500 26 ( ( PUNCT cana-1164 500 27 2004 2004 NUM cana-1164 500 28 ) ) PUNCT cana-1164 500 29 465 465 NUM cana-1164 500 30 - - SYM cana-1164 500 31 485 485 NUM cana-1164 500 32 . . PUNCT cana-1164 501 1 [ [ X cana-1164 501 2 18 18 NUM cana-1164 501 3 ] ] X cana-1164 501 4 l. l. PROPN cana-1164 501 5 somer somer PROPN cana-1164 501 6 , , PUNCT cana-1164 501 7 and and CCONJ cana-1164 501 8 m. m. NOUN cana-1164 501 9 krizek krizek PROPN cana-1164 501 10 , , PUNCT cana-1164 501 11 structure structure NOUN cana-1164 501 12 of of ADP cana-1164 501 13 digraphs digraph NOUN cana-1164 501 14 associated associate VERB cana-1164 501 15 with with ADP cana-1164 501 16 quadratic quadratic ADJ cana-1164 501 17 congruences congruence NOUN cana-1164 501 18 with with ADP cana-1164 501 19 composite composite ADJ cana-1164 501 20 moduli modulus NOUN cana-1164 501 21 , , PUNCT cana-1164 501 22 discrete discrete ADJ cana-1164 501 23 mathematics mathematic NOUN cana-1164 501 24 , , PUNCT cana-1164 501 25 306 306 NUM cana-1164 501 26 ( ( PUNCT cana-1164 501 27 2006 2006 NUM cana-1164 501 28 ) ) PUNCT cana-1164 501 29 2174 2174 NUM cana-1164 501 30 - - SYM cana-1164 501 31 2185 2185 NUM cana-1164 501 32 . . PUNCT cana-1164 502 1 [ [ X cana-1164 502 2 19 19 NUM cana-1164 502 3 ] ] X cana-1164 502 4 l. l. PROPN cana-1164 502 5 szalay szalay PROPN cana-1164 502 6 , , PUNCT cana-1164 502 7 a a DET cana-1164 502 8 discrete discrete ADJ cana-1164 502 9 iteration iteration NOUN cana-1164 502 10 in in ADP cana-1164 502 11 number number NOUN cana-1164 502 12 theory theory NOUN cana-1164 502 13 , , PUNCT cana-1164 502 14 bdtf bdtf PROPN cana-1164 502 15 tud tud PROPN cana-1164 502 16 . . PUNCT cana-1164 502 17 kรถzl kรถzl PROPN cana-1164 502 18 . . PUNCT cana-1164 502 19 , , PUNCT cana-1164 502 20 8 8 NUM cana-1164 502 21 ( ( PUNCT cana-1164 502 22 1992 1992 NUM cana-1164 502 23 ) ) PUNCT cana-1164 502 24 71 71 NUM cana-1164 502 25 - - SYM cana-1164 502 26 91 91 NUM cana-1164 502 27 ( ( PUNCT cana-1164 502 28 in in ADP cana-1164 502 29 hungarian hungarian NOUN cana-1164 502 30 ) ) PUNCT cana-1164 502 31 . . PUNCT cana-1164 503 1 [ [ X cana-1164 503 2 20 20 NUM cana-1164 503 3 ] ] PUNCT cana-1164 503 4 s. s. PROPN cana-1164 503 5 k. k. PROPN cana-1164 503 6 thakur thakur PROPN cana-1164 503 7 , , PUNCT cana-1164 503 8 p. p. NOUN cana-1164 503 9 goswami goswami NOUN cana-1164 503 10 , , PUNCT cana-1164 503 11 and and CCONJ cana-1164 503 12 g. g. PROPN cana-1164 503 13 c. c. PROPN cana-1164 503 14 ray ray PROPN cana-1164 503 15 , , PUNCT cana-1164 503 16 enumeration enumeration NOUN cana-1164 503 17 of of ADP cana-1164 503 18 cyclic cyclic ADJ cana-1164 503 19 vertices vertex NOUN cana-1164 503 20 and and CCONJ cana-1164 503 21 components component NOUN cana-1164 503 22 over over ADP cana-1164 503 23 the the DET cana-1164 503 24 congruence congruence PROPN cana-1164 503 25 ๐‘Ž11 ๐‘Ž11 PROPN cana-1164 503 26 โ‰กb โ‰กb PROPN cana-1164 503 27 ( ( PUNCT cana-1164 503 28 mod mod PROPN cana-1164 503 29 n n CCONJ cana-1164 503 30 ) ) PUNCT cana-1164 503 31 , , PUNCT cana-1164 503 32 notes note NOUN cana-1164 503 33 on on ADP cana-1164 503 34 number number NOUN cana-1164 503 35 theory theory NOUN cana-1164 503 36 and and CCONJ cana-1164 503 37 discrete discrete ADJ cana-1164 503 38 mathematics mathematic NOUN cana-1164 503 39 , , PUNCT cana-1164 503 40 29 29 NUM cana-1164 503 41 no no NOUN cana-1164 503 42 . . NOUN cana-1164 503 43 3 3 NUM cana-1164 503 44 ( ( PUNCT cana-1164 503 45 2023 2023 NUM cana-1164 503 46 ) ) PUNCT cana-1164 503 47 525 525 NUM cana-1164 503 48 - - SYM cana-1164 503 49 537 537 NUM cana-1164 503 50 . . PUNCT cana-1164 504 1 [ [ X cana-1164 504 2 21 21 NUM cana-1164 504 3 ] ] PUNCT cana-1164 504 4 s. s. PROPN cana-1164 504 5 k. k. PROPN cana-1164 504 6 thakur thakur PROPN cana-1164 504 7 , , PUNCT cana-1164 504 8 p. p. NOUN cana-1164 504 9 goswami goswami NOUN cana-1164 504 10 , , PUNCT cana-1164 504 11 and and CCONJ cana-1164 504 12 g. g. PROPN cana-1164 504 13 c. c. PROPN cana-1164 504 14 ray ray PROPN cana-1164 504 15 , , PUNCT cana-1164 504 16 some some PRON cana-1164 504 17 results result NOUN cana-1164 504 18 on on ADP cana-1164 504 19 the the DET cana-1164 504 20 degree degree NOUN cana-1164 504 21 of of ADP cana-1164 504 22 vertices vertex NOUN cana-1164 504 23 of of ADP cana-1164 504 24 the the DET cana-1164 504 25 digraphs digraph NOUN cana-1164 504 26 ( ( PUNCT cana-1164 504 27 , , PUNCT cana-1164 504 28 2)n๏‡ 2)n๏‡ NUM cana-1164 504 29 and and CCONJ cana-1164 504 30 its its PRON cana-1164 504 31 complement complement NOUN cana-1164 504 32 digraph digraph NOUN cana-1164 504 33 ( ( PUNCT cana-1164 504 34 ) ) PUNCT cana-1164 504 35 , , PUNCT cana-1164 504 36 2n๏‡ 2n๏‡ NUM cana-1164 504 37 , , PUNCT cana-1164 504 38 ( ( PUNCT cana-1164 504 39 communicated communicate VERB cana-1164 504 40 ) ) PUNCT cana-1164 504 41 . . PUNCT cana-1164 505 1 [ [ X cana-1164 505 2 22 22 NUM cana-1164 505 3 ] ] X cana-1164 505 4 c. c. PROPN cana-1164 505 5 vasudeva vasudeva PROPN cana-1164 505 6 , , PUNCT cana-1164 505 7 graph graph NOUN cana-1164 505 8 theory theory NOUN cana-1164 505 9 with with ADP cana-1164 505 10 applications application NOUN cana-1164 505 11 , , PUNCT cana-1164 505 12 new new ADJ cana-1164 505 13 age age NOUN cana-1164 505 14 international international ADJ cana-1164 505 15 ( ( PUNCT cana-1164 505 16 p p NOUN cana-1164 505 17 ) ) PUNCT cana-1164 505 18 limited limited ADJ cana-1164 505 19 publishers publisher NOUN cana-1164 505 20 , , PUNCT cana-1164 505 21 isbn isbn ADJ cana-1164 505 22 81 81 NUM cana-1164 505 23 - - PUNCT cana-1164 505 24 224 224 NUM cana-1164 505 25 - - PUNCT cana-1164 505 26 1737x 1737x NUM cana-1164 505 27 . . PUNCT