id sid tid token lemma pos cana-5226 1 1 communications communication NOUN cana-5226 1 2 on on ADP cana-5226 1 3 applied apply VERB cana-5226 1 4 nonlinear nonlinear ADJ cana-5226 1 5 analysis analysis NOUN cana-5226 1 6 issn issn NOUN cana-5226 1 7 : : PUNCT cana-5226 1 8 1074 1074 NUM cana-5226 1 9 - - PUNCT cana-5226 1 10 133x 133x NUM cana-5226 1 11 vol vol VERB cana-5226 1 12 32 32 NUM cana-5226 1 13 no no NOUN cana-5226 1 14 . . PUNCT cana-5226 2 1 10s 10 NOUN cana-5226 2 2 ( ( PUNCT cana-5226 2 3 2025 2025 NUM cana-5226 2 4 ) ) PUNCT cana-5226 2 5 1217 1217 NUM cana-5226 2 6 https://internationalpubls.com https://internationalpubls.com X cana-5226 3 1 a a DET cana-5226 3 2 general general ADJ cana-5226 3 3 characteristic characteristic ADJ cana-5226 3 4 equation equation NOUN cana-5226 3 5 for for ADP cana-5226 3 6 eigen eigen PROPN cana-5226 3 7 values value NOUN cana-5226 3 8 and and CCONJ cana-5226 3 9 energy energy NOUN cana-5226 3 10 of of ADP cana-5226 3 11 cycle cycle NOUN cana-5226 3 12 graph graph NOUN cana-5226 3 13 aadarsh aadarsh ADJ cana-5226 3 14 chaudhary1 chaudhary1 NOUN cana-5226 3 15 and and CCONJ cana-5226 3 16 kamesh kamesh ADJ cana-5226 3 17 kumar2 kumar2 PROPN cana-5226 3 18 1research 1research NUM cana-5226 3 19 scholar scholar NOUN cana-5226 3 20 , , PUNCT cana-5226 3 21 department department NOUN cana-5226 3 22 of of ADP cana-5226 3 23 mathematics mathematic NOUN cana-5226 3 24 , , PUNCT cana-5226 3 25 faculty faculty NOUN cana-5226 3 26 of of ADP cana-5226 3 27 engineering engineering NOUN cana-5226 3 28 , , PUNCT cana-5226 3 29 teerthanker teerthanker NOUN cana-5226 3 30 mahaveer mahaveer PROPN cana-5226 3 31 university university PROPN cana-5226 3 32 , , PUNCT cana-5226 3 33 moradabad moradabad PROPN cana-5226 3 34 2assistant 2assistant PROPN cana-5226 3 35 professor professor NOUN cana-5226 3 36 , , PUNCT cana-5226 3 37 department department NOUN cana-5226 3 38 of of ADP cana-5226 3 39 mathematics mathematic NOUN cana-5226 3 40 , , PUNCT cana-5226 3 41 faculty faculty NOUN cana-5226 3 42 of of ADP cana-5226 3 43 engineering engineering NOUN cana-5226 3 44 , , PUNCT cana-5226 3 45 teerthanker teerthanker NOUN cana-5226 3 46 mahaveer mahaveer PROPN cana-5226 3 47 university university PROPN cana-5226 3 48 , , PUNCT cana-5226 3 49 moradabad moradabad PROPN cana-5226 3 50 email email NOUN cana-5226 3 51 : : PUNCT cana-5226 3 52 1aadarshchaudhary277@gmail.com 1aadarshchaudhary277@gmail.com NUM cana-5226 3 53 , , PUNCT cana-5226 3 54 2drkamesh.engineering@tmu.ac.in 2drkamesh.engineering@tmu.ac.in NUM cana-5226 3 55 article article NOUN cana-5226 3 56 history history NOUN cana-5226 3 57 : : PUNCT cana-5226 3 58 received receive VERB cana-5226 3 59 : : PUNCT cana-5226 3 60 12 12 NUM cana-5226 3 61 - - SYM cana-5226 3 62 01 01 NUM cana-5226 3 63 - - PUNCT cana-5226 3 64 2025 2025 NUM cana-5226 3 65 revised revise VERB cana-5226 3 66 : : PUNCT cana-5226 3 67 15 15 NUM cana-5226 3 68 - - NUM cana-5226 3 69 02 02 NUM cana-5226 3 70 - - PUNCT cana-5226 3 71 2025 2025 NUM cana-5226 3 72 accepted accept VERB cana-5226 3 73 : : PUNCT cana-5226 3 74 01 01 NUM cana-5226 3 75 - - SYM cana-5226 3 76 03 03 NUM cana-5226 3 77 - - PUNCT cana-5226 3 78 2025 2025 NUM cana-5226 3 79 abstract abstract NOUN cana-5226 3 80 : : PUNCT cana-5226 3 81 the the DET cana-5226 3 82 characteristic characteristic ADJ cana-5226 3 83 equation equation NOUN cana-5226 3 84 of of ADP cana-5226 3 85 a a DET cana-5226 3 86 graph graph NOUN cana-5226 3 87 frequently frequently ADV cana-5226 3 88 appears appear VERB cana-5226 3 89 in in ADP cana-5226 3 90 mathematical mathematical ADJ cana-5226 3 91 sciences science NOUN cana-5226 3 92 , , PUNCT cana-5226 3 93 chemistry chemistry NOUN cana-5226 3 94 and and CCONJ cana-5226 3 95 physics physics NOUN cana-5226 3 96 . . PUNCT cana-5226 4 1 as as SCONJ cana-5226 4 2 to to PART cana-5226 4 3 graph graph VERB cana-5226 4 4 theorists theorist NOUN cana-5226 4 5 , , PUNCT cana-5226 4 6 the the DET cana-5226 4 7 characteristic characteristic ADJ cana-5226 4 8 equation equation NOUN cana-5226 4 9 tells tell VERB cana-5226 4 10 information information NOUN cana-5226 4 11 about about ADP cana-5226 4 12 the the DET cana-5226 4 13 structural structural ADJ cana-5226 4 14 properties property NOUN cana-5226 4 15 of of ADP cana-5226 4 16 a a DET cana-5226 4 17 graph graph NOUN cana-5226 4 18 . . PUNCT cana-5226 5 1 in in ADP cana-5226 5 2 this this DET cana-5226 5 3 paper paper NOUN cana-5226 5 4 , , PUNCT cana-5226 5 5 we we PRON cana-5226 5 6 find find VERB cana-5226 5 7 out out ADP cana-5226 5 8 the the DET cana-5226 5 9 characteristic characteristic ADJ cana-5226 5 10 equations equation NOUN cana-5226 5 11 & & CCONJ cana-5226 5 12 eigen eigen PROPN cana-5226 5 13 value value NOUN cana-5226 5 14 of of ADP cana-5226 5 15 cycle cycle NOUN cana-5226 5 16 cn cn PROPN cana-5226 5 17 of of ADP cana-5226 5 18 n n PRON cana-5226 5 19 vertices vertex NOUN cana-5226 5 20 . . PUNCT cana-5226 6 1 keywords keyword NOUN cana-5226 6 2 : : PUNCT cana-5226 6 3 characteristic characteristic ADJ cana-5226 6 4 equation equation NOUN cana-5226 6 5 , , PUNCT cana-5226 6 6 eigen eigen PROPN cana-5226 6 7 value value NOUN cana-5226 6 8 , , PUNCT cana-5226 6 9 cycle cycle NOUN cana-5226 6 10 graph graph NOUN cana-5226 6 11 , , PUNCT cana-5226 6 12 structural structural ADJ cana-5226 6 13 properties property NOUN cana-5226 6 14 . . PUNCT cana-5226 7 1 1 1 X cana-5226 7 2 . . X cana-5226 7 3 introduction introduction NOUN cana-5226 7 4 all all DET cana-5226 7 5 considered consider VERB cana-5226 7 6 graph graph NOUN cana-5226 7 7 in in ADP cana-5226 7 8 this this DET cana-5226 7 9 paper paper NOUN cana-5226 7 10 are be AUX cana-5226 7 11 simple simple ADJ cana-5226 7 12 , , PUNCT cana-5226 7 13 undirected undirected ADJ cana-5226 7 14 , , PUNCT cana-5226 7 15 and and CCONJ cana-5226 7 16 connected connect VERB cana-5226 7 17 . . PUNCT cana-5226 8 1 assume assume VERB cana-5226 8 2 that that SCONJ cana-5226 8 3 g g PROPN cana-5226 8 4 is be AUX cana-5226 8 5 a a DET cana-5226 8 6 simple simple ADJ cana-5226 8 7 , , PUNCT cana-5226 8 8 finite finite ADJ cana-5226 8 9 , , PUNCT cana-5226 8 10 undirected undirected ADJ cana-5226 8 11 graph graph NOUN cana-5226 8 12 with with ADP cana-5226 8 13 vertex vertex NOUN cana-5226 8 14 set set VERB cana-5226 8 15 ๐‘‰(๐บ ๐‘‰(๐บ NOUN cana-5226 8 16 ) ) PUNCT cana-5226 8 17 and and CCONJ cana-5226 8 18 edge edge VERB cana-5226 8 19 set set VERB cana-5226 8 20 ๐ธ(๐บ ๐ธ(๐บ PROPN cana-5226 8 21 ) ) PUNCT cana-5226 8 22 . . PUNCT cana-5226 9 1 g g PROPN cana-5226 9 2 has have VERB cana-5226 9 3 n n NUM cana-5226 9 4 vertices vertex NOUN cana-5226 9 5 , , PUNCT cana-5226 9 6 and and CCONJ cana-5226 9 7 each each DET cana-5226 9 8 vertex vertex NOUN cana-5226 9 9 is be AUX cana-5226 9 10 identified identify VERB cana-5226 9 11 by by ADP cana-5226 9 12 the the DET cana-5226 9 13 label label NOUN cana-5226 9 14 v1 v1 NOUN cana-5226 9 15 , , PUNCT cana-5226 9 16 v2 v2 PROPN cana-5226 9 17 , , PUNCT cana-5226 9 18 โ€ฆ โ€ฆ PUNCT cana-5226 9 19 , , PUNCT cana-5226 9 20 vn vn PROPN cana-5226 9 21 . . PUNCT cana-5226 10 1 the the DET cana-5226 10 2 adjacency adjacency PROPN cana-5226 10 3 matrix matrix NOUN cana-5226 10 4 x(g x(g PROPN cana-5226 10 5 ) ) PUNCT cana-5226 10 6 of of ADP cana-5226 10 7 the the DET cana-5226 10 8 graph graph NOUN cana-5226 10 9 g g PROPN cana-5226 10 10 is be AUX cana-5226 10 11 a a DET cana-5226 10 12 square square ADJ cana-5226 10 13 matrix matrix NOUN cana-5226 10 14 of of ADP cana-5226 10 15 order order NOUN cana-5226 10 16 n n CCONJ cana-5226 10 17 , , PUNCT cana-5226 10 18 whose whose DET cana-5226 10 19 ( ( PUNCT cana-5226 10 20 i i PROPN cana-5226 10 21 , , PUNCT cana-5226 10 22 j j NOUN cana-5226 10 23 ) ) PUNCT cana-5226 10 24 entry entry NOUN cana-5226 10 25 is be AUX cana-5226 10 26 equal equal ADJ cana-5226 10 27 to to ADP cana-5226 10 28 1 1 NUM cana-5226 10 29 if if SCONJ cana-5226 10 30 the the DET cana-5226 10 31 vertices vertex NOUN cana-5226 10 32 vi vi PROPN cana-5226 10 33 and and CCONJ cana-5226 10 34 vj vj NOUN cana-5226 10 35 are be AUX cana-5226 10 36 adjacent adjacent ADJ cana-5226 10 37 and and CCONJ cana-5226 10 38 is be AUX cana-5226 10 39 equal equal ADJ cana-5226 10 40 to to ADP cana-5226 10 41 zero zero NUM cana-5226 10 42 otherwise otherwise ADV cana-5226 10 43 . . PUNCT cana-5226 11 1 adjacent adjacent ADJ cana-5226 11 2 matrix matrix NOUN cana-5226 11 3 will will AUX cana-5226 11 4 be be AUX cana-5226 11 5 symmetric symmetric ADJ cana-5226 11 6 binary binary ADJ cana-5226 11 7 matrix matrix NOUN cana-5226 11 8 for for ADP cana-5226 11 9 simple simple ADJ cana-5226 11 10 graph graph NOUN cana-5226 11 11 and and CCONJ cana-5226 11 12 all all DET cana-5226 11 13 entries entry NOUN cana-5226 11 14 along along ADP cana-5226 11 15 principal principal ADJ cana-5226 11 16 diagonal diagonal NOUN cana-5226 11 17 of of ADP cana-5226 11 18 x(g x(g NOUN cana-5226 11 19 ) ) PUNCT cana-5226 11 20 are be AUX cana-5226 11 21 all all PRON cana-5226 11 22 0 0 NUM cana-5226 11 23 โ€™s โ€™s NOUN cana-5226 11 24 for for ADP cana-5226 11 25 simple simple ADJ cana-5226 11 26 graph graph NOUN cana-5226 11 27 . . PUNCT cana-5226 12 1 ( ( PUNCT cana-5226 12 2 brooks brooks PROPN cana-5226 12 3 ) ) PUNCT cana-5226 12 4 the the DET cana-5226 12 5 characteristic characteristic ADJ cana-5226 12 6 polynomial polynomial NOUN cana-5226 12 7 of of ADP cana-5226 12 8 the the DET cana-5226 12 9 graph graph NOUN cana-5226 12 10 g g PROPN cana-5226 12 11 along along ADP cana-5226 12 12 the the DET cana-5226 12 13 adjacency adjacency NOUN cana-5226 12 14 matrix matrix NOUN cana-5226 12 15 x(g x(g PROPN cana-5226 12 16 ) ) PUNCT cana-5226 12 17 is be AUX cana-5226 12 18 detโก(ฮปin detโก(ฮปin ADJ cana-5226 12 19 โˆ’ โˆ’ PROPN cana-5226 12 20 x(g x(g NOUN cana-5226 12 21 ) ) PUNCT cana-5226 12 22 ) ) PUNCT cana-5226 12 23 , , PUNCT cana-5226 12 24 where where SCONJ cana-5226 12 25 in in ADP cana-5226 12 26 is be AUX cana-5226 12 27 the the DET cana-5226 12 28 unit unit NOUN cana-5226 12 29 matrix matrix NOUN cana-5226 12 30 of of ADP cana-5226 12 31 order order NOUN cana-5226 12 32 n n NOUN cana-5226 12 33 and and CCONJ cana-5226 12 34 will will AUX cana-5226 12 35 be be AUX cana-5226 12 36 denoted denote VERB cana-5226 12 37 by by ADP cana-5226 12 38 ฮด(g ฮด(g PROPN cana-5226 12 39 , , PUNCT cana-5226 12 40 ฮป ฮป NOUN cana-5226 12 41 ) ) PUNCT cana-5226 12 42 . . PUNCT cana-5226 13 1 detโก(ฮปin detโก(ฮปin NOUN cana-5226 14 1 โˆ’ โˆ’ NUM cana-5226 14 2 x(g x(g NOUN cana-5226 14 3 ) ) PUNCT cana-5226 14 4 ) ) PUNCT cana-5226 15 1 = = SYM cana-5226 15 2 0 0 NUM cana-5226 15 3 is be AUX cana-5226 15 4 the the DET cana-5226 15 5 characteristic characteristic ADJ cana-5226 15 6 equation equation NOUN cana-5226 15 7 of of ADP cana-5226 15 8 graph graph NOUN cana-5226 15 9 g g PROPN cana-5226 15 10 and and CCONJ cana-5226 15 11 the the DET cana-5226 15 12 eigenvalues eigenvalue NOUN cana-5226 15 13 of of ADP cana-5226 15 14 a a DET cana-5226 15 15 graph graph NOUN cana-5226 15 16 g g NOUN cana-5226 15 17 are be AUX cana-5226 15 18 defined define VERB cana-5226 15 19 as as ADP cana-5226 15 20 the the DET cana-5226 15 21 eigenvalues eigenvalue NOUN cana-5226 15 22 of of ADP cana-5226 15 23 its its PRON cana-5226 15 24 adjacency adjacency NOUN cana-5226 15 25 matrix matrix NOUN cana-5226 15 26 x(g x(g PROPN cana-5226 15 27 ) ) PUNCT cana-5226 15 28 . . PUNCT cana-5226 16 1 ( ( PUNCT cana-5226 16 2 kumar kumar PROPN cana-5226 16 3 et et PROPN cana-5226 16 4 . . PUNCT cana-5226 17 1 al al PROPN cana-5226 17 2 ) ) PUNCT cana-5226 18 1 so so SCONJ cana-5226 18 2 they they PRON cana-5226 18 3 are be AUX cana-5226 18 4 just just ADV cana-5226 18 5 the the DET cana-5226 18 6 roots root NOUN cana-5226 18 7 of of ADP cana-5226 18 8 the the DET cana-5226 18 9 characteristic characteristic ADJ cana-5226 18 10 equation equation NOUN cana-5226 18 11 det det NOUN cana-5226 18 12 โก(ฮปin โก(ฮปin NOUN cana-5226 18 13 โˆ’ โˆ’ PROPN cana-5226 18 14 x(g x(g NOUN cana-5226 18 15 ) ) PUNCT cana-5226 18 16 ) ) PUNCT cana-5226 19 1 = = SYM cana-5226 19 2 0 0 NUM cana-5226 19 3 i.e. i.e. X cana-5226 19 4 ฮด(g ฮด(g X cana-5226 19 5 , , PUNCT cana-5226 19 6 ฮป ฮป NOUN cana-5226 19 7 ) ) PUNCT cana-5226 19 8 = = SYM cana-5226 19 9 0 0 X cana-5226 19 10 . . PUNCT cana-5226 19 11 since since SCONJ cana-5226 19 12 x(g x(g PROPN cana-5226 19 13 ) ) PUNCT cana-5226 19 14 is be AUX cana-5226 19 15 a a DET cana-5226 19 16 real real ADJ cana-5226 19 17 symmetric symmetric ADJ cana-5226 19 18 matrix matrix NOUN cana-5226 19 19 , , PUNCT cana-5226 19 20 so so CCONJ cana-5226 19 21 its its PRON cana-5226 19 22 eigenvalues eigenvalue NOUN cana-5226 19 23 are be AUX cana-5226 19 24 all all ADV cana-5226 19 25 real real ADJ cana-5226 19 26 with with ADP cana-5226 19 27 sum sum NOUN cana-5226 19 28 equal equal ADJ cana-5226 19 29 to to ADP cana-5226 19 30 zero zero NUM cana-5226 19 31 . . PUNCT cana-5226 20 1 denoting denote VERB cana-5226 20 2 them they PRON cana-5226 20 3 by by ADP cana-5226 20 4 ฮป1 ฮป1 PROPN cana-5226 20 5 , , PUNCT cana-5226 20 6 ฮป2 ฮป2 NOUN cana-5226 20 7 , , PUNCT cana-5226 20 8 โ€ฆ โ€ฆ PUNCT cana-5226 20 9 , , PUNCT cana-5226 20 10 ฮปn ฮปn NOUN cana-5226 20 11 and and CCONJ cana-5226 20 12 as as ADP cana-5226 20 13 a a DET cana-5226 20 14 whole whole NOUN cana-5226 20 15 , , PUNCT cana-5226 20 16 they they PRON cana-5226 20 17 are be AUX cana-5226 20 18 called call VERB cana-5226 20 19 the the DET cana-5226 20 20 spectrum spectrum NOUN cana-5226 20 21 of of ADP cana-5226 20 22 g. g. PROPN cana-5226 20 23 for for ADP cana-5226 20 24 specifics specific NOUN cana-5226 20 25 , , PUNCT cana-5226 20 26 spectral spectral ADJ cana-5226 20 27 aspects aspect NOUN cana-5226 20 28 of of ADP cana-5226 20 29 graphs graph NOUN cana-5226 20 30 , , PUNCT cana-5226 20 31 including include VERB cana-5226 20 32 characteristics characteristic NOUN cana-5226 20 33 of of ADP cana-5226 20 34 the the DET cana-5226 20 35 characteristic characteristic ADJ cana-5226 20 36 polynomial polynomial NOUN cana-5226 20 37 , , PUNCT cana-5226 20 38 have have AUX cana-5226 20 39 been be AUX cana-5226 20 40 thoroughly thoroughly ADV cana-5226 20 41 investigated investigate VERB cana-5226 20 42 , , PUNCT cana-5226 20 43 see see VERB cana-5226 20 44 ( ( PUNCT cana-5226 20 45 brouwer brouwer PROPN cana-5226 20 46 & & CCONJ cana-5226 20 47 haemers haemer NOUN cana-5226 20 48 ) ) PUNCT cana-5226 20 49 characteristic characteristic ADJ cana-5226 20 50 polynomials polynomial NOUN cana-5226 20 51 ฮด(๐บ ฮด(๐บ NOUN cana-5226 20 52 , , PUNCT cana-5226 20 53 ๐œ† ๐œ† NOUN cana-5226 20 54 ) ) PUNCT cana-5226 20 55 = = SYM cana-5226 21 1 ๐œ†๐‘› ๐œ†๐‘› ADP cana-5226 22 1 โˆ’ โˆ’ NOUN cana-5226 22 2 ๐‘†1,๐‘›๐œ† ๐‘†1,๐‘›๐œ† PROPN cana-5226 22 3 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-5226 22 4 + + CCONJ cana-5226 22 5 ๐‘†2,๐‘›๐œ† ๐‘†2,๐‘›๐œ† PROPN cana-5226 22 6 ๐‘›โˆ’2 ๐‘›โˆ’2 PROPN cana-5226 22 7 โˆ’ โˆ’ NOUN cana-5226 22 8 ๐‘†3,๐‘›๐œ† ๐‘†3,๐‘›๐œ† PROPN cana-5226 22 9 ๐‘›โˆ’3 ๐‘›โˆ’3 PROPN cana-5226 22 10 + + PROPN cana-5226 22 11 . . PUNCT cana-5226 22 12 . . PUNCT cana-5226 22 13 . . PUNCT cana-5226 23 1 + + ADV cana-5226 23 2 ( ( PUNCT cana-5226 23 3 โˆ’1)๐‘˜๐‘†๐‘˜,๐‘›๐œ† โˆ’1)๐‘˜๐‘†๐‘˜,๐‘›๐œ† PROPN cana-5226 23 4 ๐‘›โˆ’๐‘˜+ ๐‘›โˆ’๐‘˜+ PROPN cana-5226 23 5 . . PUNCT cana-5226 23 6 . . PUNCT cana-5226 23 7 . . PUNCT cana-5226 24 1 + + ADV cana-5226 24 2 ( ( PUNCT cana-5226 24 3 โˆ’1)๐‘›๐‘†๐‘›,๐‘› โˆ’1)๐‘›๐‘†๐‘›,๐‘› PROPN cana-5226 24 4 mailto:aadarshchaudhary277@gmail.com mailto:aadarshchaudhary277@gmail.com PROPN cana-5226 24 5 mailto:drkamesh.engineering@tmu.ac.in mailto:drkamesh.engineering@tmu.ac.in PROPN cana-5226 24 6 communications communication NOUN cana-5226 24 7 on on ADP cana-5226 24 8 applied apply VERB cana-5226 24 9 nonlinear nonlinear ADJ cana-5226 24 10 analysis analysis NOUN cana-5226 24 11 issn issn NOUN cana-5226 24 12 : : PUNCT cana-5226 24 13 1074 1074 NUM cana-5226 24 14 - - PUNCT cana-5226 24 15 133x 133x NUM cana-5226 24 16 vol vol VERB cana-5226 24 17 32 32 NUM cana-5226 24 18 no no NOUN cana-5226 24 19 . . PUNCT cana-5226 25 1 10s 10 NOUN cana-5226 25 2 ( ( PUNCT cana-5226 25 3 2025 2025 NUM cana-5226 25 4 ) ) PUNCT cana-5226 25 5 1218 1218 NUM cana-5226 25 6 https://internationalpubls.com https://internationalpubls.com X cana-5226 25 7 where where SCONJ cana-5226 25 8 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 25 9 is be AUX cana-5226 25 10 the the DET cana-5226 25 11 sum sum NOUN cana-5226 25 12 of of ADP cana-5226 25 13 the the DET cana-5226 25 14 principal principal ADJ cana-5226 25 15 minors minor NOUN cana-5226 25 16 of of ADP cana-5226 25 17 order order NOUN cana-5226 25 18 k k X cana-5226 25 19 in in ADP cana-5226 25 20 a a DET cana-5226 25 21 matrix matrix NOUN cana-5226 25 22 of of ADP cana-5226 25 23 order order NOUN cana-5226 25 24 n. n. NOUN cana-5226 25 25 since since SCONJ cana-5226 25 26 cycle cycle NOUN cana-5226 25 27 graphs graph NOUN cana-5226 25 28 have have VERB cana-5226 25 29 so so ADV cana-5226 25 30 many many ADJ cana-5226 25 31 uses use NOUN cana-5226 25 32 in in ADP cana-5226 25 33 domains domain NOUN cana-5226 25 34 like like ADP cana-5226 25 35 computer computer NOUN cana-5226 25 36 science science NOUN cana-5226 25 37 , , PUNCT cana-5226 25 38 biology biology NOUN cana-5226 25 39 , , PUNCT cana-5226 25 40 and and CCONJ cana-5226 25 41 chemistry chemistry NOUN cana-5226 25 42 , , PUNCT cana-5226 25 43 graph graph NOUN cana-5226 25 44 theory theory NOUN cana-5226 25 45 has have AUX cana-5226 25 46 explored explore VERB cana-5226 25 47 cycle cycle NOUN cana-5226 25 48 graphs graph NOUN cana-5226 25 49 extensively extensively ADV cana-5226 25 50 . . PUNCT cana-5226 26 1 even even ADV cana-5226 26 2 with with ADP cana-5226 26 3 its its PRON cana-5226 26 4 significance significance NOUN cana-5226 26 5 , , PUNCT cana-5226 26 6 figuring figure VERB cana-5226 26 7 out out ADP cana-5226 26 8 cycle cycle NOUN cana-5226 26 9 graph graph NOUN cana-5226 26 10 eigenvalues eigenvalue NOUN cana-5226 26 11 and and CCONJ cana-5226 26 12 energy energy NOUN cana-5226 26 13 has have AUX cana-5226 26 14 proven prove VERB cana-5226 26 15 to to PART cana-5226 26 16 be be AUX cana-5226 26 17 a a DET cana-5226 26 18 difficult difficult ADJ cana-5226 26 19 undertaking undertaking NOUN cana-5226 26 20 . . PUNCT cana-5226 27 1 in in ADP cana-5226 27 2 this this DET cana-5226 27 3 study study NOUN cana-5226 27 4 , , PUNCT cana-5226 27 5 we we PRON cana-5226 27 6 provide provide VERB cana-5226 27 7 an an DET cana-5226 27 8 equation equation NOUN cana-5226 27 9 for for ADP cana-5226 27 10 the the DET cana-5226 27 11 general general ADJ cana-5226 27 12 characteristics characteristic NOUN cana-5226 27 13 that that PRON cana-5226 27 14 can can AUX cana-5226 27 15 be be AUX cana-5226 27 16 applied apply VERB cana-5226 27 17 to to ADP cana-5226 27 18 any any DET cana-5226 27 19 cycle cycle NOUN cana-5226 27 20 graph graph NOUN cana-5226 27 21 to to PART cana-5226 27 22 find find VERB cana-5226 27 23 its its PRON cana-5226 27 24 energy energy NOUN cana-5226 27 25 and and CCONJ cana-5226 27 26 eigenvalues eigenvalue NOUN cana-5226 27 27 . . PUNCT cana-5226 28 1 the the DET cana-5226 28 2 goal goal NOUN cana-5226 28 3 of of ADP cana-5226 28 4 this this DET cana-5226 28 5 research research NOUN cana-5226 28 6 is be AUX cana-5226 28 7 to to PART cana-5226 28 8 present present VERB cana-5226 28 9 a a DET cana-5226 28 10 thorough thorough ADJ cana-5226 28 11 explanation explanation NOUN cana-5226 28 12 of of ADP cana-5226 28 13 the the DET cana-5226 28 14 equation equation NOUN cana-5226 28 15 's 's PART cana-5226 28 16 development development NOUN cana-5226 28 17 and and CCONJ cana-5226 28 18 its its PRON cana-5226 28 19 numerous numerous ADJ cana-5226 28 20 uses use NOUN cana-5226 28 21 . . PUNCT cana-5226 29 1 this this DET cana-5226 29 2 work work NOUN cana-5226 29 3 aims aim VERB cana-5226 29 4 to to PART cana-5226 29 5 address address VERB cana-5226 29 6 the the DET cana-5226 29 7 research research NOUN cana-5226 29 8 challenge challenge NOUN cana-5226 29 9 of of ADP cana-5226 29 10 not not PART cana-5226 29 11 having have VERB cana-5226 29 12 a a DET cana-5226 29 13 generic generic ADJ cana-5226 29 14 equation equation NOUN cana-5226 29 15 for for ADP cana-5226 29 16 calculating calculate VERB cana-5226 29 17 the the DET cana-5226 29 18 energy energy NOUN cana-5226 29 19 and and CCONJ cana-5226 29 20 eigenvalues eigenvalue NOUN cana-5226 29 21 of of ADP cana-5226 29 22 cycle cycle NOUN cana-5226 29 23 graphs graph NOUN cana-5226 29 24 . . PUNCT cana-5226 30 1 we we PRON cana-5226 30 2 can can AUX cana-5226 30 3 improve improve VERB cana-5226 30 4 our our PRON cana-5226 30 5 knowledge knowledge NOUN cana-5226 30 6 of of ADP cana-5226 30 7 cycle cycle NOUN cana-5226 30 8 graphs graph NOUN cana-5226 30 9 and and CCONJ cana-5226 30 10 their their PRON cana-5226 30 11 applications application NOUN cana-5226 30 12 by by ADP cana-5226 30 13 proving prove VERB cana-5226 30 14 this this DET cana-5226 30 15 equation equation NOUN cana-5226 30 16 . . PUNCT cana-5226 31 1 the the DET cana-5226 31 2 steps step NOUN cana-5226 31 3 taken take VERB cana-5226 31 4 to to PART cana-5226 31 5 construct construct VERB cana-5226 31 6 the the DET cana-5226 31 7 characteristic characteristic ADJ cana-5226 31 8 equation equation NOUN cana-5226 31 9 , , PUNCT cana-5226 31 10 its its PRON cana-5226 31 11 derivation derivation NOUN cana-5226 31 12 , , PUNCT cana-5226 31 13 and and CCONJ cana-5226 31 14 its its PRON cana-5226 31 15 applications application NOUN cana-5226 31 16 in in ADP cana-5226 31 17 diverse diverse ADJ cana-5226 31 18 sectors sector NOUN cana-5226 31 19 will will AUX cana-5226 31 20 be be AUX cana-5226 31 21 the the DET cana-5226 31 22 key key ADJ cana-5226 31 23 points point NOUN cana-5226 31 24 of of ADP cana-5226 31 25 contention contention NOUN cana-5226 31 26 in in ADP cana-5226 31 27 this this DET cana-5226 31 28 work work NOUN cana-5226 31 29 . . PUNCT cana-5226 32 1 in in ADP cana-5226 32 2 summary summary NOUN cana-5226 32 3 , , PUNCT cana-5226 32 4 this this DET cana-5226 32 5 study study NOUN cana-5226 32 6 will will AUX cana-5226 32 7 advance advance VERB cana-5226 32 8 the the DET cana-5226 32 9 subject subject NOUN cana-5226 32 10 of of ADP cana-5226 32 11 graph graph NOUN cana-5226 32 12 theory theory NOUN cana-5226 32 13 by by ADP cana-5226 32 14 offering offer VERB cana-5226 32 15 a a DET cana-5226 32 16 general general ADJ cana-5226 32 17 formula formula NOUN cana-5226 32 18 for for ADP cana-5226 32 19 calculating calculate VERB cana-5226 32 20 cycle cycle NOUN cana-5226 32 21 graph graph NOUN cana-5226 32 22 energy energy NOUN cana-5226 32 23 and and CCONJ cana-5226 32 24 eigenvalues eigenvalue NOUN cana-5226 32 25 , , PUNCT cana-5226 32 26 which which PRON cana-5226 32 27 can can AUX cana-5226 32 28 have have VERB cana-5226 32 29 important important ADJ cana-5226 32 30 applications application NOUN cana-5226 32 31 across across ADP cana-5226 32 32 a a DET cana-5226 32 33 range range NOUN cana-5226 32 34 of of ADP cana-5226 32 35 domains domain NOUN cana-5226 32 36 . . PUNCT cana-5226 33 1 a a DET cana-5226 33 2 graph graph NOUN cana-5226 33 3 's 's PART cana-5226 33 4 energy energy NOUN cana-5226 33 5 can can AUX cana-5226 33 6 be be AUX cana-5226 33 7 expressed express VERB cana-5226 33 8 as as ADP cana-5226 33 9 the the DET cana-5226 33 10 total total NOUN cana-5226 33 11 of of ADP cana-5226 33 12 the the DET cana-5226 33 13 absolute absolute ADJ cana-5226 33 14 values value NOUN cana-5226 33 15 of of ADP cana-5226 33 16 its its PRON cana-5226 33 17 adjacency adjacency NOUN cana-5226 33 18 matrix matrix NOUN cana-5226 33 19 's 's PART cana-5226 33 20 eigenvalues eigenvalue NOUN cana-5226 33 21 . . PUNCT cana-5226 34 1 stated state VERB cana-5226 34 2 otherwise otherwise ADV cana-5226 34 3 , , PUNCT cana-5226 34 4 the the DET cana-5226 34 5 total total NOUN cana-5226 34 6 of of ADP cana-5226 34 7 the the DET cana-5226 34 8 eigenvalues eigenvalue NOUN cana-5226 34 9 ' ' PART cana-5226 34 10 magnitudes magnitude NOUN cana-5226 34 11 represents represent VERB cana-5226 34 12 it it PRON cana-5226 34 13 . . PUNCT cana-5226 35 1 in in ADP cana-5226 35 2 mathematical mathematical ADJ cana-5226 35 3 terms term NOUN cana-5226 35 4 , , PUNCT cana-5226 35 5 the the DET cana-5226 35 6 energy energy NOUN cana-5226 35 7 e(g e(g PROPN cana-5226 35 8 ) ) PUNCT cana-5226 35 9 of of ADP cana-5226 35 10 the the DET cana-5226 35 11 graph graph NOUN cana-5226 35 12 g g NOUN cana-5226 35 13 is be AUX cana-5226 35 14 given give VERB cana-5226 35 15 by by ADP cana-5226 35 16 : : PUNCT cana-5226 35 17 if if SCONJ cana-5226 35 18 the the DET cana-5226 35 19 eigenvalues eigenvalue NOUN cana-5226 35 20 of of ADP cana-5226 35 21 the the DET cana-5226 35 22 adjacency adjacency NOUN cana-5226 35 23 matrix matrix NOUN cana-5226 35 24 a a PRON cana-5226 35 25 are be AUX cana-5226 35 26 ฮปโ‚ ฮปโ‚ PROPN cana-5226 35 27 , , PUNCT cana-5226 35 28 ฮปโ‚‚ ฮปโ‚‚ PROPN cana-5226 35 29 , , PUNCT cana-5226 35 30 . . PUNCT cana-5226 35 31 . . PUNCT cana-5226 36 1 . . PUNCT cana-5226 37 1 , , PUNCT cana-5226 37 2 ฮปโ‚™. ฮปโ‚™. PROPN cana-5226 37 3 then then ADV cana-5226 37 4 graph graph VERB cana-5226 37 5 energy energy NOUN cana-5226 37 6 ๐ธ(๐บ ๐ธ(๐บ PROPN cana-5226 37 7 ) ) PUNCT cana-5226 38 1 = = PROPN cana-5226 38 2 โˆฃ โˆฃ ADJ cana-5226 38 3 ๐œ†1 ๐œ†1 NOUN cana-5226 38 4 โˆฃ โˆฃ VERB cana-5226 38 5 + + PROPN cana-5226 38 6 โˆฃ โˆฃ ADJ cana-5226 38 7 ๐œ†2 ๐œ†2 NOUN cana-5226 38 8 โˆฃ โˆฃ ADJ cana-5226 38 9 + + PROPN cana-5226 38 10 . . PUNCT cana-5226 38 11 . . PUNCT cana-5226 38 12 . . PUNCT cana-5226 39 1 + + ADJ cana-5226 39 2 โˆฃ โˆฃ PROPN cana-5226 39 3 ๐œ†๐‘› ๐œ†๐‘› PROPN cana-5226 39 4 โˆฃ โˆฃ PROPN cana-5226 39 5 ( ( PUNCT cana-5226 39 6 estrada estrada PROPN cana-5226 39 7 & & CCONJ cana-5226 39 8 benzi benzi PROPN cana-5226 39 9 ) ) PUNCT cana-5226 39 10 when when SCONJ cana-5226 39 11 examining examine VERB cana-5226 39 12 the the DET cana-5226 39 13 properties property NOUN cana-5226 39 14 of of ADP cana-5226 39 15 cycle cycle NOUN cana-5226 39 16 graphs graph NOUN cana-5226 39 17 , , PUNCT cana-5226 39 18 many many ADJ cana-5226 39 19 matrices matrix NOUN cana-5226 39 20 are be AUX cana-5226 39 21 crucial crucial ADJ cana-5226 39 22 . . PUNCT cana-5226 40 1 among among ADP cana-5226 40 2 these these PRON cana-5226 40 3 are be AUX cana-5226 40 4 the the DET cana-5226 40 5 seidel seidel PROPN cana-5226 40 6 adjacency adjacency PROPN cana-5226 40 7 matrix matrix NOUN cana-5226 40 8 laplacian laplacian ADJ cana-5226 40 9 matrix matrix NOUN cana-5226 40 10 , , PUNCT cana-5226 40 11 singles single NOUN cana-5226 40 12 laplacian laplacian ADJ cana-5226 40 13 matrix matrix NOUN cana-5226 40 14 , , PUNCT cana-5226 40 15 and and CCONJ cana-5226 40 16 normalized normalize VERB cana-5226 40 17 laplacian laplacian ADJ cana-5226 40 18 matrix matrix NOUN cana-5226 40 19 . . PUNCT cana-5226 41 1 the the DET cana-5226 41 2 invention invention NOUN cana-5226 41 3 of of ADP cana-5226 41 4 a a DET cana-5226 41 5 general general ADJ cana-5226 41 6 characteristic characteristic ADJ cana-5226 41 7 equation equation NOUN cana-5226 41 8 that that PRON cana-5226 41 9 makes make VERB cana-5226 41 10 it it PRON cana-5226 41 11 possible possible ADJ cana-5226 41 12 to to PART cana-5226 41 13 calculate calculate VERB cana-5226 41 14 the the DET cana-5226 41 15 energy energy NOUN cana-5226 41 16 and and CCONJ cana-5226 41 17 eigenvalues eigenvalue NOUN cana-5226 41 18 of of ADP cana-5226 41 19 cycle cycle NOUN cana-5226 41 20 graphs graph NOUN cana-5226 41 21 is be AUX cana-5226 41 22 particularly particularly ADV cana-5226 41 23 intriguing intriguing ADJ cana-5226 41 24 . . PUNCT cana-5226 42 1 this this PRON cana-5226 42 2 emphasizes emphasize VERB cana-5226 42 3 the the DET cana-5226 42 4 generic generic ADJ cana-5226 42 5 formulas formula NOUN cana-5226 42 6 for for ADP cana-5226 42 7 a a DET cana-5226 42 8 cycle cycle NOUN cana-5226 42 9 graph graph NOUN cana-5226 42 10 's 's PART cana-5226 42 11 spectra spectra NOUN cana-5226 42 12 , , PUNCT cana-5226 42 13 which which PRON cana-5226 42 14 are be AUX cana-5226 42 15 based base VERB cana-5226 42 16 on on ADP cana-5226 42 17 the the DET cana-5226 42 18 cycle cycle NOUN cana-5226 42 19 's 's PART cana-5226 42 20 length length NOUN cana-5226 42 21 and and CCONJ cana-5226 42 22 vertex vertex NOUN cana-5226 42 23 count count NOUN cana-5226 42 24 ( ( PUNCT cana-5226 42 25 stin stin PROPN cana-5226 42 26 et et PROPN cana-5226 42 27 . . PUNCT cana-5226 43 1 al al PROPN cana-5226 43 2 ) ) PUNCT cana-5226 43 3 . . PUNCT cana-5226 44 1 a a DET cana-5226 44 2 crucial crucial ADJ cana-5226 44 3 source source NOUN cana-5226 44 4 of of ADP cana-5226 44 5 information information NOUN cana-5226 44 6 about about ADP cana-5226 44 7 a a DET cana-5226 44 8 graph graph NOUN cana-5226 44 9 's 's PART cana-5226 44 10 structure structure NOUN cana-5226 44 11 and and CCONJ cana-5226 44 12 connectivity connectivity NOUN cana-5226 44 13 is be AUX cana-5226 44 14 its its PRON cana-5226 44 15 energy energy NOUN cana-5226 44 16 and and CCONJ cana-5226 44 17 eigenvalues eigenvalue NOUN cana-5226 44 18 . . PUNCT cana-5226 45 1 researchers researcher NOUN cana-5226 45 2 can can AUX cana-5226 45 3 gain gain VERB cana-5226 45 4 a a DET cana-5226 45 5 better well ADJ cana-5226 45 6 understanding understanding NOUN cana-5226 45 7 of of ADP cana-5226 45 8 the the DET cana-5226 45 9 connections connection NOUN cana-5226 45 10 between between ADP cana-5226 45 11 these these DET cana-5226 45 12 attributes attribute NOUN cana-5226 45 13 and and CCONJ cana-5226 45 14 the the DET cana-5226 45 15 general general ADJ cana-5226 45 16 dynamics dynamic NOUN cana-5226 45 17 of of ADP cana-5226 45 18 the the DET cana-5226 45 19 graph graph NOUN cana-5226 45 20 by by ADP cana-5226 45 21 creating create VERB cana-5226 45 22 a a DET cana-5226 45 23 general general ADJ cana-5226 45 24 characteristic characteristic ADJ cana-5226 45 25 equation equation NOUN cana-5226 45 26 for for ADP cana-5226 45 27 cycle cycle NOUN cana-5226 45 28 graphs graph NOUN cana-5226 45 29 . . PUNCT cana-5226 46 1 ( ( PUNCT cana-5226 46 2 rowlinson rowlinson NOUN cana-5226 46 3 ) ) PUNCT cana-5226 46 4 addresses address VERB cana-5226 46 5 spectral spectral ADJ cana-5226 46 6 properties property NOUN cana-5226 46 7 of of ADP cana-5226 46 8 non non ADJ cana-5226 46 9 - - ADJ cana-5226 46 10 bipartite bipartite ADJ cana-5226 46 11 graphs graph NOUN cana-5226 46 12 with with ADP cana-5226 46 13 three three NUM cana-5226 46 14 distinct distinct ADJ cana-5226 46 15 eigenvalues eigenvalue NOUN cana-5226 46 16 , , PUNCT cana-5226 46 17 focusing focus VERB cana-5226 46 18 on on ADP cana-5226 46 19 conditions condition NOUN cana-5226 46 20 for for SCONJ cana-5226 46 21 g g NOUN cana-5226 46 22 to to PART cana-5226 46 23 be be AUX cana-5226 46 24 the the DET cana-5226 46 25 cone cone NOUN cana-5226 46 26 over over ADP cana-5226 46 27 a a DET cana-5226 46 28 strongly strongly ADV cana-5226 46 29 regular regular ADJ cana-5226 46 30 graph graph NOUN cana-5226 46 31 and and CCONJ cana-5226 46 32 analyzing analyze VERB cana-5226 46 33 scenarios scenario NOUN cana-5226 46 34 involving involve VERB cana-5226 46 35 one one NUM cana-5226 46 36 non non ADJ cana-5226 46 37 - - ADJ cana-5226 46 38 main main ADJ cana-5226 46 39 eigenvalue eigenvalue NOUN cana-5226 46 40 and and CCONJ cana-5226 46 41 certain certain ADJ cana-5226 46 42 degree degree NOUN cana-5226 46 43 conditions condition NOUN cana-5226 46 44 . . PUNCT cana-5226 47 1 with with ADP cana-5226 47 2 precisely precisely ADV cana-5226 47 3 three three NUM cana-5226 47 4 different different ADJ cana-5226 47 5 eigenvalues eigenvalue NOUN cana-5226 47 6 and and CCONJ cana-5226 47 7 a a DET cana-5226 47 8 maximum maximum NOUN cana-5226 47 9 of of ADP cana-5226 47 10 one one NUM cana-5226 47 11 for for ADP cana-5226 47 12 the the DET cana-5226 47 13 second second ADJ cana-5226 47 14 greatest great ADJ cana-5226 47 15 eigenvalue eigenvalue NOUN cana-5226 47 16 , , PUNCT cana-5226 47 17 [ [ X cana-5226 47 18 ( ( PUNCT cana-5226 47 19 cheng cheng PROPN cana-5226 47 20 et et PROPN cana-5226 47 21 . . PUNCT cana-5226 47 22 al al PROPN cana-5226 47 23 ) ) PUNCT cana-5226 47 24 , , PUNCT cana-5226 47 25 ( ( PUNCT cana-5226 47 26 qi qi X cana-5226 47 27 et et PROPN cana-5226 47 28 al al PROPN cana-5226 47 29 ) ) PUNCT cana-5226 47 30 ] ] PUNCT cana-5226 47 31 categorize categorize VERB cana-5226 47 32 the the DET cana-5226 47 33 connected connected ADJ cana-5226 47 34 graphs graph NOUN cana-5226 47 35 . . PUNCT cana-5226 48 1 by by ADP cana-5226 48 2 shedding shed VERB cana-5226 48 3 light light NOUN cana-5226 48 4 on on ADP cana-5226 48 5 the the DET cana-5226 48 6 structure structure NOUN cana-5226 48 7 and and CCONJ cana-5226 48 8 features feature NOUN cana-5226 48 9 of of ADP cana-5226 48 10 connected connected ADJ cana-5226 48 11 regular regular ADJ cana-5226 48 12 graphs graph NOUN cana-5226 48 13 with with ADP cana-5226 48 14 four four NUM cana-5226 48 15 different different ADJ cana-5226 48 16 eigenvalues eigenvalue NOUN cana-5226 48 17 , , PUNCT cana-5226 48 18 ( ( PUNCT cana-5226 48 19 huang huang PROPN cana-5226 48 20 & & CCONJ cana-5226 48 21 huang huang PROPN cana-5226 48 22 ) ) PUNCT cana-5226 48 23 advanced advance VERB cana-5226 48 24 the the DET cana-5226 48 25 knowledge knowledge NOUN cana-5226 48 26 of of ADP cana-5226 48 27 spectral spectral ADJ cana-5226 48 28 qualities quality NOUN cana-5226 48 29 and and CCONJ cana-5226 48 30 validates validate VERB cana-5226 48 31 some some DET cana-5226 48 32 discoveries discovery NOUN cana-5226 48 33 regarding regard VERB cana-5226 48 34 their their PRON cana-5226 48 35 non non ADJ cana-5226 48 36 - - NOUN cana-5226 48 37 existence existence NOUN cana-5226 48 38 ( ( PUNCT cana-5226 48 39 li li PROPN cana-5226 48 40 et et PROPN cana-5226 48 41 al al PROPN cana-5226 48 42 . . PROPN cana-5226 48 43 ) ) PUNCT cana-5226 48 44 . . PUNCT cana-5226 49 1 2 2 X cana-5226 49 2 . . X cana-5226 49 3 characteristics characteristic NOUN cana-5226 49 4 of of ADP cana-5226 49 5 cyclic cyclic ADJ cana-5226 49 6 graph graph NOUN cana-5226 49 7 : : PUNCT cana-5226 49 8 let let VERB cana-5226 49 9 cn cn PROPN cana-5226 49 10 be be AUX cana-5226 49 11 a a DET cana-5226 49 12 cycle cycle NOUN cana-5226 49 13 of of ADP cana-5226 49 14 n n NUM cana-5226 49 15 vertices vertex NOUN cana-5226 49 16 v1 v1 NOUN cana-5226 49 17 , , PUNCT cana-5226 49 18 v2 v2 PROPN cana-5226 49 19 , , PUNCT cana-5226 49 20 ... ... PUNCT cana-5226 49 21 , , PUNCT cana-5226 49 22 vn vn NOUN cana-5226 49 23 and and CCONJ cana-5226 49 24 adjacent adjacent ADJ cana-5226 49 25 matrix matrix NOUN cana-5226 49 26 of of ADP cana-5226 49 27 cn cn PROPN cana-5226 49 28 denoted denote VERB cana-5226 49 29 by by ADP cana-5226 49 30 x(cn x(cn PROPN cana-5226 49 31 ) ) PUNCT cana-5226 49 32 defined define VERB cana-5226 49 33 as as ADP cana-5226 49 34 : : PUNCT cana-5226 49 35 x(cn x(cn NUM cana-5226 49 36 ) ) PUNCT cana-5226 49 37 โก= โก= NOUN cana-5226 49 38 [ [ PUNCT cana-5226 49 39 0 0 NUM cana-5226 49 40 1 1 NUM cana-5226 49 41 0 0 NUM cana-5226 49 42 1 1 NUM cana-5226 49 43 0 0 NUM cana-5226 49 44 1 1 NUM cana-5226 49 45 0 0 NUM cana-5226 49 46 1 1 NUM cana-5226 49 47 0 0 NUM cana-5226 49 48 โ‹ฏ โ‹ฏ ADP cana-5226 49 49 0 0 NUM cana-5226 49 50 0 0 NUM cana-5226 49 51 1 1 NUM cana-5226 49 52 0 0 NUM cana-5226 49 53 0 0 NUM cana-5226 49 54 0 0 NUM cana-5226 49 55 0 0 NUM cana-5226 49 56 0 0 NUM cana-5226 49 57 0 0 NUM cana-5226 49 58 โ‹ฎ โ‹ฎ NOUN cana-5226 49 59 โ‹ฑ โ‹ฑ PUNCT cana-5226 49 60 โ‹ฎ โ‹ฎ NOUN cana-5226 49 61 0 0 NOUN cana-5226 49 62 0 0 NUM cana-5226 49 63 0 0 NUM cana-5226 49 64 0 0 NUM cana-5226 49 65 0 0 NUM cana-5226 49 66 0 0 NUM cana-5226 49 67 1 1 NUM cana-5226 49 68 0 0 NUM cana-5226 49 69 0 0 NUM cana-5226 50 1 โ‹ฏ โ‹ฏ VERB cana-5226 50 2 0 0 NUM cana-5226 50 3 1 1 NUM cana-5226 50 4 0 0 NUM cana-5226 50 5 1 1 NUM cana-5226 50 6 0 0 NUM cana-5226 50 7 1 1 NUM cana-5226 50 8 0 0 NUM cana-5226 50 9 1 1 NUM cana-5226 50 10 0 0 NUM cana-5226 50 11 ] ] PUNCT cana-5226 50 12 โกโกโก๐‘›ร—๐‘› โกโกโก๐‘›ร—๐‘› ADJ cana-5226 50 13 communications communication NOUN cana-5226 50 14 on on ADP cana-5226 50 15 applied apply VERB cana-5226 50 16 nonlinear nonlinear ADJ cana-5226 50 17 analysis analysis NOUN cana-5226 50 18 issn issn NOUN cana-5226 50 19 : : PUNCT cana-5226 50 20 1074 1074 NUM cana-5226 50 21 - - PUNCT cana-5226 50 22 133x 133x NUM cana-5226 50 23 vol vol VERB cana-5226 50 24 32 32 NUM cana-5226 50 25 no no NOUN cana-5226 50 26 . . PUNCT cana-5226 51 1 10s 10 NOUN cana-5226 51 2 ( ( PUNCT cana-5226 51 3 2025 2025 NUM cana-5226 51 4 ) ) PUNCT cana-5226 51 5 1219 1219 NUM cana-5226 51 6 https://internationalpubls.com https://internationalpubls.com X cana-5226 52 1 its its PRON cana-5226 52 2 characteristic characteristic ADJ cana-5226 52 3 polynomials polynomial NOUN cana-5226 52 4 is be AUX cana-5226 52 5 โˆ†(๐ถ๐‘› โˆ†(๐ถ๐‘› NOUN cana-5226 52 6 , , PUNCT cana-5226 52 7 ๐œ† ๐œ† X cana-5226 52 8 ) ) PUNCT cana-5226 52 9 = = SYM cana-5226 52 10 detโก(๐œ†๐ผ๐‘› detโก(๐œ†๐ผ๐‘› PROPN cana-5226 52 11 โˆ’ โˆ’ PROPN cana-5226 52 12 ๐‘‹(๐ถ๐‘› ๐‘‹(๐ถ๐‘› NOUN cana-5226 52 13 ) ) PUNCT cana-5226 52 14 ) ) PUNCT cana-5226 53 1 โˆ†(๐ถ๐‘› โˆ†(๐ถ๐‘› PROPN cana-5226 53 2 , , PUNCT cana-5226 53 3 ๐œ† ๐œ† X cana-5226 53 4 ) ) PUNCT cana-5226 53 5 = = PUNCT cana-5226 54 1 | | ADV cana-5226 54 2 | | ADV cana-5226 54 3 โกโก๐œ† โกโก๐œ† ADP cana-5226 54 4 โˆ’1 โˆ’1 NOUN cana-5226 54 5 โกโก0 โกโก0 VERB cana-5226 54 6 โกโกโˆ’1 โกโกโˆ’1 NOUN cana-5226 54 7 โกโก๐œ† โกโก๐œ† SYM cana-5226 54 8 โˆ’1 โˆ’1 NOUN cana-5226 54 9 โกโก0 โกโก0 NOUN cana-5226 54 10 โˆ’1 โˆ’1 NOUN cana-5226 54 11 โกโก๐œ† โกโก๐œ† PUNCT cana-5226 54 12 โ‹ฏ โ‹ฏ NOUN cana-5226 54 13 0 0 NUM cana-5226 54 14 โก0 โก0 NOUN cana-5226 54 15 โกโกโˆ’1 โกโกโˆ’1 NOUN cana-5226 54 16 0 0 NUM cana-5226 54 17 โก0 โก0 NOUN cana-5226 54 18 โกโกโกโกโก0 โกโกโกโกโก0 X cana-5226 54 19 0 0 NUM cana-5226 54 20 โก0 โก0 NOUN cana-5226 54 21 โกโกโกโกโก0 โกโกโกโกโก0 PUNCT cana-5226 54 22 โ‹ฎ โ‹ฎ NOUN cana-5226 54 23 โ‹ฑ โ‹ฑ PUNCT cana-5226 54 24 โ‹ฎ โ‹ฎ NOUN cana-5226 54 25 โกโก0 โกโก0 PUNCT cana-5226 54 26 โก0 โก0 NOUN cana-5226 54 27 โกโกโกโกโกโก0 โกโกโกโกโกโก0 NOUN cana-5226 54 28 โกโก0 โกโก0 ADJ cana-5226 54 29 โก0 โก0 NOUN cana-5226 54 30 โกโกโกโกโก0 โกโกโกโกโก0 NOUN cana-5226 54 31 โˆ’1 โˆ’1 NOUN cana-5226 54 32 โก0 โก0 NOUN cana-5226 54 33 โกโกโกโกโกโก0 โกโกโกโกโกโก0 PUNCT cana-5226 54 34 โ‹ฏ โ‹ฏ PROPN cana-5226 54 35 โก๐œ† โก๐œ† NOUN cana-5226 54 36 โˆ’1 โˆ’1 NOUN cana-5226 54 37 โกโก0 โกโก0 X cana-5226 54 38 โˆ’1 โˆ’1 NOUN cana-5226 54 39 โกโก๐œ† โกโก๐œ† ADP cana-5226 54 40 โˆ’1 โˆ’1 NOUN cana-5226 54 41 โกโก0 โกโก0 NOUN cana-5226 54 42 โˆ’1 โˆ’1 NOUN cana-5226 54 43 โกโก๐œ† โกโก๐œ† PUNCT cana-5226 54 44 | | ADV cana-5226 54 45 | | ADV cana-5226 54 46 โˆ†(๐ถ๐‘› โˆ†(๐ถ๐‘› NOUN cana-5226 54 47 , , PUNCT cana-5226 54 48 ๐œ† ๐œ† X cana-5226 54 49 ) ) PUNCT cana-5226 55 1 = = SYM cana-5226 55 2 ๐œ†๐‘› ๐œ†๐‘› ADP cana-5226 55 3 โˆ’ โˆ’ NOUN cana-5226 56 1 ๐‘†1,๐‘›๐œ† ๐‘†1,๐‘›๐œ† PROPN cana-5226 56 2 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-5226 56 3 + + CCONJ cana-5226 56 4 ๐‘†2,๐‘›๐œ† ๐‘†2,๐‘›๐œ† PROPN cana-5226 56 5 ๐‘›โˆ’2 ๐‘›โˆ’2 PROPN cana-5226 56 6 โˆ’ โˆ’ NOUN cana-5226 56 7 ๐‘†3,๐‘›๐œ† ๐‘†3,๐‘›๐œ† PROPN cana-5226 56 8 ๐‘›โˆ’3 ๐‘›โˆ’3 X cana-5226 56 9 + + ADJ cana-5226 56 10 โ‹ฏ+ โ‹ฏ+ NOUN cana-5226 56 11 ( ( PUNCT cana-5226 56 12 โˆ’1)๐‘˜๐‘†๐‘˜,๐‘›๐œ† โˆ’1)๐‘˜๐‘†๐‘˜,๐‘›๐œ† NOUN cana-5226 56 13 ๐‘›โˆ’๐‘˜ ๐‘›โˆ’๐‘˜ NOUN cana-5226 56 14 + + NOUN cana-5226 56 15 โ‹ฏ+ โ‹ฏ+ NOUN cana-5226 56 16 ( ( PUNCT cana-5226 56 17 โˆ’1)๐‘›๐‘†๐‘›,๐‘› โˆ’1)๐‘›๐‘†๐‘›,๐‘› NOUN cana-5226 56 18 and and CCONJ cana-5226 56 19 characteristic characteristic ADJ cana-5226 56 20 equation equation NOUN cana-5226 56 21 is be AUX cana-5226 56 22 โˆ†(๐ถ๐‘› โˆ†(๐ถ๐‘› NOUN cana-5226 56 23 , , PUNCT cana-5226 56 24 ๐œ† ๐œ† X cana-5226 56 25 ) ) PUNCT cana-5226 56 26 = = SYM cana-5226 56 27 detโก(๐œ†๐ผ๐‘› detโก(๐œ†๐ผ๐‘› PROPN cana-5226 57 1 โˆ’ โˆ’ PROPN cana-5226 57 2 ๐‘‹(๐ถ๐‘› ๐‘‹(๐ถ๐‘› PROPN cana-5226 57 3 ) ) PUNCT cana-5226 57 4 ) ) PUNCT cana-5226 58 1 = = SYM cana-5226 58 2 0โกโก 0โกโก PUNCT cana-5226 59 1 i.e. i.e. X cana-5226 59 2 ๐œ†๐‘› ๐œ†๐‘› ADP cana-5226 59 3 โˆ’ โˆ’ PUNCT cana-5226 59 4 ๐‘†1,๐‘›๐œ† ๐‘†1,๐‘›๐œ† PROPN cana-5226 59 5 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-5226 59 6 + + CCONJ cana-5226 59 7 ๐‘†2,๐‘›๐œ† ๐‘†2,๐‘›๐œ† PROPN cana-5226 59 8 ๐‘›โˆ’2 ๐‘›โˆ’2 PROPN cana-5226 59 9 โˆ’ โˆ’ NOUN cana-5226 59 10 ๐‘†3,๐‘›๐œ† ๐‘†3,๐‘›๐œ† PROPN cana-5226 59 11 ๐‘›โˆ’3 ๐‘›โˆ’3 X cana-5226 59 12 + + ADJ cana-5226 59 13 โ‹ฏ+ โ‹ฏ+ NOUN cana-5226 59 14 ( ( PUNCT cana-5226 59 15 โˆ’1)๐‘˜๐‘†๐‘˜,๐‘›๐œ† โˆ’1)๐‘˜๐‘†๐‘˜,๐‘›๐œ† NOUN cana-5226 59 16 ๐‘›โˆ’๐‘˜ ๐‘›โˆ’๐‘˜ NOUN cana-5226 59 17 + + NOUN cana-5226 59 18 โ‹ฏ+ โ‹ฏ+ NOUN cana-5226 59 19 ( ( PUNCT cana-5226 59 20 โˆ’1)๐‘›๐‘†๐‘›,๐‘› โˆ’1)๐‘›๐‘†๐‘›,๐‘› SYM cana-5226 59 21 = = NOUN cana-5226 59 22 0 0 NUM cana-5226 59 23 where where SCONJ cana-5226 59 24 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 59 25 is be AUX cana-5226 59 26 the the DET cana-5226 59 27 sum sum NOUN cana-5226 59 28 of of ADP cana-5226 59 29 the the DET cana-5226 59 30 principal principal ADJ cana-5226 59 31 minors minor NOUN cana-5226 59 32 of of ADP cana-5226 59 33 order order NOUN cana-5226 60 1 k k PROPN cana-5226 60 2 of of ADP cana-5226 60 3 cycle cycle NOUN cana-5226 61 1 ๐ถ๐‘› ๐ถ๐‘› PROPN cana-5226 61 2 in in ADP cana-5226 61 3 an an DET cana-5226 61 4 adjacent adjacent ADJ cana-5226 61 5 matrix matrix NOUN cana-5226 61 6 x. x. NOUN cana-5226 62 1 now now ADV cana-5226 62 2 we we PRON cana-5226 62 3 will will AUX cana-5226 62 4 try try VERB cana-5226 62 5 to to PART cana-5226 62 6 find find VERB cana-5226 62 7 the the DET cana-5226 62 8 values value NOUN cana-5226 62 9 of of ADP cana-5226 62 10 ๐‘†๐‘˜,๐‘›. ๐‘†๐‘˜,๐‘›. PROPN cana-5226 62 11 there there PRON cana-5226 62 12 are be VERB cana-5226 62 13 four four NUM cana-5226 62 14 cases case NOUN cana-5226 62 15 to to PART cana-5226 62 16 find find VERB cana-5226 62 17 ๐‘†๐‘˜,๐‘›. ๐‘†๐‘˜,๐‘›. ADJ cana-5226 62 18 case case NOUN cana-5226 63 1 i i PRON cana-5226 63 2 : : PUNCT cana-5226 63 3 if if SCONJ cana-5226 63 4 n n PRON cana-5226 63 5 โ‰ก โ‰ก PROPN cana-5226 63 6 0(modโก4 0(modโก4 PROPN cana-5226 63 7 ) ) PUNCT cana-5226 63 8 , , PUNCT cana-5226 63 9 then then ADV cana-5226 63 10 sk sk VERB cana-5226 63 11 , , PUNCT cana-5226 63 12 n n NOUN cana-5226 63 13 = = PRON cana-5226 63 14 { { PUNCT cana-5226 63 15 0,โกโกโกkโกisโกoddโกorโกk 0,โกโกโกkโกisโกoddโกorโกk NOUN cana-5226 63 16 = = SYM cana-5226 63 17 n n CCONJ cana-5226 63 18 , , PUNCT cana-5226 63 19 โˆ’nโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk โˆ’nโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk X cana-5226 63 20 = = SYM cana-5226 63 21 2 2 NUM cana-5226 63 22 , , PUNCT cana-5226 63 23 and and CCONJ cana-5226 63 24 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 63 25 = = SYM cana-5226 63 26 ( ( PUNCT cana-5226 63 27 โˆ’1)๐‘˜ โˆ’1)๐‘˜ NOUN cana-5226 63 28 2โ„ 2โ„ NUM cana-5226 63 29 [ [ X cana-5226 63 30 ( ( PUNCT cana-5226 63 31 ๐‘˜ ๐‘˜ X cana-5226 63 32 + + NOUN cana-5226 63 33 1 1 NUM cana-5226 63 34 ) ) PUNCT cana-5226 63 35 + + CCONJ cana-5226 63 36 ( ( PUNCT cana-5226 63 37 โˆ’1)(๐‘˜โˆ’2 โˆ’1)(๐‘˜โˆ’2 PROPN cana-5226 63 38 ) ) PUNCT cana-5226 63 39 2โ„ 2โ„ NUM cana-5226 63 40 โก โก NOUN cana-5226 63 41 โˆ‘ โˆ‘ ADP cana-5226 63 42 ๐‘†(๐‘˜ ๐‘†(๐‘˜ ADV cana-5226 63 43 โˆ’ โˆ’ PROPN cana-5226 63 44 2 2 NUM cana-5226 63 45 , , PUNCT cana-5226 63 46 ๐‘˜ ๐‘˜ PROPN cana-5226 63 47 + + NOUN cana-5226 63 48 ๐‘š ๐‘š NOUN cana-5226 63 49 ) ) PUNCT cana-5226 63 50 ๐‘›โˆ’2โˆ’๐‘˜ ๐‘›โˆ’2โˆ’๐‘˜ NOUN cana-5226 63 51 ๐‘š=0 ๐‘š=0 PUNCT cana-5226 63 52 ] ] PUNCT cana-5226 63 53 where where SCONJ cana-5226 63 54 k k PROPN cana-5226 63 55 is be AUX cana-5226 63 56 even even ADV cana-5226 63 57 , , PUNCT cana-5226 63 58 ๐‘˜ ๐‘˜ DET cana-5226 63 59 โ‰  โ‰  PROPN cana-5226 63 60 2 2 NUM cana-5226 63 61 and and CCONJ cana-5226 63 62 ๐‘˜ ๐‘˜ X cana-5226 63 63 < < X cana-5226 63 64 ๐‘› ๐‘› PRON cana-5226 63 65 โˆ’ โˆ’ PROPN cana-5226 63 66 1 1 NUM cana-5226 63 67 . . PUNCT cana-5226 64 1 for for ADP cana-5226 64 2 example example NOUN cana-5226 64 3 : : PUNCT cana-5226 64 4 let let VERB cana-5226 64 5 ๐ถ4 ๐ถ4 PROPN cana-5226 64 6 be be AUX cana-5226 64 7 a a DET cana-5226 64 8 cycle cycle NOUN cana-5226 64 9 of of ADP cana-5226 64 10 4 4 NUM cana-5226 64 11 vertices vertex NOUN cana-5226 64 12 , , PUNCT cana-5226 64 13 adjacent adjacent ADJ cana-5226 64 14 matrix matrix NOUN cana-5226 64 15 of of ADP cana-5226 64 16 ๐ถ4 ๐ถ4 NOUN cana-5226 64 17 is be AUX cana-5226 64 18 . . PUNCT cana-5226 65 1 x(๐ถ4 x(๐ถ4 NOUN cana-5226 65 2 ) ) PUNCT cana-5226 65 3 โก= โก= NOUN cana-5226 65 4 [ [ PUNCT cana-5226 65 5 0 0 NUM cana-5226 65 6 1 1 NUM cana-5226 65 7 1 1 NUM cana-5226 65 8 0 0 NUM cana-5226 65 9 0 0 NUM cana-5226 65 10 1 1 NUM cana-5226 65 11 1 1 NUM cana-5226 65 12 0 0 NUM cana-5226 65 13 0 0 NUM cana-5226 65 14 1 1 NUM cana-5226 65 15 1 1 NUM cana-5226 65 16 0 0 NUM cana-5226 65 17 0 0 NUM cana-5226 65 18 1 1 NUM cana-5226 65 19 1 1 NUM cana-5226 65 20 0 0 NUM cana-5226 65 21 ] ] PUNCT cana-5226 65 22 โกโกโก4ร—4 โกโกโก4ร—4 ADJ cana-5226 65 23 characteristic characteristic ADJ cana-5226 65 24 equation equation NOUN cana-5226 65 25 is be AUX cana-5226 65 26 โˆ†(๐ถ4 โˆ†(๐ถ4 NOUN cana-5226 65 27 , , PUNCT cana-5226 65 28 ๐œ† ๐œ† X cana-5226 65 29 ) ) PUNCT cana-5226 65 30 = = SYM cana-5226 65 31 det(๐œ†๐ผ4 det(๐œ†๐ผ4 VERB cana-5226 65 32 โˆ’ โˆ’ NOUN cana-5226 65 33 ๐‘‹(๐ถ4 ๐‘‹(๐ถ4 NOUN cana-5226 65 34 ) ) PUNCT cana-5226 65 35 ) ) PUNCT cana-5226 66 1 = = SYM cana-5226 66 2 0 0 NUM cana-5226 66 3 โ‡’ โ‡’ PROPN cana-5226 66 4 ๐œ†4 ๐œ†4 NOUN cana-5226 66 5 โˆ’ โˆ’ PROPN cana-5226 66 6 ๐‘†1,4๐œ† ๐‘†1,4๐œ† PROPN cana-5226 66 7 3 3 NUM cana-5226 66 8 + + NUM cana-5226 66 9 ๐‘†2,4๐œ† ๐‘†2,4๐œ† PROPN cana-5226 66 10 2 2 NUM cana-5226 66 11 โˆ’ โˆ’ NOUN cana-5226 66 12 ๐‘†3,4๐œ† ๐‘†3,4๐œ† PROPN cana-5226 66 13 + + CCONJ cana-5226 66 14 ๐‘†4,4 ๐‘†4,4 PROPN cana-5226 66 15 = = SYM cana-5226 66 16 0 0 PROPN cana-5226 66 17 โ‡’ โ‡’ PROPN cana-5226 66 18 ๐œ†4 ๐œ†4 NOUN cana-5226 66 19 โˆ’ โˆ’ PROPN cana-5226 66 20 4๐œ†2 4๐œ†2 NUM cana-5226 67 1 = = NOUN cana-5226 67 2 0 0 PUNCT cana-5226 67 3 as as SCONJ cana-5226 67 4 ๐‘†1,4 ๐‘†1,4 NOUN cana-5226 67 5 = = PROPN cana-5226 67 6 0 0 NUM cana-5226 67 7 , , PUNCT cana-5226 67 8 ๐‘†2,4 ๐‘†2,4 NOUN cana-5226 67 9 = = SYM cana-5226 67 10 โˆ’4 โˆ’4 ADJ cana-5226 67 11 , , PUNCT cana-5226 67 12 ๐‘†3,4 ๐‘†3,4 ADJ cana-5226 67 13 = = SYM cana-5226 67 14 0โก๐‘Ž๐‘›๐‘‘โก๐‘†4,4 0โก๐‘Ž๐‘›๐‘‘โก๐‘†4,4 VERB cana-5226 67 15 = = SYM cana-5226 67 16 0 0 NUM cana-5226 67 17 โ‡’ โ‡’ NOUN cana-5226 67 18 ๐œ† ๐œ† X cana-5226 68 1 = = SYM cana-5226 68 2 0 0 NUM cana-5226 68 3 , , PUNCT cana-5226 68 4 0 0 NUM cana-5226 68 5 , , PUNCT cana-5226 68 6 โˆ’2โก๐‘Ž๐‘›๐‘‘โก2 โˆ’2โก๐‘Ž๐‘›๐‘‘โก2 PROPN cana-5226 68 7 case case NOUN cana-5226 68 8 ii ii NOUN cana-5226 68 9 : : PUNCT cana-5226 68 10 if if SCONJ cana-5226 68 11 , , PUNCT cana-5226 68 12 โกn โกn PROPN cana-5226 68 13 โ‰ก โ‰ก PROPN cana-5226 68 14 1(modโก4 1(modโก4 PROPN cana-5226 68 15 ) ) PUNCT cana-5226 68 16 , , PUNCT cana-5226 68 17 then then ADV cana-5226 68 18 communications communication NOUN cana-5226 68 19 on on ADP cana-5226 68 20 applied apply VERB cana-5226 68 21 nonlinear nonlinear ADJ cana-5226 68 22 analysis analysis NOUN cana-5226 68 23 issn issn NOUN cana-5226 68 24 : : PUNCT cana-5226 68 25 1074 1074 NUM cana-5226 68 26 - - PUNCT cana-5226 68 27 133x 133x NUM cana-5226 68 28 vol vol VERB cana-5226 68 29 32 32 NUM cana-5226 68 30 no no NOUN cana-5226 68 31 . . PUNCT cana-5226 69 1 10s 10 NOUN cana-5226 69 2 ( ( PUNCT cana-5226 69 3 2025 2025 NUM cana-5226 69 4 ) ) PUNCT cana-5226 69 5 1220 1220 NUM cana-5226 69 6 https://internationalpubls.com https://internationalpubls.com X cana-5226 70 1 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 70 2 = = SYM cana-5226 70 3 { { PUNCT cana-5226 70 4 0,โกโกโกโกโกโกโกkโกisโกoddโกandโกk 0,โกโกโกโกโกโกโกkโกisโกoddโกandโกk NUM cana-5226 70 5 โ‰  โ‰  NOUN cana-5226 70 6 n n PRON cana-5226 70 7 โˆ’n,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk โˆ’n,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk NOUN cana-5226 70 8 = = SYM cana-5226 70 9 2 2 NUM cana-5226 70 10 2,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk 2,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk NUM cana-5226 70 11 = = SYM cana-5226 70 12 nโก nโก NOUN cana-5226 70 13 n,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk n,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกk NOUN cana-5226 70 14 = = SYM cana-5226 70 15 n n CCONJ cana-5226 70 16 โˆ’ โˆ’ PROPN cana-5226 70 17 1 1 NUM cana-5226 70 18 and and CCONJ cana-5226 70 19 sk sk INTJ cana-5226 70 20 , , PUNCT cana-5226 70 21 n=(โˆ’1 n=(โˆ’1 PROPN cana-5226 70 22 ) ) PUNCT cana-5226 71 1 ๐‘˜ ๐‘˜ ADP cana-5226 71 2 2โ„ 2โ„ NUM cana-5226 71 3 [ [ X cana-5226 71 4 ( ( PUNCT cana-5226 71 5 ๐‘˜ ๐‘˜ X cana-5226 71 6 + + NOUN cana-5226 71 7 1 1 NUM cana-5226 71 8 ) ) PUNCT cana-5226 71 9 + + CCONJ cana-5226 71 10 ( ( PUNCT cana-5226 71 11 โˆ’1)(๐‘˜โˆ’2 โˆ’1)(๐‘˜โˆ’2 PROPN cana-5226 71 12 ) ) PUNCT cana-5226 71 13 2โ„ 2โ„ NUM cana-5226 71 14 โก โก NOUN cana-5226 71 15 โˆ‘ โˆ‘ ADP cana-5226 71 16 ๐‘†(๐‘˜ ๐‘†(๐‘˜ ADV cana-5226 71 17 โˆ’ โˆ’ PROPN cana-5226 71 18 2 2 NUM cana-5226 71 19 , , PUNCT cana-5226 71 20 ๐‘˜ ๐‘˜ PROPN cana-5226 71 21 + + NOUN cana-5226 71 22 ๐‘š ๐‘š NOUN cana-5226 71 23 ) ) PUNCT cana-5226 71 24 ๐‘›โˆ’2โˆ’๐‘˜ ๐‘›โˆ’2โˆ’๐‘˜ NOUN cana-5226 71 25 ๐‘š=0 ๐‘š=0 PUNCT cana-5226 71 26 ] ] PUNCT cana-5226 71 27 where where SCONJ cana-5226 71 28 k k PROPN cana-5226 71 29 is be AUX cana-5226 71 30 even even ADV cana-5226 71 31 , , PUNCT cana-5226 71 32 ๐‘˜ ๐‘˜ DET cana-5226 71 33 โ‰  โ‰  PROPN cana-5226 71 34 2 2 NUM cana-5226 71 35 and and CCONJ cana-5226 71 36 ๐‘˜ ๐‘˜ X cana-5226 71 37 < < X cana-5226 71 38 ๐‘› ๐‘› PRON cana-5226 71 39 โˆ’ โˆ’ PROPN cana-5226 71 40 1 1 NUM cana-5226 71 41 . . PUNCT cana-5226 72 1 for for ADP cana-5226 72 2 example example NOUN cana-5226 72 3 : : PUNCT cana-5226 72 4 let let VERB cana-5226 72 5 ๐ถ5 ๐ถ5 NOUN cana-5226 72 6 be be AUX cana-5226 72 7 a a DET cana-5226 72 8 cycle cycle NOUN cana-5226 72 9 of of ADP cana-5226 72 10 5 5 NUM cana-5226 72 11 vertices vertex NOUN cana-5226 72 12 , , PUNCT cana-5226 72 13 adjacent adjacent ADJ cana-5226 72 14 matrix matrix NOUN cana-5226 72 15 of of ADP cana-5226 72 16 ๐ถ5 ๐ถ5 NOUN cana-5226 72 17 is be AUX cana-5226 72 18 . . PUNCT cana-5226 72 19 ๐‘‹(๐ถ5 ๐‘‹(๐ถ5 PUNCT cana-5226 72 20 ) ) PUNCT cana-5226 73 1 = = PUNCT cana-5226 73 2 [ [ PUNCT cana-5226 73 3 0 0 NUM cana-5226 73 4 1 1 NUM cana-5226 73 5 0 0 NUM cana-5226 73 6 0 0 NUM cana-5226 73 7 1 1 NUM cana-5226 73 8 1 1 NUM cana-5226 73 9 0 0 NUM cana-5226 73 10 1 1 NUM cana-5226 73 11 0 0 NUM cana-5226 73 12 0 0 NUM cana-5226 73 13 0 0 NUM cana-5226 73 14 0 0 NUM cana-5226 73 15 1 1 NUM cana-5226 73 16 1 1 NUM cana-5226 73 17 0 0 NUM cana-5226 73 18 0 0 NUM cana-5226 73 19 0 0 NUM cana-5226 73 20 1 1 NUM cana-5226 73 21 0 0 NUM cana-5226 73 22 1 1 NUM cana-5226 73 23 0 0 NUM cana-5226 73 24 1 1 NUM cana-5226 73 25 0 0 NUM cana-5226 73 26 1 1 NUM cana-5226 73 27 0 0 NUM cana-5226 73 28 ] ] PUNCT cana-5226 73 29 โกโกโก5ร—5 โกโกโก5ร—5 VERB cana-5226 73 30 characteristic characteristic ADJ cana-5226 73 31 equation equation NOUN cana-5226 73 32 is be AUX cana-5226 73 33 โˆ†(๐ถ5 โˆ†(๐ถ5 PROPN cana-5226 73 34 , , PUNCT cana-5226 73 35 ๐œ† ๐œ† X cana-5226 73 36 ) ) PUNCT cana-5226 74 1 = = PRON cana-5226 74 2 det(๐œ†๐ผ5 det(๐œ†๐ผ5 NOUN cana-5226 74 3 โˆ’ โˆ’ PROPN cana-5226 74 4 ๐‘‹(๐ถ5 ๐‘‹(๐ถ5 PROPN cana-5226 74 5 ) ) PUNCT cana-5226 74 6 ) ) PUNCT cana-5226 75 1 = = SYM cana-5226 75 2 0 0 NUM cana-5226 75 3 โ‡’ โ‡’ NOUN cana-5226 75 4 ๐œ†5 ๐œ†5 ADV cana-5226 76 1 โˆ’ โˆ’ PROPN cana-5226 76 2 ๐‘†1,5๐œ† ๐‘†1,5๐œ† PROPN cana-5226 76 3 4 4 NUM cana-5226 76 4 + + NUM cana-5226 76 5 ๐‘†2,5๐œ† ๐‘†2,5๐œ† PROPN cana-5226 76 6 3 3 NUM cana-5226 76 7 โˆ’ โˆ’ NOUN cana-5226 76 8 ๐‘†3,5๐œ† ๐‘†3,5๐œ† NOUN cana-5226 76 9 2 2 NUM cana-5226 76 10 + + NUM cana-5226 76 11 ๐‘†4,5๐œ† ๐‘†4,5๐œ† PROPN cana-5226 76 12 โˆ’ โˆ’ PROPN cana-5226 76 13 ๐‘†5,5 ๐‘†5,5 X cana-5226 76 14 = = SYM cana-5226 76 15 0 0 NUM cana-5226 76 16 โ‡’ โ‡’ NOUN cana-5226 76 17 ๐œ†5 ๐œ†5 ADV cana-5226 76 18 โˆ’ โˆ’ PROPN cana-5226 76 19 5๐œ†3 5๐œ†3 NUM cana-5226 77 1 + + CCONJ cana-5226 77 2 5๐œ† 5๐œ† NUM cana-5226 77 3 โˆ’ โˆ’ NOUN cana-5226 77 4 2 2 NUM cana-5226 77 5 = = SYM cana-5226 77 6 0 0 NUM cana-5226 77 7 as as ADP cana-5226 77 8 ๐‘†1,5 ๐‘†1,5 NOUN cana-5226 77 9 = = SYM cana-5226 77 10 0 0 NUM cana-5226 77 11 , , PUNCT cana-5226 77 12 ๐‘†2,5 ๐‘†2,5 NOUN cana-5226 77 13 = = SYM cana-5226 77 14 โˆ’5 โˆ’5 ADJ cana-5226 77 15 , , PUNCT cana-5226 77 16 ๐‘†3,5 ๐‘†3,5 NOUN cana-5226 77 17 = = SYM cana-5226 77 18 0 0 NUM cana-5226 77 19 , , PUNCT cana-5226 77 20 ๐‘†4,5 ๐‘†4,5 NOUN cana-5226 77 21 = = SYM cana-5226 77 22 5โก๐‘Ž๐‘›๐‘‘โก๐‘†5,5 5โก๐‘Ž๐‘›๐‘‘โก๐‘†5,5 NUM cana-5226 77 23 = = SYM cana-5226 77 24 2 2 NUM cana-5226 77 25 โ‡’ โ‡’ NOUN cana-5226 77 26 ๐œ† ๐œ† X cana-5226 77 27 = = SYM cana-5226 77 28 โˆ’1.6180,โˆ’1.6180 โˆ’1.6180,โˆ’1.6180 PROPN cana-5226 77 29 , , PUNCT cana-5226 77 30 0.6180 0.6180 NUM cana-5226 77 31 , , PUNCT cana-5226 77 32 0.6180 0.6180 NUM cana-5226 77 33 , , PUNCT cana-5226 77 34 2 2 NUM cana-5226 77 35 case case NOUN cana-5226 77 36 iii iii NOUN cana-5226 77 37 : : PUNCT cana-5226 77 38 if if SCONJ cana-5226 77 39 2(mod 2(mod NUM cana-5226 77 40 4)n 4)n ADJ cana-5226 77 41 ๏‚บ ๏‚บ PROPN cana-5226 77 42 and and CCONJ cana-5226 77 43 ๐‘› ๐‘› DET cana-5226 77 44 โ‰  โ‰  PROPN cana-5226 77 45 2 2 NUM cana-5226 77 46 , , PUNCT cana-5226 77 47 then then ADV cana-5226 77 48 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 77 49 = = SYM cana-5226 77 50 { { PUNCT cana-5226 77 51 0 0 NUM cana-5226 77 52 , , PUNCT cana-5226 77 53 ๐‘˜โก๐‘–๐‘ โก๐‘œ๐‘‘๐‘‘โก๐‘Ž๐‘›๐‘‘โก๐‘˜ ๐‘˜โก๐‘–๐‘ โก๐‘œ๐‘‘๐‘‘โก๐‘Ž๐‘›๐‘‘โก๐‘˜ ADJ cana-5226 77 54 โ‰  โ‰  PROPN cana-5226 77 55 ๐‘› ๐‘› PROPN cana-5226 77 56 โˆ’๐‘›,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ โˆ’๐‘›,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ NOUN cana-5226 77 57 = = SYM cana-5226 77 58 2 2 NUM cana-5226 77 59 โˆ’4,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ โˆ’4,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ NOUN cana-5226 77 60 = = PUNCT cana-5226 77 61 ๐‘› ๐‘› PROPN cana-5226 77 62 and and CCONJ cana-5226 77 63 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 77 64 = = SYM cana-5226 77 65 ( ( PUNCT cana-5226 77 66 โˆ’1)๐‘˜ โˆ’1)๐‘˜ NOUN cana-5226 77 67 2โ„ 2โ„ NUM cana-5226 77 68 [ [ X cana-5226 77 69 ( ( PUNCT cana-5226 77 70 ๐‘˜ ๐‘˜ X cana-5226 77 71 + + NOUN cana-5226 77 72 1 1 NUM cana-5226 77 73 ) ) PUNCT cana-5226 77 74 + + CCONJ cana-5226 77 75 ( ( PUNCT cana-5226 77 76 โˆ’1)(๐‘˜โˆ’2 โˆ’1)(๐‘˜โˆ’2 PROPN cana-5226 77 77 ) ) PUNCT cana-5226 77 78 2โ„ 2โ„ NUM cana-5226 77 79 โก โก NOUN cana-5226 77 80 โˆ‘ โˆ‘ ADP cana-5226 77 81 ๐‘†(๐‘˜ ๐‘†(๐‘˜ ADV cana-5226 77 82 โˆ’ โˆ’ PROPN cana-5226 77 83 2 2 NUM cana-5226 77 84 , , PUNCT cana-5226 77 85 ๐‘˜ ๐‘˜ PROPN cana-5226 77 86 + + NOUN cana-5226 77 87 ๐‘š ๐‘š NOUN cana-5226 77 88 ) ) PUNCT cana-5226 77 89 ๐‘›โˆ’2โˆ’๐‘˜ ๐‘›โˆ’2โˆ’๐‘˜ NOUN cana-5226 77 90 ๐‘š=0 ๐‘š=0 PUNCT cana-5226 77 91 ] ] PUNCT cana-5226 77 92 where where SCONJ cana-5226 77 93 k k PROPN cana-5226 77 94 is be AUX cana-5226 77 95 even even ADV cana-5226 77 96 , , PUNCT cana-5226 77 97 ๐‘˜ ๐‘˜ DET cana-5226 77 98 โ‰  โ‰  PROPN cana-5226 77 99 2 2 NUM cana-5226 77 100 and and CCONJ cana-5226 77 101 ๐‘˜ ๐‘˜ X cana-5226 78 1 < < X cana-5226 78 2 ๐‘› ๐‘› PRON cana-5226 78 3 โˆ’ โˆ’ PROPN cana-5226 78 4 1 1 NUM cana-5226 78 5 . . PUNCT cana-5226 79 1 for for ADP cana-5226 79 2 example example NOUN cana-5226 79 3 : : PUNCT cana-5226 79 4 let let VERB cana-5226 79 5 ๐ถ6 ๐ถ6 NOUN cana-5226 79 6 be be AUX cana-5226 79 7 a a DET cana-5226 79 8 cycle cycle NOUN cana-5226 79 9 of of ADP cana-5226 79 10 6 6 NUM cana-5226 79 11 vertices vertex NOUN cana-5226 79 12 , , PUNCT cana-5226 79 13 adjacent adjacent ADJ cana-5226 79 14 matrix matrix NOUN cana-5226 79 15 of of ADP cana-5226 79 16 ๐ถ6 ๐ถ6 NOUN cana-5226 79 17 is be AUX cana-5226 79 18 ๐‘‹(๐ถ6 ๐‘‹(๐ถ6 VERB cana-5226 79 19 ) ) PUNCT cana-5226 79 20 โก= โก= NOUN cana-5226 79 21 [ [ PUNCT cana-5226 79 22 0 0 NUM cana-5226 79 23 1 1 NUM cana-5226 79 24 0 0 NUM cana-5226 79 25 0 0 NUM cana-5226 79 26 0 0 NUM cana-5226 79 27 1 1 NUM cana-5226 79 28 1 1 NUM cana-5226 79 29 0 0 NUM cana-5226 79 30 1 1 NUM cana-5226 79 31 0 0 NUM cana-5226 79 32 0 0 NUM cana-5226 79 33 0 0 NUM cana-5226 79 34 0 0 NUM cana-5226 79 35 0 0 NUM cana-5226 79 36 0 0 NUM cana-5226 79 37 1 1 NUM cana-5226 79 38 1 1 NUM cana-5226 79 39 0 0 NUM cana-5226 79 40 0 0 NUM cana-5226 79 41 0 0 NUM cana-5226 79 42 0 0 NUM cana-5226 79 43 1 1 NUM cana-5226 79 44 0 0 NUM cana-5226 79 45 0 0 NUM cana-5226 79 46 1 1 NUM cana-5226 79 47 0 0 NUM cana-5226 79 48 1 1 NUM cana-5226 79 49 0 0 NUM cana-5226 79 50 0 0 NUM cana-5226 79 51 1 1 NUM cana-5226 79 52 0 0 NUM cana-5226 79 53 1 1 NUM cana-5226 79 54 0 0 NUM cana-5226 79 55 0 0 NUM cana-5226 79 56 1 1 NUM cana-5226 79 57 0 0 NUM cana-5226 79 58 ] ] PUNCT cana-5226 79 59 โกโกโก6ร—6 โกโกโก6ร—6 ADJ cana-5226 79 60 characteristic characteristic ADJ cana-5226 79 61 equation equation NOUN cana-5226 79 62 is be AUX cana-5226 79 63 โˆ†(๐ถ6 โˆ†(๐ถ6 VERB cana-5226 79 64 , , PUNCT cana-5226 79 65 ๐œ† ๐œ† X cana-5226 79 66 ) ) PUNCT cana-5226 79 67 = = NOUN cana-5226 79 68 det(๐œ†๐ผ6 det(๐œ†๐ผ6 NOUN cana-5226 79 69 โˆ’ โˆ’ PROPN cana-5226 79 70 ๐‘‹(๐ถ6 ๐‘‹(๐ถ6 PUNCT cana-5226 79 71 ) ) PUNCT cana-5226 79 72 ) ) PUNCT cana-5226 80 1 = = SYM cana-5226 80 2 0 0 NUM cana-5226 80 3 communications communication NOUN cana-5226 80 4 on on ADP cana-5226 80 5 applied apply VERB cana-5226 80 6 nonlinear nonlinear ADJ cana-5226 80 7 analysis analysis NOUN cana-5226 80 8 issn issn NOUN cana-5226 80 9 : : PUNCT cana-5226 80 10 1074 1074 NUM cana-5226 80 11 - - PUNCT cana-5226 80 12 133x 133x NUM cana-5226 80 13 vol vol VERB cana-5226 80 14 32 32 NUM cana-5226 80 15 no no NOUN cana-5226 80 16 . . PUNCT cana-5226 81 1 10s 10 NOUN cana-5226 81 2 ( ( PUNCT cana-5226 81 3 2025 2025 NUM cana-5226 81 4 ) ) PUNCT cana-5226 81 5 1221 1221 NUM cana-5226 81 6 https://internationalpubls.com https://internationalpubls.com X cana-5226 81 7 โ‡’ โ‡’ PROPN cana-5226 81 8 ฮป6 ฮป6 PROPN cana-5226 81 9 โˆ’ โˆ’ PROPN cana-5226 81 10 s1,6ฮป s1,6ฮป VERB cana-5226 81 11 5 5 NUM cana-5226 81 12 + + CCONJ cana-5226 81 13 s2,6ฮป s2,6ฮป VERB cana-5226 81 14 4 4 NUM cana-5226 81 15 โˆ’ โˆ’ NOUN cana-5226 81 16 s3,6ฮป s3,6ฮป NUM cana-5226 81 17 3 3 NUM cana-5226 81 18 + + CCONJ cana-5226 81 19 s4,6ฮป s4,6ฮป PROPN cana-5226 81 20 2 2 NUM cana-5226 81 21 โˆ’ โˆ’ NOUN cana-5226 82 1 s5,6ฮป s5,6ฮป PROPN cana-5226 82 2 + + PROPN cana-5226 82 3 s6,6 s6,6 PROPN cana-5226 82 4 = = SYM cana-5226 82 5 0 0 NUM cana-5226 82 6 โ‡’ โ‡’ PROPN cana-5226 82 7 ฮป6 ฮป6 PROPN cana-5226 82 8 โˆ’ โˆ’ PROPN cana-5226 82 9 6ฮป4 6ฮป4 NUM cana-5226 82 10 + + CCONJ cana-5226 82 11 9ฮป2 9ฮป2 NUM cana-5226 82 12 โˆ’ โˆ’ NOUN cana-5226 82 13 4 4 NUM cana-5226 82 14 = = SYM cana-5226 82 15 0 0 PUNCT cana-5226 82 16 as as ADP cana-5226 82 17 s1,6 s1,6 PROPN cana-5226 82 18 = = SYM cana-5226 82 19 0 0 PROPN cana-5226 82 20 , , PUNCT cana-5226 82 21 s2,6 s2,6 PROPN cana-5226 82 22 = = SYM cana-5226 82 23 โˆ’6 โˆ’6 PROPN cana-5226 82 24 , , PUNCT cana-5226 82 25 s3,6 s3,6 PROPN cana-5226 82 26 = = SYM cana-5226 82 27 0 0 NUM cana-5226 82 28 , , PUNCT cana-5226 82 29 s4,6 s4,6 NOUN cana-5226 82 30 = = NOUN cana-5226 82 31 9 9 NUM cana-5226 82 32 , , PUNCT cana-5226 82 33 s5,6 s5,6 PROPN cana-5226 82 34 = = SYM cana-5226 82 35 0โกandโกs6,6 0โกandโกs6,6 NUM cana-5226 82 36 = = SYM cana-5226 82 37 โˆ’4 โˆ’4 X cana-5226 82 38 โ‡’ โ‡’ X cana-5226 82 39 ฮป ฮป PROPN cana-5226 82 40 = = SYM cana-5226 82 41 โˆ’2,โˆ’1,โˆ’1 โˆ’2,โˆ’1,โˆ’1 NOUN cana-5226 82 42 , , PUNCT cana-5226 82 43 1 1 NUM cana-5226 82 44 , , PUNCT cana-5226 82 45 1 1 NUM cana-5226 82 46 , , PUNCT cana-5226 82 47 2 2 NUM cana-5226 82 48 case case NOUN cana-5226 82 49 iv iv X cana-5226 82 50 : : PUNCT cana-5226 82 51 if if SCONJ cana-5226 82 52 n n PRON cana-5226 82 53 โ‰ก โ‰ก PROPN cana-5226 82 54 3(modโก4),โกthen 3(modโก4),โกthen PUNCT cana-5226 83 1 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 83 2 = = SYM cana-5226 83 3 { { PUNCT cana-5226 83 4 0 0 NUM cana-5226 83 5 , , PUNCT cana-5226 83 6 ๐‘˜โก๐‘–๐‘ โก๐‘œ๐‘‘๐‘‘โก๐‘Ž๐‘›๐‘‘โก๐‘˜ ๐‘˜โก๐‘–๐‘ โก๐‘œ๐‘‘๐‘‘โก๐‘Ž๐‘›๐‘‘โก๐‘˜ ADV cana-5226 83 7 โ‰  โ‰  PROPN cana-5226 83 8 ๐‘› ๐‘› PRON cana-5226 83 9 โˆ’๐‘›,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ โˆ’๐‘›,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ NOUN cana-5226 83 10 = = SYM cana-5226 83 11 2 2 NUM cana-5226 83 12 2,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ 2,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ NUM cana-5226 83 13 = = SYM cana-5226 83 14 ๐‘› ๐‘› PROPN cana-5226 83 15 โˆ’๐‘›,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ โˆ’๐‘›,โกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโกโก๐‘˜ NOUN cana-5226 83 16 = = SYM cana-5226 83 17 ๐‘› ๐‘› PRON cana-5226 83 18 โˆ’ โˆ’ NUM cana-5226 83 19 1 1 NUM cana-5226 83 20 and and CCONJ cana-5226 83 21 ๐‘†๐‘˜,๐‘› ๐‘†๐‘˜,๐‘› PROPN cana-5226 83 22 = = SYM cana-5226 83 23 ( ( PUNCT cana-5226 83 24 โˆ’1)๐‘˜ โˆ’1)๐‘˜ NOUN cana-5226 83 25 2โ„ 2โ„ NUM cana-5226 83 26 [ [ X cana-5226 83 27 ( ( PUNCT cana-5226 83 28 ๐‘˜ ๐‘˜ X cana-5226 83 29 + + NOUN cana-5226 83 30 1 1 NUM cana-5226 83 31 ) ) PUNCT cana-5226 83 32 + + CCONJ cana-5226 83 33 ( ( PUNCT cana-5226 83 34 โˆ’1)(๐‘˜โˆ’2 โˆ’1)(๐‘˜โˆ’2 PROPN cana-5226 83 35 ) ) PUNCT cana-5226 83 36 2โ„ 2โ„ NUM cana-5226 83 37 โก โก NOUN cana-5226 83 38 โˆ‘ โˆ‘ ADP cana-5226 83 39 ๐‘†(๐‘˜ ๐‘†(๐‘˜ ADV cana-5226 83 40 โˆ’ โˆ’ PROPN cana-5226 83 41 2 2 NUM cana-5226 83 42 , , PUNCT cana-5226 83 43 ๐‘˜ ๐‘˜ PROPN cana-5226 83 44 + + NOUN cana-5226 83 45 ๐‘š ๐‘š NOUN cana-5226 83 46 ) ) PUNCT cana-5226 83 47 ๐‘›โˆ’2โˆ’๐‘˜ ๐‘›โˆ’2โˆ’๐‘˜ NOUN cana-5226 83 48 ๐‘š=0 ๐‘š=0 PUNCT cana-5226 83 49 ] ] PUNCT cana-5226 83 50 where where SCONJ cana-5226 83 51 k k PROPN cana-5226 83 52 is be AUX cana-5226 83 53 even even ADV cana-5226 83 54 , , PUNCT cana-5226 83 55 ๐‘˜ ๐‘˜ DET cana-5226 83 56 โ‰  โ‰  PROPN cana-5226 83 57 2 2 NUM cana-5226 83 58 and and CCONJ cana-5226 83 59 ๐‘˜ ๐‘˜ X cana-5226 83 60 < < X cana-5226 83 61 ๐‘› ๐‘› PRON cana-5226 83 62 โˆ’ โˆ’ PROPN cana-5226 83 63 1 1 NUM cana-5226 83 64 . . PUNCT cana-5226 84 1 for for ADP cana-5226 84 2 example example NOUN cana-5226 84 3 : : PUNCT cana-5226 84 4 let let VERB cana-5226 84 5 ๐ถ7 ๐ถ7 NOUN cana-5226 84 6 be be AUX cana-5226 84 7 a a DET cana-5226 84 8 cycle cycle NOUN cana-5226 84 9 of of ADP cana-5226 84 10 7 7 NUM cana-5226 84 11 vertices vertex NOUN cana-5226 84 12 , , PUNCT cana-5226 84 13 adjacent adjacent ADJ cana-5226 84 14 matrix matrix NOUN cana-5226 84 15 of of ADP cana-5226 84 16 c7 c7 PROPN cana-5226 84 17 is be AUX cana-5226 84 18 x(c7 x(c7 PROPN cana-5226 84 19 ) ) PUNCT cana-5226 84 20 โก= โก= NOUN cana-5226 84 21 [ [ PUNCT cana-5226 84 22 0 0 NUM cana-5226 84 23 1 1 NUM cana-5226 84 24 0 0 NUM cana-5226 84 25 0 0 NUM cana-5226 84 26 0 0 NUM cana-5226 84 27 0 0 NUM cana-5226 84 28 1 1 NUM cana-5226 84 29 1 1 NUM cana-5226 84 30 0 0 NUM cana-5226 84 31 1 1 NUM cana-5226 84 32 0 0 NUM cana-5226 84 33 0 0 NUM cana-5226 84 34 0 0 NUM cana-5226 84 35 0 0 NUM cana-5226 84 36 0 0 NUM cana-5226 84 37 0 0 NUM cana-5226 84 38 0 0 NUM cana-5226 84 39 0 0 NUM cana-5226 84 40 1 1 NUM cana-5226 84 41 1 1 NUM cana-5226 84 42 0 0 NUM cana-5226 84 43 0 0 NUM cana-5226 84 44 0 0 NUM cana-5226 84 45 0 0 NUM cana-5226 84 46 0 0 NUM cana-5226 84 47 1 1 NUM cana-5226 84 48 0 0 NUM cana-5226 84 49 0 0 NUM cana-5226 84 50 0 0 NUM cana-5226 84 51 1 1 NUM cana-5226 84 52 0 0 NUM cana-5226 84 53 1 1 NUM cana-5226 84 54 0 0 NUM cana-5226 84 55 0 0 NUM cana-5226 84 56 0 0 NUM cana-5226 84 57 1 1 NUM cana-5226 84 58 0 0 NUM cana-5226 84 59 1 1 NUM cana-5226 84 60 0 0 NUM cana-5226 84 61 0 0 NUM cana-5226 84 62 0 0 NUM cana-5226 84 63 1 1 NUM cana-5226 84 64 0 0 NUM cana-5226 84 65 1 1 NUM cana-5226 84 66 0 0 NUM cana-5226 84 67 0 0 NUM cana-5226 84 68 0 0 NUM cana-5226 84 69 1 1 NUM cana-5226 84 70 0 0 NUM cana-5226 84 71 ] ] PUNCT cana-5226 84 72 โกโกโก7ร—7 โกโกโก7ร—7 NOUN cana-5226 84 73 characteristic characteristic ADJ cana-5226 84 74 equation equation NOUN cana-5226 84 75 is be AUX cana-5226 84 76 โˆ†(๐ถ7 โˆ†(๐ถ7 ADJ cana-5226 84 77 , , PUNCT cana-5226 84 78 ๐œ† ๐œ† X cana-5226 84 79 ) ) PUNCT cana-5226 84 80 = = SYM cana-5226 84 81 det(๐œ†๐ผ7 det(๐œ†๐ผ7 X cana-5226 85 1 โˆ’ โˆ’ X cana-5226 85 2 ๐‘‹(๐ถ7 ๐‘‹(๐ถ7 ADJ cana-5226 85 3 ) ) PUNCT cana-5226 85 4 ) ) PUNCT cana-5226 86 1 = = SYM cana-5226 86 2 0 0 NUM cana-5226 86 3 โ‡’ โ‡’ NOUN cana-5226 86 4 ๐œ†7 ๐œ†7 PROPN cana-5226 86 5 โˆ’ โˆ’ PROPN cana-5226 86 6 ๐‘†1,7๐œ† ๐‘†1,7๐œ† NOUN cana-5226 86 7 6 6 NUM cana-5226 86 8 + + CCONJ cana-5226 86 9 ๐‘†2,7๐œ† ๐‘†2,7๐œ† NOUN cana-5226 86 10 5 5 NUM cana-5226 86 11 โˆ’ โˆ’ NOUN cana-5226 86 12 ๐‘†3,7๐œ† ๐‘†3,7๐œ† NOUN cana-5226 86 13 4 4 NUM cana-5226 86 14 + + NOUN cana-5226 86 15 ๐‘†4,7๐œ† ๐‘†4,7๐œ† NOUN cana-5226 86 16 3 3 NUM cana-5226 86 17 โˆ’ โˆ’ NOUN cana-5226 86 18 ๐‘†5,7๐œ† ๐‘†5,7๐œ† NOUN cana-5226 86 19 2 2 NUM cana-5226 86 20 + + CCONJ cana-5226 86 21 ๐‘†6,7๐œ† ๐‘†6,7๐œ† NOUN cana-5226 86 22 โˆ’ โˆ’ PROPN cana-5226 86 23 ๐‘†7,7 ๐‘†7,7 NOUN cana-5226 86 24 = = SYM cana-5226 86 25 0 0 NUM cana-5226 86 26 โ‡’ โ‡’ NOUN cana-5226 87 1 ๐œ†7 ๐œ†7 NOUN cana-5226 87 2 โˆ’ โˆ’ PROPN cana-5226 88 1 7๐œ†5 7๐œ†5 NUM cana-5226 89 1 + + CCONJ cana-5226 90 1 14๐œ†3 14๐œ†3 NUM cana-5226 90 2 โˆ’ โˆ’ NOUN cana-5226 90 3 7๐œ† 7๐œ† NOUN cana-5226 90 4 โˆ’ โˆ’ PROPN cana-5226 90 5 2 2 NUM cana-5226 90 6 = = SYM cana-5226 90 7 0 0 PUNCT cana-5226 90 8 as as ADP cana-5226 90 9 ๐‘†1,7 ๐‘†1,7 PROPN cana-5226 90 10 = = SYM cana-5226 90 11 0 0 PROPN cana-5226 90 12 , , PUNCT cana-5226 90 13 ๐‘†2,7 ๐‘†2,7 NOUN cana-5226 90 14 = = SYM cana-5226 90 15 โˆ’7 โˆ’7 PROPN cana-5226 90 16 , , PUNCT cana-5226 90 17 ๐‘†3,7 ๐‘†3,7 PROPN cana-5226 90 18 = = SYM cana-5226 90 19 0 0 NUM cana-5226 90 20 , , PUNCT cana-5226 90 21 ๐‘†4,7 ๐‘†4,7 NOUN cana-5226 90 22 = = SYM cana-5226 90 23 14 14 NUM cana-5226 90 24 , , PUNCT cana-5226 90 25 ๐‘†5,7 ๐‘†5,7 NUM cana-5226 90 26 = = SYM cana-5226 90 27 0 0 NUM cana-5226 90 28 , , PUNCT cana-5226 90 29 ๐‘†6,7 ๐‘†6,7 NOUN cana-5226 90 30 = = SYM cana-5226 90 31 โˆ’7โก๐‘Ž๐‘›๐‘‘โก๐‘†7,7 โˆ’7โก๐‘Ž๐‘›๐‘‘โก๐‘†7,7 ADJ cana-5226 90 32 = = SYM cana-5226 90 33 2 2 NUM cana-5226 90 34 โ‡’ โ‡’ NOUN cana-5226 90 35 ๐œ† ๐œ† X cana-5226 90 36 = = SYM cana-5226 90 37 โˆ’1.8019,โˆ’1.8019 โˆ’1.8019,โˆ’1.8019 NOUN cana-5226 90 38 , , PUNCT cana-5226 90 39 โˆ’0.4450,โˆ’0.4450 โˆ’0.4450,โˆ’0.4450 NUM cana-5226 90 40 , , PUNCT cana-5226 90 41 1.2470 1.2470 NUM cana-5226 90 42 , , PUNCT cana-5226 90 43 1.2470 1.2470 NUM cana-5226 90 44 , , PUNCT cana-5226 90 45 2 2 NUM cana-5226 90 46 table table NOUN cana-5226 90 47 : : PUNCT cana-5226 90 48 for for ADP cana-5226 90 49 ๐‘บ๐’Œ,๐’โกof ๐‘บ๐’Œ,๐’โกof PROPN cana-5226 90 50 cycle cycle NOUN cana-5226 90 51 ๐‘ช๐’ ๐‘ช๐’ PROPN cana-5226 90 52 n n CCONJ cana-5226 90 53 ๐‘†1,๐‘› ๐‘†1,๐‘› PROPN cana-5226 90 54 ๐‘†2,๐‘› ๐‘†2,๐‘› ADJ cana-5226 90 55 ๐‘†3,๐‘› ๐‘†3,๐‘› PROPN cana-5226 90 56 ๐‘†4,๐‘› ๐‘†4,๐‘› PROPN cana-5226 90 57 ๐‘†5,๐‘› ๐‘†5,๐‘› PROPN cana-5226 90 58 ๐‘†6,๐‘› ๐‘†6,๐‘› PROPN cana-5226 90 59 ๐‘†7,๐‘› ๐‘†7,๐‘› PROPN cana-5226 90 60 ๐‘†8,๐‘› ๐‘†8,๐‘› NUM cana-5226 90 61 ๐‘†9,๐‘› ๐‘†9,๐‘› PROPN cana-5226 91 1 ๐‘†10,๐‘› ๐‘†10,๐‘› NUM cana-5226 91 2 ๐‘†12,๐‘› ๐‘†12,๐‘› X cana-5226 91 3 ๐‘†14,๐‘› ๐‘†14,๐‘› X cana-5226 91 4 ๐‘†16,๐‘› ๐‘†16,๐‘› ADV cana-5226 91 5 ๐‘†18,๐‘› ๐‘†18,๐‘› PUNCT cana-5226 91 6 ๐‘†20,๐‘› ๐‘†20,๐‘› NOUN cana-5226 91 7 3 3 NUM cana-5226 91 8 0 0 NUM cana-5226 91 9 -3 -3 SYM cana-5226 91 10 2 2 NUM cana-5226 91 11 4 4 NUM cana-5226 91 12 0 0 NUM cana-5226 91 13 -4 -4 SYM cana-5226 91 14 0 0 NUM cana-5226 91 15 0 0 NUM cana-5226 91 16 5 5 NUM cana-5226 91 17 0 0 NUM cana-5226 91 18 -5 -5 NOUN cana-5226 91 19 0 0 NUM cana-5226 91 20 5 5 NUM cana-5226 91 21 2 2 NUM cana-5226 91 22 6 6 NUM cana-5226 91 23 0 0 NUM cana-5226 92 1 -6 -6 NOUN cana-5226 92 2 0 0 NUM cana-5226 93 1 9 9 NUM cana-5226 93 2 0 0 NUM cana-5226 93 3 -4 -4 SYM cana-5226 93 4 7 7 NUM cana-5226 93 5 0 0 NUM cana-5226 93 6 -7 -7 NOUN cana-5226 93 7 0 0 NUM cana-5226 93 8 14 14 NUM cana-5226 93 9 0 0 NUM cana-5226 94 1 -7 -7 NOUN cana-5226 94 2 2 2 NUM cana-5226 94 3 8 8 NUM cana-5226 94 4 0 0 NUM cana-5226 94 5 -8 -8 NOUN cana-5226 94 6 0 0 NUM cana-5226 94 7 20 20 NUM cana-5226 94 8 0 0 NUM cana-5226 94 9 -16 -16 SYM cana-5226 94 10 0 0 NUM cana-5226 94 11 0 0 NUM cana-5226 94 12 9 9 NUM cana-5226 94 13 0 0 NUM cana-5226 94 14 -9 -9 NOUN cana-5226 94 15 0 0 NUM cana-5226 94 16 27 27 NUM cana-5226 94 17 0 0 NUM cana-5226 94 18 -30 -30 SYM cana-5226 94 19 0 0 NUM cana-5226 94 20 9 9 NUM cana-5226 94 21 2 2 NUM cana-5226 94 22 10 10 NUM cana-5226 94 23 0 0 NUM cana-5226 94 24 -10 -10 SYM cana-5226 94 25 0 0 NUM cana-5226 94 26 35 35 NUM cana-5226 94 27 0 0 NUM cana-5226 94 28 -50 -50 PUNCT cana-5226 94 29 0 0 NUM cana-5226 94 30 25 25 NUM cana-5226 94 31 0 0 NUM cana-5226 94 32 -4 -4 NOUN cana-5226 94 33 communications communication NOUN cana-5226 94 34 on on ADP cana-5226 94 35 applied apply VERB cana-5226 94 36 nonlinear nonlinear ADJ cana-5226 94 37 analysis analysis NOUN cana-5226 94 38 issn issn NOUN cana-5226 94 39 : : PUNCT cana-5226 94 40 1074 1074 NUM cana-5226 94 41 - - PUNCT cana-5226 94 42 133x 133x NUM cana-5226 94 43 vol vol VERB cana-5226 94 44 32 32 NUM cana-5226 94 45 no no NOUN cana-5226 94 46 . . PUNCT cana-5226 95 1 10s 10 NOUN cana-5226 95 2 ( ( PUNCT cana-5226 95 3 2025 2025 NUM cana-5226 95 4 ) ) PUNCT cana-5226 95 5 1222 1222 NUM cana-5226 95 6 https://internationalpubls.com https://internationalpubls.com X cana-5226 95 7 11 11 NUM cana-5226 95 8 0 0 NUM cana-5226 95 9 -11 -11 SYM cana-5226 95 10 0 0 NUM cana-5226 95 11 44 44 NUM cana-5226 95 12 0 0 NUM cana-5226 95 13 -77 -77 SYM cana-5226 95 14 0 0 NUM cana-5226 96 1 55 55 NUM cana-5226 96 2 0 0 NUM cana-5226 96 3 -11 -11 SYM cana-5226 96 4 12 12 NUM cana-5226 96 5 0 0 NUM cana-5226 96 6 -12 -12 SYM cana-5226 96 7 0 0 NUM cana-5226 96 8 54 54 NUM cana-5226 96 9 0 0 NUM cana-5226 97 1 -112 -112 PROPN cana-5226 97 2 0 0 NUM cana-5226 97 3 105 105 NUM cana-5226 97 4 0 0 NUM cana-5226 97 5 -36 -36 SYM cana-5226 97 6 0 0 NUM cana-5226 97 7 13 13 NUM cana-5226 97 8 0 0 NUM cana-5226 97 9 -13 -13 SYM cana-5226 97 10 0 0 NUM cana-5226 98 1 65 65 NUM cana-5226 98 2 0 0 NUM cana-5226 99 1 -156 -156 PROPN cana-5226 99 2 0 0 NUM cana-5226 99 3 182 182 NUM cana-5226 99 4 0 0 NUM cana-5226 99 5 -91 -91 SYM cana-5226 99 6 13 13 NUM cana-5226 99 7 14 14 NUM cana-5226 99 8 0 0 NUM cana-5226 99 9 -14 -14 SYM cana-5226 99 10 0 0 NUM cana-5226 100 1 77 77 NUM cana-5226 100 2 0 0 NUM cana-5226 101 1 -210 -210 NOUN cana-5226 101 2 0 0 NUM cana-5226 101 3 294 294 NUM cana-5226 101 4 0 0 NUM cana-5226 101 5 -196 -196 NUM cana-5226 101 6 49 49 NUM cana-5226 101 7 -4 -4 SYM cana-5226 101 8 15 15 NUM cana-5226 101 9 0 0 NUM cana-5226 101 10 -15 -15 SYM cana-5226 101 11 0 0 NUM cana-5226 102 1 90 90 NUM cana-5226 102 2 0 0 NUM cana-5226 102 3 -275 -275 PROPN cana-5226 102 4 0 0 NUM cana-5226 103 1 450 450 NUM cana-5226 103 2 0 0 NUM cana-5226 103 3 -378 -378 NUM cana-5226 103 4 140 140 NUM cana-5226 103 5 -15 -15 NUM cana-5226 103 6 16 16 NUM cana-5226 103 7 0 0 NUM cana-5226 103 8 -16 -16 PROPN cana-5226 103 9 0 0 NUM cana-5226 104 1 104 104 NUM cana-5226 104 2 0 0 NUM cana-5226 105 1 -352 -352 NUM cana-5226 105 2 0 0 NUM cana-5226 106 1 660 660 NUM cana-5226 106 2 0 0 NUM cana-5226 106 3 -672 -672 NUM cana-5226 106 4 336 336 NUM cana-5226 106 5 -64 -64 NUM cana-5226 106 6 0 0 NUM cana-5226 107 1 17 17 NUM cana-5226 107 2 0 0 NUM cana-5226 107 3 -17 -17 SYM cana-5226 107 4 0 0 NUM cana-5226 108 1 119 119 NUM cana-5226 108 2 0 0 NUM cana-5226 109 1 -442 -442 PROPN cana-5226 109 2 0 0 NUM cana-5226 109 3 935 935 NUM cana-5226 109 4 0 0 NUM cana-5226 110 1 -1122 -1122 NOUN cana-5226 110 2 714 714 NUM cana-5226 110 3 -204 -204 PROPN cana-5226 110 4 17 17 NUM cana-5226 110 5 18 18 NUM cana-5226 110 6 0 0 NUM cana-5226 110 7 -18 -18 SYM cana-5226 110 8 0 0 NUM cana-5226 110 9 135 135 NUM cana-5226 110 10 0 0 NUM cana-5226 111 1 -546 -546 PROPN cana-5226 111 2 0 0 NUM cana-5226 111 3 1287 1287 NUM cana-5226 111 4 0 0 NUM cana-5226 111 5 -1782 -1782 PROPN cana-5226 111 6 1386 1386 NUM cana-5226 111 7 -540 -540 PROPN cana-5226 111 8 81 81 NUM cana-5226 111 9 -4 -4 SYM cana-5226 111 10 19 19 NUM cana-5226 111 11 0 0 NUM cana-5226 111 12 -19 -19 SYM cana-5226 111 13 0 0 NUM cana-5226 111 14 152 152 NUM cana-5226 111 15 0 0 NUM cana-5226 112 1 -665 -665 NUM cana-5226 112 2 0 0 NUM cana-5226 112 3 1729 1729 NUM cana-5226 112 4 0 0 NUM cana-5226 112 5 -2717 -2717 PROPN cana-5226 112 6 2508 2508 NUM cana-5226 112 7 -1254 -1254 NOUN cana-5226 112 8 285 285 NUM cana-5226 112 9 -19 -19 NUM cana-5226 112 10 20 20 NUM cana-5226 112 11 0 0 NUM cana-5226 112 12 -20 -20 NUM cana-5226 112 13 0 0 NUM cana-5226 112 14 170 170 NUM cana-5226 112 15 0 0 NUM cana-5226 113 1 -800 -800 NUM cana-5226 113 2 0 0 NUM cana-5226 113 3 2275 2275 NUM cana-5226 113 4 0 0 NUM cana-5226 113 5 -4004 -4004 X cana-5226 113 6 4290 4290 NUM cana-5226 113 7 -2640 -2640 ADJ cana-5226 113 8 825 825 NUM cana-5226 113 9 -100 -100 PROPN cana-5226 113 10 0 0 NUM cana-5226 113 11 table table NOUN cana-5226 113 12 : : PUNCT cana-5226 113 13 for for ADP cana-5226 113 14 eigen eigen PROPN cana-5226 113 15 values value NOUN cana-5226 113 16 and and CCONJ cana-5226 113 17 energy energy NOUN cana-5226 113 18 of of ADP cana-5226 113 19 cycle cycle NOUN cana-5226 113 20 nc nc PROPN cana-5226 113 21 n n PROPN cana-5226 113 22 eigen eigen PROPN cana-5226 113 23 values value NOUN cana-5226 113 24 energy energy NOUN cana-5226 113 25 3 3 NUM cana-5226 113 26 -1 -1 NOUN cana-5226 113 27 , , PUNCT cana-5226 113 28 1 1 NUM cana-5226 113 29 , , PUNCT cana-5226 113 30 2 2 NUM cana-5226 113 31 4 4 NUM cana-5226 113 32 4 4 NUM cana-5226 113 33 -2 -2 NOUN cana-5226 113 34 , , PUNCT cana-5226 113 35 0 0 NUM cana-5226 113 36 , , PUNCT cana-5226 113 37 0 0 NUM cana-5226 113 38 , , PUNCT cana-5226 113 39 2 2 NUM cana-5226 113 40 4 4 NUM cana-5226 113 41 5 5 NUM cana-5226 113 42 -1.6180 -1.6180 NOUN cana-5226 113 43 , , PUNCT cana-5226 113 44 -1.6180 -1.6180 PROPN cana-5226 113 45 , , PUNCT cana-5226 113 46 0.6180 0.6180 NUM cana-5226 113 47 , , PUNCT cana-5226 113 48 0.6180 0.6180 NUM cana-5226 113 49 , , PUNCT cana-5226 113 50 2 2 NUM cana-5226 113 51 6.472 6.472 NUM cana-5226 113 52 6 6 NUM cana-5226 113 53 -2 -2 NOUN cana-5226 113 54 , , PUNCT cana-5226 113 55 -1 -1 INTJ cana-5226 113 56 , , PUNCT cana-5226 113 57 -1 -1 INTJ cana-5226 113 58 , , PUNCT cana-5226 113 59 1 1 NUM cana-5226 113 60 , , PUNCT cana-5226 113 61 1 1 NUM cana-5226 113 62 , , PUNCT cana-5226 113 63 2 2 NUM cana-5226 113 64 8 8 NUM cana-5226 113 65 7 7 NUM cana-5226 113 66 -1.8019 -1.8019 NOUN cana-5226 113 67 , , PUNCT cana-5226 113 68 -1.8019 -1.8019 X cana-5226 113 69 , , PUNCT cana-5226 113 70 -0.4450 -0.4450 X cana-5226 113 71 , , PUNCT cana-5226 113 72 -0.4450 -0.4450 X cana-5226 113 73 , , PUNCT cana-5226 113 74 1.2470 1.2470 NUM cana-5226 113 75 , , PUNCT cana-5226 113 76 1.2470 1.2470 NUM cana-5226 113 77 , , PUNCT cana-5226 113 78 2 2 NUM cana-5226 113 79 8.9878 8.9878 NUM cana-5226 113 80 8 8 NUM cana-5226 113 81 -2 -2 NUM cana-5226 113 82 , , PUNCT cana-5226 113 83 -1.4142 -1.4142 PROPN cana-5226 113 84 , , PUNCT cana-5226 113 85 -1.4142 -1.4142 PROPN cana-5226 113 86 , , PUNCT cana-5226 113 87 0 0 NUM cana-5226 113 88 , , PUNCT cana-5226 113 89 0 0 NUM cana-5226 113 90 , , PUNCT cana-5226 113 91 1.4142 1.4142 NUM cana-5226 113 92 , , PUNCT cana-5226 113 93 1.4142 1.4142 NUM cana-5226 113 94 , , PUNCT cana-5226 113 95 2 2 NUM cana-5226 113 96 9.6568 9.6568 NUM cana-5226 113 97 9 9 NUM cana-5226 113 98 -1.8794 -1.8794 NOUN cana-5226 113 99 , , PUNCT cana-5226 113 100 -1.8794 -1.8794 NOUN cana-5226 113 101 , , PUNCT cana-5226 113 102 -1 -1 INTJ cana-5226 113 103 , , PUNCT cana-5226 113 104 -1 -1 INTJ cana-5226 113 105 , , PUNCT cana-5226 113 106 0.3473 0.3473 NUM cana-5226 113 107 , , PUNCT cana-5226 113 108 0.3473 0.3473 NUM cana-5226 113 109 , , PUNCT cana-5226 113 110 1.5321 1.5321 NUM cana-5226 113 111 , , PUNCT cana-5226 113 112 1.5321 1.5321 NUM cana-5226 113 113 , , PUNCT cana-5226 113 114 2 2 NUM cana-5226 113 115 11.5176 11.5176 NUM cana-5226 113 116 10 10 NUM cana-5226 113 117 -2 -2 NOUN cana-5226 113 118 , , PUNCT cana-5226 113 119 -1.6180 -1.6180 PROPN cana-5226 113 120 , , PUNCT cana-5226 113 121 -1.6180 -1.6180 PROPN cana-5226 113 122 , , PUNCT cana-5226 113 123 -0.6180 -0.6180 PROPN cana-5226 113 124 , , PUNCT cana-5226 113 125 -0.6180 -0.6180 PROPN cana-5226 113 126 , , PUNCT cana-5226 113 127 0.6180 0.6180 NUM cana-5226 113 128 , , PUNCT cana-5226 113 129 0.6180 0.6180 NUM cana-5226 113 130 , , PUNCT cana-5226 113 131 1.6180 1.6180 NUM cana-5226 113 132 , , PUNCT cana-5226 113 133 1.6180 1.6180 NUM cana-5226 113 134 , , PUNCT cana-5226 113 135 2 2 NUM cana-5226 113 136 12.944 12.944 NUM cana-5226 113 137 11 11 NUM cana-5226 113 138 -1.9190 -1.9190 NOUN cana-5226 113 139 , , PUNCT cana-5226 113 140 -1.9190 -1.9190 NOUN cana-5226 113 141 , , PUNCT cana-5226 113 142 -1.3097 -1.3097 PROPN cana-5226 113 143 , , PUNCT cana-5226 113 144 -1.3097 -1.3097 PROPN cana-5226 113 145 , , PUNCT cana-5226 113 146 -0.2846 -0.2846 PROPN cana-5226 113 147 , , PUNCT cana-5226 113 148 -0.2846 -0.2846 PROPN cana-5226 113 149 , , PUNCT cana-5226 113 150 0.8308 0.8308 NUM cana-5226 113 151 , , PUNCT cana-5226 113 152 0.8308 0.8308 NUM cana-5226 113 153 , , PUNCT cana-5226 113 154 1.6825 1.6825 NUM cana-5226 113 155 , , PUNCT cana-5226 113 156 1.6825 1.6825 NUM cana-5226 113 157 , , PUNCT cana-5226 113 158 2 2 NUM cana-5226 113 159 14.0532 14.0532 NUM cana-5226 113 160 12 12 NUM cana-5226 113 161 -2 -2 NOUN cana-5226 113 162 , , PUNCT cana-5226 113 163 -1.732 -1.732 PRON cana-5226 113 164 , , PUNCT cana-5226 113 165 -1.732 -1.732 PRON cana-5226 113 166 , , PUNCT cana-5226 113 167 -1 -1 NOUN cana-5226 113 168 , , PUNCT cana-5226 113 169 -1 -1 INTJ cana-5226 113 170 , , PUNCT cana-5226 113 171 0 0 NUM cana-5226 113 172 , , PUNCT cana-5226 113 173 0 0 NUM cana-5226 113 174 , , PUNCT cana-5226 113 175 1 1 NUM cana-5226 113 176 , , PUNCT cana-5226 113 177 1 1 NUM cana-5226 113 178 , , PUNCT cana-5226 113 179 1.732 1.732 NUM cana-5226 113 180 , , PUNCT cana-5226 113 181 1.732 1.732 NUM cana-5226 113 182 , , PUNCT cana-5226 113 183 2 2 NUM cana-5226 113 184 14.928 14.928 NUM cana-5226 113 185 15 15 NUM cana-5226 113 186 -1.9563 -1.9563 NOUN cana-5226 113 187 , , PUNCT cana-5226 113 188 -1.9563 -1.9563 PROPN cana-5226 113 189 , , PUNCT cana-5226 113 190 -1.6180 -1.6180 PROPN cana-5226 113 191 , , PUNCT cana-5226 113 192 -1.6180 -1.6180 PROPN cana-5226 113 193 , , PUNCT cana-5226 113 194 -1 -1 INTJ cana-5226 113 195 , , PUNCT cana-5226 113 196 -1 -1 INTJ cana-5226 113 197 , , PUNCT cana-5226 113 198 -0.2091 -0.2091 INTJ cana-5226 113 199 , , PUNCT cana-5226 113 200 -0.2091 -0.2091 PROPN cana-5226 113 201 , , PUNCT cana-5226 113 202 0.6180 0.6180 NUM cana-5226 113 203 , , PUNCT cana-5226 113 204 0.6180 0.6180 NUM cana-5226 113 205 , , PUNCT cana-5226 113 206 1.3383 1.3383 NUM cana-5226 113 207 , , PUNCT cana-5226 113 208 1.3383 1.3383 NUM cana-5226 113 209 , , PUNCT cana-5226 113 210 1.8271 1.8271 NUM cana-5226 113 211 , , PUNCT cana-5226 113 212 1.8271 1.8271 NUM cana-5226 113 213 , , PUNCT cana-5226 113 214 2 2 NUM cana-5226 113 215 19.1336 19.1336 NUM cana-5226 113 216 16 16 NUM cana-5226 113 217 -2 -2 NOUN cana-5226 113 218 , , PUNCT cana-5226 113 219 -1.8478 -1.8478 PROPN cana-5226 113 220 , , PUNCT cana-5226 113 221 -1.8478 -1.8478 PROPN cana-5226 113 222 , , PUNCT cana-5226 113 223 -1.4142 -1.4142 PROPN cana-5226 113 224 , , PUNCT cana-5226 113 225 -1.4142 -1.4142 PROPN cana-5226 113 226 , , PUNCT cana-5226 113 227 -0.7654 -0.7654 PROPN cana-5226 113 228 , , PUNCT cana-5226 113 229 -0.7654 -0.7654 PROPN cana-5226 113 230 , , PUNCT cana-5226 113 231 0 0 NUM cana-5226 113 232 , , PUNCT cana-5226 113 233 0 0 NUM cana-5226 113 234 , , PUNCT cana-5226 113 235 0.7654 0.7654 NUM cana-5226 113 236 , , PUNCT cana-5226 113 237 0.7654 0.7654 NUM cana-5226 113 238 , , PUNCT cana-5226 113 239 1.4142 1.4142 NUM cana-5226 113 240 , , PUNCT cana-5226 113 241 1.4142 1.4142 NUM cana-5226 113 242 , , PUNCT cana-5226 113 243 1.8478 1.8478 NUM cana-5226 113 244 , , PUNCT cana-5226 113 245 1.8478 1.8478 NUM cana-5226 113 246 , , PUNCT cana-5226 113 247 2 2 NUM cana-5226 113 248 20.1096 20.1096 NUM cana-5226 113 249 conclusion conclusion NOUN cana-5226 114 1 the the DET cana-5226 114 2 characteristic characteristic ADJ cana-5226 114 3 polynomial polynomial NOUN cana-5226 114 4 in in ADP cana-5226 114 5 the the DET cana-5226 114 6 above above ADJ cana-5226 114 7 tables table NOUN cana-5226 114 8 is be AUX cana-5226 114 9 easy easy ADJ cana-5226 114 10 to to PART cana-5226 114 11 compute compute VERB cana-5226 114 12 . . PUNCT cana-5226 115 1 when when SCONJ cana-5226 115 2 working work VERB cana-5226 115 3 with with ADP cana-5226 115 4 matrices matrix NOUN cana-5226 115 5 that that PRON cana-5226 115 6 have have VERB cana-5226 115 7 complex complex ADJ cana-5226 115 8 expressions expression NOUN cana-5226 115 9 with with ADP cana-5226 115 10 parameters parameter NOUN cana-5226 115 11 , , PUNCT cana-5226 115 12 which which PRON cana-5226 115 13 are be AUX cana-5226 115 14 frequently frequently ADV cana-5226 115 15 encountered encounter VERB cana-5226 115 16 in in ADP cana-5226 115 17 different different ADJ cana-5226 115 18 mathematical mathematical ADJ cana-5226 115 19 models model NOUN cana-5226 115 20 , , PUNCT cana-5226 115 21 the the DET cana-5226 115 22 actual actual ADJ cana-5226 115 23 usefulness usefulness NOUN cana-5226 115 24 of of ADP cana-5226 115 25 this this DET cana-5226 115 26 idea idea NOUN cana-5226 115 27 becomes become VERB cana-5226 115 28 evident evident ADJ cana-5226 115 29 . . PUNCT cana-5226 116 1 by by ADP cana-5226 116 2 expanding expand VERB cana-5226 116 3 to to ADP cana-5226 116 4 first first ADJ cana-5226 116 5 - - PUNCT cana-5226 116 6 order order NOUN cana-5226 116 7 three three NUM cana-5226 116 8 - - PUNCT cana-5226 116 9 dimensional dimensional ADJ cana-5226 116 10 discrete discrete ADJ cana-5226 116 11 dynamics dynamic NOUN cana-5226 116 12 , , PUNCT cana-5226 116 13 this this DET cana-5226 116 14 method method NOUN cana-5226 116 15 is be AUX cana-5226 116 16 useful useful ADJ cana-5226 116 17 in in ADP cana-5226 116 18 determining determine VERB cana-5226 116 19 stability stability NOUN cana-5226 116 20 criteria criterion NOUN cana-5226 116 21 for for ADP cana-5226 116 22 first first ADJ cana-5226 116 23 - - PUNCT cana-5226 116 24 order order NOUN cana-5226 116 25 two two NUM cana-5226 116 26 - - PUNCT cana-5226 116 27 dimensional dimensional ADJ cana-5226 116 28 discrete discrete ADJ cana-5226 116 29 dynamics dynamic NOUN cana-5226 116 30 . . PUNCT cana-5226 117 1 building build VERB cana-5226 117 2 upon upon SCONJ cana-5226 117 3 these these DET cana-5226 117 4 findings finding NOUN cana-5226 117 5 , , PUNCT cana-5226 117 6 our our PRON cana-5226 117 7 future future ADJ cana-5226 117 8 work work NOUN cana-5226 117 9 will will AUX cana-5226 117 10 expand expand VERB cana-5226 117 11 these these DET cana-5226 117 12 investigations investigation NOUN cana-5226 117 13 to to PART cana-5226 117 14 derive derive VERB cana-5226 117 15 stability stability NOUN cana-5226 117 16 criteria criterion NOUN cana-5226 117 17 for for ADP cana-5226 117 18 first first ADJ cana-5226 117 19 - - PUNCT cana-5226 117 20 order order NOUN cana-5226 117 21 four four NUM cana-5226 117 22 - - PUNCT cana-5226 117 23 dimensional dimensional ADJ cana-5226 117 24 discrete discrete ADJ cana-5226 117 25 dynamics dynamic NOUN cana-5226 117 26 and and CCONJ cana-5226 117 27 , , PUNCT cana-5226 117 28 more more ADV cana-5226 117 29 generally generally ADV cana-5226 117 30 , , PUNCT cana-5226 117 31 for for ADP cana-5226 117 32 first first ADJ cana-5226 117 33 - - PUNCT cana-5226 117 34 order order NOUN cana-5226 117 35 ndimensional ndimensional ADJ cana-5226 117 36 discrete discrete ADJ cana-5226 117 37 dynamics dynamic NOUN cana-5226 117 38 . . PUNCT cana-5226 118 1 communications communication NOUN cana-5226 118 2 on on ADP cana-5226 118 3 applied apply VERB cana-5226 118 4 nonlinear nonlinear ADJ cana-5226 118 5 analysis analysis NOUN cana-5226 118 6 issn issn NOUN cana-5226 118 7 : : PUNCT cana-5226 118 8 1074 1074 NUM cana-5226 118 9 - - PUNCT cana-5226 118 10 133x 133x NUM cana-5226 118 11 vol vol VERB cana-5226 118 12 32 32 NUM cana-5226 118 13 no no NOUN cana-5226 118 14 . . PUNCT cana-5226 119 1 10s 10 NOUN cana-5226 119 2 ( ( PUNCT cana-5226 119 3 2025 2025 NUM cana-5226 119 4 ) ) PUNCT cana-5226 119 5 1223 1223 NUM cana-5226 119 6 https://internationalpubls.com https://internationalpubls.com X cana-5226 119 7 refrences refrence VERB cana-5226 119 8 [ [ X cana-5226 119 9 1 1 NUM cana-5226 119 10 ] ] X cana-5226 119 11 brooks brooks PROPN cana-5226 119 12 , , PUNCT cana-5226 119 13 b. b. PROPN cana-5226 119 14 p. p. PROPN cana-5226 119 15 ( ( PUNCT cana-5226 119 16 2006 2006 NUM cana-5226 119 17 ) ) PUNCT cana-5226 119 18 . . PUNCT cana-5226 120 1 the the DET cana-5226 120 2 coefficients coefficient NOUN cana-5226 120 3 of of ADP cana-5226 120 4 the the DET cana-5226 120 5 characteristic characteristic ADJ cana-5226 120 6 polynomial polynomial NOUN cana-5226 120 7 in in ADP cana-5226 120 8 terms term NOUN cana-5226 120 9 of of ADP cana-5226 120 10 the the DET cana-5226 120 11 eigenvalues eigenvalue NOUN cana-5226 120 12 and and CCONJ cana-5226 120 13 the the DET cana-5226 120 14 elements element NOUN cana-5226 120 15 of of ADP cana-5226 120 16 an an DET cana-5226 120 17 nร— nร— NOUN cana-5226 120 18 n n NOUN cana-5226 120 19 matrix matrix NOUN cana-5226 120 20 . . PUNCT cana-5226 121 1 applied apply VERB cana-5226 121 2 mathematics mathematics NOUN cana-5226 121 3 letters letter NOUN cana-5226 121 4 , , PUNCT cana-5226 121 5 19(6 19(6 NUM cana-5226 121 6 ) ) PUNCT cana-5226 121 7 , , PUNCT cana-5226 121 8 511 511 NUM cana-5226 121 9 - - SYM cana-5226 121 10 515 515 NUM cana-5226 121 11 . . PUNCT cana-5226 122 1 [ [ X cana-5226 122 2 2 2 NUM cana-5226 122 3 ] ] X cana-5226 122 4 kumar kumar PROPN cana-5226 122 5 , , PUNCT cana-5226 122 6 s. s. PROPN cana-5226 122 7 , , PUNCT cana-5226 122 8 sarkar sarkar PROPN cana-5226 122 9 , , PUNCT cana-5226 122 10 p. p. PROPN cana-5226 122 11 , , PUNCT cana-5226 122 12 & & CCONJ cana-5226 122 13 pal pal NOUN cana-5226 122 14 , , PUNCT cana-5226 122 15 a. a. NOUN cana-5226 122 16 ( ( PUNCT cana-5226 122 17 2024 2024 NUM cana-5226 122 18 ) ) PUNCT cana-5226 122 19 . . PUNCT cana-5226 123 1 a a DET cana-5226 123 2 study study NOUN cana-5226 123 3 on on ADP cana-5226 123 4 the the DET cana-5226 123 5 energy energy NOUN cana-5226 123 6 of of ADP cana-5226 123 7 graphs graph NOUN cana-5226 123 8 and and CCONJ cana-5226 123 9 its its PRON cana-5226 123 10 applications application NOUN cana-5226 123 11 . . PUNCT cana-5226 124 1 polycyclic polycyclic ADJ cana-5226 124 2 aromatic aromatic ADJ cana-5226 124 3 compounds compound NOUN cana-5226 124 4 , , PUNCT cana-5226 124 5 44(6 44(6 NOUN cana-5226 124 6 ) ) PUNCT cana-5226 124 7 , , PUNCT cana-5226 124 8 4127 4127 NUM cana-5226 124 9 - - SYM cana-5226 124 10 4136 4136 NUM cana-5226 124 11 . . PUNCT cana-5226 125 1 [ [ X cana-5226 125 2 3 3 NUM cana-5226 125 3 ] ] PUNCT cana-5226 125 4 a. a. PROPN cana-5226 125 5 e. e. PROPN cana-5226 125 6 brouwer brouwer PROPN cana-5226 125 7 and and CCONJ cana-5226 125 8 w. w. PROPN cana-5226 125 9 h. h. PROPN cana-5226 125 10 haemers haemers PROPN cana-5226 125 11 , , PUNCT cana-5226 125 12 spectra spectra NOUN cana-5226 125 13 of of ADP cana-5226 125 14 graphs graph NOUN cana-5226 125 15 , , PUNCT cana-5226 125 16 springer springer NOUN cana-5226 125 17 , , PUNCT cana-5226 125 18 new new PROPN cana-5226 125 19 york york PROPN cana-5226 125 20 , , PUNCT cana-5226 125 21 ny ny PROPN cana-5226 125 22 , , PUNCT cana-5226 125 23 usa usa PROPN cana-5226 125 24 , , PUNCT cana-5226 125 25 2012 2012 NUM cana-5226 125 26 [ [ X cana-5226 125 27 4 4 NUM cana-5226 125 28 ] ] X cana-5226 125 29 estrada estrada PROPN cana-5226 125 30 , , PUNCT cana-5226 125 31 e. e. PROPN cana-5226 125 32 , , PUNCT cana-5226 125 33 & & CCONJ cana-5226 125 34 benzi benzi PROPN cana-5226 125 35 , , PUNCT cana-5226 125 36 m. m. NOUN cana-5226 125 37 ( ( PUNCT cana-5226 125 38 2017 2017 NUM cana-5226 125 39 ) ) PUNCT cana-5226 125 40 . . PUNCT cana-5226 126 1 what what PRON cana-5226 126 2 is be AUX cana-5226 126 3 the the DET cana-5226 126 4 meaning meaning NOUN cana-5226 126 5 of of ADP cana-5226 126 6 the the DET cana-5226 126 7 graph graph NOUN cana-5226 126 8 energy energy NOUN cana-5226 126 9 after after ADV cana-5226 126 10 all all ADV cana-5226 126 11 ? ? PUNCT cana-5226 126 12 . . PUNCT cana-5226 127 1 discrete discrete ADJ cana-5226 127 2 applied applied ADJ cana-5226 127 3 mathematics mathematic NOUN cana-5226 127 4 , , PUNCT cana-5226 127 5 230 230 NUM cana-5226 127 6 , , PUNCT cana-5226 127 7 71 71 NUM cana-5226 127 8 - - SYM cana-5226 127 9 77 77 NUM cana-5226 127 10 . . PUNCT cana-5226 128 1 [ [ X cana-5226 128 2 5 5 NUM cana-5226 128 3 ] ] X cana-5226 128 4 stin stin PROPN cana-5226 128 5 , , PUNCT cana-5226 128 6 r. r. PROPN cana-5226 128 7 , , PUNCT cana-5226 128 8 aminah aminah PROPN cana-5226 128 9 , , PUNCT cana-5226 128 10 s. s. PROPN cana-5226 128 11 , , PUNCT cana-5226 128 12 utama utama PROPN cana-5226 128 13 , , PUNCT cana-5226 128 14 s. s. PROPN cana-5226 128 15 , , PUNCT cana-5226 128 16 & & CCONJ cana-5226 128 17 silaban silaban NOUN cana-5226 128 18 , , PUNCT cana-5226 128 19 d. d. PROPN cana-5226 128 20 r. r. PROPN cana-5226 128 21 ( ( PUNCT cana-5226 128 22 2020 2020 NUM cana-5226 128 23 , , PUNCT cana-5226 128 24 june june PROPN cana-5226 128 25 ) ) PUNCT cana-5226 128 26 . . PUNCT cana-5226 129 1 characteristic characteristic ADJ cana-5226 129 2 polynomials polynomial NOUN cana-5226 129 3 and and CCONJ cana-5226 129 4 eigenvalues eigenvalue NOUN cana-5226 129 5 of of ADP cana-5226 129 6 the the DET cana-5226 129 7 adjacency adjacency NOUN cana-5226 129 8 matrix matrix NOUN cana-5226 129 9 and and CCONJ cana-5226 129 10 the the DET cana-5226 129 11 laplacian laplacian ADJ cana-5226 129 12 matrix matrix NOUN cana-5226 129 13 of of ADP cana-5226 129 14 cyclic cyclic ADJ cana-5226 129 15 directed direct VERB cana-5226 129 16 prism prism NOUN cana-5226 129 17 graph graph NOUN cana-5226 129 18 . . PUNCT cana-5226 130 1 in in ADP cana-5226 130 2 aip aip PROPN cana-5226 130 3 conference conference NOUN cana-5226 130 4 proceedings proceeding NOUN cana-5226 130 5 ( ( PUNCT cana-5226 130 6 vol vol NOUN cana-5226 130 7 . . NOUN cana-5226 130 8 2242 2242 NUM cana-5226 130 9 , , PUNCT cana-5226 130 10 no no INTJ cana-5226 130 11 . . NOUN cana-5226 130 12 1 1 NUM cana-5226 130 13 ) ) PUNCT cana-5226 130 14 . . PUNCT cana-5226 131 1 aip aip PROPN cana-5226 131 2 publishing publishing PROPN cana-5226 131 3 . . PUNCT cana-5226 132 1 [ [ X cana-5226 132 2 6 6 NUM cana-5226 132 3 ] ] X cana-5226 132 4 rowlinson rowlinson NOUN cana-5226 132 5 , , PUNCT cana-5226 132 6 p. p. NOUN cana-5226 132 7 ( ( PUNCT cana-5226 132 8 2016 2016 NUM cana-5226 132 9 ) ) PUNCT cana-5226 132 10 . . PUNCT cana-5226 133 1 on on ADP cana-5226 133 2 graphs graph NOUN cana-5226 133 3 with with ADP cana-5226 133 4 just just ADV cana-5226 133 5 three three NUM cana-5226 133 6 distinct distinct ADJ cana-5226 133 7 eigenvalues eigenvalue NOUN cana-5226 133 8 . . PUNCT cana-5226 134 1 linear linear ADJ cana-5226 134 2 algebra algebra NOUN cana-5226 134 3 and and CCONJ cana-5226 134 4 its its PRON cana-5226 134 5 applications application NOUN cana-5226 134 6 , , PUNCT cana-5226 134 7 507 507 NUM cana-5226 134 8 , , PUNCT cana-5226 134 9 462 462 NUM cana-5226 134 10 - - SYM cana-5226 134 11 473 473 NUM cana-5226 134 12 . . PUNCT cana-5226 135 1 [ [ X cana-5226 135 2 7 7 NUM cana-5226 135 3 ] ] X cana-5226 135 4 cheng cheng PROPN cana-5226 135 5 , , PUNCT cana-5226 135 6 x. x. PROPN cana-5226 135 7 m. m. PROPN cana-5226 135 8 , , PUNCT cana-5226 135 9 greaves greave NOUN cana-5226 135 10 , , PUNCT cana-5226 135 11 g. g. PROPN cana-5226 135 12 r. r. PROPN cana-5226 135 13 , , PUNCT cana-5226 135 14 & & CCONJ cana-5226 135 15 koolen koolen PROPN cana-5226 135 16 , , PUNCT cana-5226 135 17 j. j. PROPN cana-5226 135 18 h. h. PROPN cana-5226 135 19 ( ( PUNCT cana-5226 135 20 2018 2018 NUM cana-5226 135 21 ) ) PUNCT cana-5226 135 22 . . PUNCT cana-5226 136 1 graphs graph NOUN cana-5226 136 2 with with ADP cana-5226 136 3 three three NUM cana-5226 136 4 eigenvalues eigenvalue NOUN cana-5226 136 5 and and CCONJ cana-5226 136 6 second second ADV cana-5226 136 7 largest large ADJ cana-5226 136 8 eigenvalue eigenvalue NOUN cana-5226 136 9 at at ADP cana-5226 136 10 most most ADJ cana-5226 136 11 1 1 NUM cana-5226 136 12 . . PUNCT cana-5226 136 13 journal journal NOUN cana-5226 136 14 of of ADP cana-5226 136 15 combinatorial combinatorial ADJ cana-5226 136 16 theory theory NOUN cana-5226 136 17 , , PUNCT cana-5226 136 18 series series PROPN cana-5226 136 19 b b PROPN cana-5226 136 20 , , PUNCT cana-5226 136 21 129 129 NUM cana-5226 136 22 , , PUNCT cana-5226 136 23 55 55 NUM cana-5226 136 24 - - SYM cana-5226 136 25 78 78 NUM cana-5226 136 26 . . PUNCT cana-5226 137 1 [ [ X cana-5226 137 2 8 8 NUM cana-5226 137 3 ] ] SYM cana-5226 137 4 qi qi PROPN cana-5226 137 5 , , PUNCT cana-5226 137 6 l. l. PROPN cana-5226 137 7 , , PUNCT cana-5226 137 8 miao miao PROPN cana-5226 137 9 , , PUNCT cana-5226 137 10 l. l. PROPN cana-5226 137 11 , , PUNCT cana-5226 137 12 zhao zhao PROPN cana-5226 137 13 , , PUNCT cana-5226 137 14 w. w. PROPN cana-5226 137 15 , , PUNCT cana-5226 137 16 & & CCONJ cana-5226 137 17 liu liu PROPN cana-5226 137 18 , , PUNCT cana-5226 137 19 l. l. PROPN cana-5226 137 20 ( ( PUNCT cana-5226 137 21 2020 2020 NUM cana-5226 137 22 ) ) PUNCT cana-5226 137 23 . . PUNCT cana-5226 138 1 characterization characterization NOUN cana-5226 138 2 of of ADP cana-5226 138 3 graphs graph NOUN cana-5226 138 4 with with ADP cana-5226 138 5 an an DET cana-5226 138 6 eigenvalue eigenvalue NOUN cana-5226 138 7 of of ADP cana-5226 138 8 large large ADJ cana-5226 138 9 multiplicity multiplicity NOUN cana-5226 138 10 . . PUNCT cana-5226 139 1 advances advance NOUN cana-5226 139 2 in in ADP cana-5226 139 3 mathematical mathematical ADJ cana-5226 139 4 physics physics NOUN cana-5226 139 5 , , PUNCT cana-5226 139 6 2020 2020 NUM cana-5226 139 7 , , PUNCT cana-5226 139 8 1 1 NUM cana-5226 139 9 - - SYM cana-5226 139 10 5 5 NUM cana-5226 139 11 . . PUNCT cana-5226 140 1 [ [ X cana-5226 140 2 9 9 NUM cana-5226 140 3 ] ] X cana-5226 140 4 huang huang PROPN cana-5226 140 5 , , PUNCT cana-5226 140 6 x. x. PROPN cana-5226 140 7 , , PUNCT cana-5226 140 8 & & CCONJ cana-5226 140 9 huang huang PROPN cana-5226 140 10 , , PUNCT cana-5226 140 11 q. q. PROPN cana-5226 140 12 ( ( PUNCT cana-5226 140 13 2017 2017 NUM cana-5226 140 14 ) ) PUNCT cana-5226 140 15 . . PUNCT cana-5226 141 1 on on ADP cana-5226 141 2 regular regular ADJ cana-5226 141 3 graphs graph NOUN cana-5226 141 4 with with ADP cana-5226 141 5 four four NUM cana-5226 141 6 distinct distinct ADJ cana-5226 141 7 eigenvalues eigenvalue NOUN cana-5226 141 8 . . PUNCT cana-5226 142 1 linear linear ADJ cana-5226 142 2 algebra algebra NOUN cana-5226 142 3 and and CCONJ cana-5226 142 4 its its PRON cana-5226 142 5 applications application NOUN cana-5226 142 6 , , PUNCT cana-5226 142 7 512 512 NUM cana-5226 142 8 , , PUNCT cana-5226 142 9 219 219 NUM cana-5226 142 10 - - SYM cana-5226 142 11 233 233 NUM cana-5226 142 12 . . PUNCT cana-5226 143 1 [ [ X cana-5226 143 2 10 10 NUM cana-5226 143 3 ] ] X cana-5226 143 4 li li PROPN cana-5226 143 5 , , PUNCT cana-5226 143 6 x. x. PROPN cana-5226 143 7 , , PUNCT cana-5226 143 8 shi shi PROPN cana-5226 143 9 , , PUNCT cana-5226 143 10 y. y. PROPN cana-5226 143 11 , , PUNCT cana-5226 143 12 & & CCONJ cana-5226 143 13 gutman gutman PROPN cana-5226 143 14 , , PUNCT cana-5226 143 15 i. i. PROPN cana-5226 143 16 ( ( PUNCT cana-5226 143 17 2012 2012 NUM cana-5226 143 18 ) ) PUNCT cana-5226 143 19 . . PUNCT cana-5226 144 1 graph graph NOUN cana-5226 144 2 energy energy NOUN cana-5226 144 3 . . PUNCT cana-5226 145 1 springer springer NOUN cana-5226 145 2 science science PROPN cana-5226 145 3 & & CCONJ cana-5226 145 4 business business NOUN cana-5226 145 5 media medium NOUN cana-5226 145 6 . . PUNCT