id sid tid token lemma pos cana-1013 1 1 communications communication NOUN cana-1013 1 2 on on ADP cana-1013 1 3 applied apply VERB cana-1013 1 4 nonlinear nonlinear ADJ cana-1013 1 5 analysis analysis NOUN cana-1013 1 6 issn issn NOUN cana-1013 1 7 : : PUNCT cana-1013 1 8 1074 1074 NUM cana-1013 1 9 - - PUNCT cana-1013 1 10 133x 133x NUM cana-1013 1 11 vol vol NOUN cana-1013 1 12 31 31 NUM cana-1013 1 13 no no NOUN cana-1013 1 14 . . PUNCT cana-1013 2 1 5s 5s NUM cana-1013 2 2 ( ( PUNCT cana-1013 2 3 2024 2024 NUM cana-1013 2 4 ) ) PUNCT cana-1013 2 5 197 197 NUM cana-1013 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1013 2 7 the the DET cana-1013 2 8 structure structure NOUN cana-1013 2 9 of of ADP cana-1013 2 10 generalized generalized ADJ cana-1013 2 11 cayley cayley ADJ cana-1013 2 12 graph graph NOUN cana-1013 2 13 when when SCONJ cana-1013 2 14 ๐‘ช๐’‚๐’š(๐‘ฎ ๐‘ช๐’‚๐’š(๐‘ฎ NOUN cana-1013 2 15 , , PUNCT cana-1013 2 16 ๐‘บ ๐‘บ NOUN cana-1013 2 17 ) ) PUNCT cana-1013 2 18 = = PUNCT cana-1013 2 19 ๐‘ท๐Ÿ ๐‘ท๐Ÿ PROPN cana-1013 2 20 ร— ร— PROPN cana-1013 2 21 ๐‘ท๐Ÿ ๐‘ท๐Ÿ PROPN cana-1013 2 22 and and CCONJ cana-1013 2 23 ๐‘ท๐Ÿ ๐‘ท๐Ÿ PROPN cana-1013 2 24 ร— ร— PROPN cana-1013 2 25 ๐‘ช๐Ÿ‘ ๐‘ช๐Ÿ‘ PROPN cana-1013 2 26 ayat ayat PROPN cana-1013 2 27 a. a. PROPN cana-1013 2 28 neamah neamah PROPN cana-1013 2 29 department department PROPN cana-1013 2 30 of of ADP cana-1013 2 31 mathematics mathematics PROPN cana-1013 2 32 , , PUNCT cana-1013 2 33 al al PROPN cana-1013 2 34 - - PUNCT cana-1013 2 35 nahrain nahrain PROPN cana-1013 2 36 university university NOUN cana-1013 2 37 , , PUNCT cana-1013 2 38 baฯˆhdad baฯˆhdad PROPN cana-1013 2 39 , , PUNCT cana-1013 2 40 iraq iraq PROPN cana-1013 2 41 ayatneamah@nahrainuniv.edu.iq ayatneamah@nahrainuniv.edu.iq ADJ cana-1013 2 42 article article NOUN cana-1013 2 43 history history NOUN cana-1013 2 44 : : PUNCT cana-1013 2 45 received receive VERB cana-1013 2 46 : : PUNCT cana-1013 2 47 15 15 NUM cana-1013 2 48 - - SYM cana-1013 2 49 05 05 NUM cana-1013 2 50 - - PUNCT cana-1013 2 51 2024 2024 NUM cana-1013 2 52 revised revise VERB cana-1013 2 53 : : PUNCT cana-1013 2 54 24 24 NUM cana-1013 2 55 - - PUNCT cana-1013 2 56 06 06 NUM cana-1013 2 57 - - PUNCT cana-1013 2 58 2024 2024 NUM cana-1013 2 59 accepted accept VERB cana-1013 2 60 : : PUNCT cana-1013 2 61 03 03 NUM cana-1013 2 62 - - PUNCT cana-1013 2 63 07 07 NUM cana-1013 2 64 - - PUNCT cana-1013 2 65 2024 2024 NUM cana-1013 2 66 abstract abstract NOUN cana-1013 2 67 this this DET cana-1013 2 68 work work NOUN cana-1013 2 69 aims aim VERB cana-1013 2 70 to to PART cana-1013 2 71 present present VERB cana-1013 2 72 the the DET cana-1013 2 73 generalized generalized ADJ cana-1013 2 74 cayley cayley ADJ cana-1013 2 75 graph graph NOUN cana-1013 2 76 and and CCONJ cana-1013 2 77 identify identify VERB cana-1013 2 78 its its PRON cana-1013 2 79 structure structure NOUN cana-1013 2 80 in in ADP cana-1013 2 81 a a DET cana-1013 2 82 few few ADJ cana-1013 2 83 specific specific ADJ cana-1013 2 84 scenarios scenario NOUN cana-1013 2 85 . . PUNCT cana-1013 3 1 assume assume VERB cana-1013 3 2 that that SCONJ cana-1013 3 3 ฯˆ ฯˆ NOUN cana-1013 3 4 is be AUX cana-1013 3 5 a a DET cana-1013 3 6 finite finite NOUN cana-1013 3 7 - - NOUN cana-1013 3 8 group group NOUN cana-1013 3 9 and and CCONJ cana-1013 3 10 that that PRON cana-1013 3 11 s s VERB cana-1013 3 12 is be AUX cana-1013 3 13 a a DET cana-1013 3 14 non non ADJ cana-1013 3 15 - - ADJ cana-1013 3 16 empty empty ADJ cana-1013 3 17 subset subset NOUN cana-1013 3 18 of of ADP cana-1013 3 19 ฯˆ ฯˆ PROPN cana-1013 3 20 . . PUNCT cana-1013 3 21 ๐‘’ ๐‘’ PROPN cana-1013 3 22 โˆ‰ โˆ‰ PROPN cana-1013 3 23 ๐‘† ๐‘† PROPN cana-1013 3 24 and and CCONJ cana-1013 3 25 . . PUNCT cana-1013 4 1 as as ADP cana-1013 4 2 a a DET cana-1013 4 3 result result NOUN cana-1013 4 4 , , PUNCT cana-1013 4 5 the the DET cana-1013 4 6 vertices vertex NOUN cana-1013 4 7 of of ADP cana-1013 4 8 the the DET cana-1013 4 9 cayley cayley ADJ cana-1013 4 10 graph graph NOUN cana-1013 4 11 cay cay PROPN cana-1013 4 12 ( ( PUNCT cana-1013 4 13 ฯˆ ฯˆ X cana-1013 4 14 , , PUNCT cana-1013 4 15 s s PART cana-1013 4 16 ) ) PUNCT cana-1013 4 17 are be AUX cana-1013 4 18 all all PRON cana-1013 4 19 members member NOUN cana-1013 4 20 of of ADP cana-1013 4 21 ฯˆ ฯˆ NOUN cana-1013 4 22 , , PUNCT cana-1013 4 23 and and CCONJ cana-1013 4 24 two two NUM cana-1013 4 25 nearby nearby ADJ cana-1013 4 26 vertices vertex NOUN cana-1013 4 27 , , PUNCT cana-1013 4 28 x x PUNCT cana-1013 4 29 and and CCONJ cana-1013 4 30 y y PROPN cana-1013 4 31 , , PUNCT cana-1013 4 32 are be AUX cana-1013 4 33 only only ADV cana-1013 4 34 adjacent adjacent ADJ cana-1013 4 35 if if SCONJ cana-1013 4 36 ๐‘ฅ๐‘ฆโˆ’1 ๐‘ฅ๐‘ฆโˆ’1 PROPN cana-1013 4 37 โˆˆ โˆˆ PROPN cana-1013 4 38 ๐‘†. ๐‘†. VERB cana-1013 4 39 the the DET cana-1013 4 40 given give VERB cana-1013 4 41 generalized generalize VERB cana-1013 4 42 cayley cayley ADJ cana-1013 4 43 graph graph NOUN cana-1013 4 44 is be AUX cana-1013 4 45 defined define VERB cana-1013 4 46 as as ADP cana-1013 4 47 ๐ถ๐‘Ž๐‘ฆ๐‘š ๐ถ๐‘Ž๐‘ฆ๐‘š PROPN cana-1013 4 48 this this PRON cana-1013 4 49 is be AUX cana-1013 4 50 a a DET cana-1013 4 51 graph graph NOUN cana-1013 4 52 whose whose DET cana-1013 4 53 vertex vertex NOUN cana-1013 4 54 set set NOUN cana-1013 4 55 is be AUX cana-1013 4 56 made make VERB cana-1013 4 57 up up ADP cana-1013 4 58 of of ADP cana-1013 4 59 every every DET cana-1013 4 60 column column NOUN cana-1013 4 61 matrix matrix NOUN cana-1013 4 62 ๐‘‹๐‘š ๐‘‹๐‘š PROPN cana-1013 4 63 it it PRON cana-1013 4 64 has have VERB cana-1013 4 65 two two NUM cana-1013 4 66 vertices vertex NOUN cana-1013 4 67 and and CCONJ cana-1013 4 68 all all PRON cana-1013 4 69 of of ADP cana-1013 4 70 its its PRON cana-1013 4 71 components component NOUN cana-1013 4 72 in in ADP cana-1013 4 73 ฯˆ ฯˆ NOUN cana-1013 4 74 . . PUNCT cana-1013 5 1 ๐‘‹๐‘š ๐‘‹๐‘š PROPN cana-1013 5 2 and and CCONJ cana-1013 5 3 ๐‘Œ๐‘š ๐‘Œ๐‘š PROPN cana-1013 5 4 are be AUX cana-1013 5 5 adjacent adjacent ADJ cana-1013 5 6 โ†” โ†” PROPN cana-1013 5 7 , , PUNCT cana-1013 5 8 where where SCONJ cana-1013 5 9 ๐‘Œ๐‘š ๐‘Œ๐‘š PROPN cana-1013 5 10 โˆ’1 โˆ’1 NOUN cana-1013 5 11 is be AUX cana-1013 5 12 a a DET cana-1013 5 13 column column NOUN cana-1013 5 14 matrix matrix NOUN cana-1013 5 15 in in ADP cana-1013 5 16 which which PRON cana-1013 5 17 โˆ€ โˆ€ NOUN cana-1013 5 18 entry entry NOUN cana-1013 5 19 correlates correlate VERB cana-1013 5 20 to to ADP cana-1013 5 21 an an DET cana-1013 5 22 associated associated ADJ cana-1013 5 23 element element NOUN cana-1013 5 24 's 's PART cana-1013 5 25 inverse inverse NOUN cana-1013 5 26 . . PUNCT cana-1013 6 1 and and CCONJ cana-1013 6 2 ๐‘€(๐‘† ๐‘€(๐‘† NOUN cana-1013 6 3 ) ) PUNCT cana-1013 6 4 is be AUX cana-1013 6 5 a a DET cana-1013 6 6 mร—m mร—m ADJ cana-1013 6 7 matrix matrix NOUN cana-1013 6 8 where where SCONJ cana-1013 6 9 every every DET cana-1013 6 10 entry entry NOUN cana-1013 6 11 is be AUX cana-1013 6 12 in in ADP cana-1013 6 13 s s PRON cana-1013 6 14 , , PUNCT cana-1013 6 15 is be AUX cana-1013 6 16 the the DET cana-1013 6 17 opposite opposite NOUN cana-1013 6 18 of of ADP cana-1013 6 19 and and CCONJ cana-1013 6 20 . . PUNCT cana-1013 7 1 in in ADP cana-1013 7 2 this this DET cana-1013 7 3 study study NOUN cana-1013 7 4 , , PUNCT cana-1013 7 5 we we PRON cana-1013 7 6 assign assign VERB cana-1013 7 7 the the DET cana-1013 7 8 structure structure NOUN cana-1013 7 9 of of ADP cana-1013 7 10 the the DET cana-1013 7 11 new new ADJ cana-1013 7 12 graph graph NOUN cana-1013 7 13 and and CCONJ cana-1013 7 14 highlight highlight VERB cana-1013 7 15 some some PRON cana-1013 7 16 of of ADP cana-1013 7 17 its its PRON cana-1013 7 18 fundamental fundamental ADJ cana-1013 7 19 aspects aspect NOUN cana-1013 7 20 ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ NOUN cana-1013 7 21 , , PUNCT cana-1013 7 22 ๐‘† ๐‘† PROPN cana-1013 7 23 ) ) PUNCT cana-1013 7 24 when when SCONJ cana-1013 7 25 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 7 26 , , PUNCT cana-1013 7 27 ๐‘† ๐‘† PROPN cana-1013 7 28 ) ) PUNCT cana-1013 7 29 is be AUX cana-1013 7 30 the the DET cana-1013 7 31 ๐‘ƒ2 ๐‘ƒ2 ADJ cana-1013 7 32 ร— ร— NOUN cana-1013 7 33 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 7 34 and and CCONJ cana-1013 7 35 ๐‘ƒ2 ๐‘ƒ2 DET cana-1013 7 36 ร— ร— PROPN cana-1013 7 37 ๐ถ3 ๐ถ3 NOUN cana-1013 7 38 . . PUNCT cana-1013 8 1 keywords keyword NOUN cana-1013 8 2 : : PUNCT cana-1013 8 3 cayley cayley ADJ cana-1013 8 4 graph graph NOUN cana-1013 8 5 , , PUNCT cana-1013 8 6 algebraic algebraic ADJ cana-1013 8 7 graph graph NOUN cana-1013 8 8 theory theory NOUN cana-1013 8 9 etc etc X cana-1013 8 10 . . PUNCT cana-1013 9 1 introduction introduction NOUN cana-1013 9 2 . . PUNCT cana-1013 10 1 algebraic algebraic ADJ cana-1013 10 2 graph graph NOUN cana-1013 10 3 theory theory NOUN cana-1013 10 4 has have AUX cana-1013 10 5 emerged emerge VERB cana-1013 10 6 as as ADP cana-1013 10 7 a a DET cana-1013 10 8 prominent prominent ADJ cana-1013 10 9 mathematical mathematical ADJ cana-1013 10 10 topic topic NOUN cana-1013 10 11 of of ADP cana-1013 10 12 interest interest NOUN cana-1013 10 13 to to ADP cana-1013 10 14 specialists specialist NOUN cana-1013 10 15 in in ADP cana-1013 10 16 the the DET cana-1013 10 17 domains domain NOUN cana-1013 10 18 of of ADP cana-1013 10 19 algebra algebra NOUN cana-1013 10 20 and and CCONJ cana-1013 10 21 graph graph NOUN cana-1013 10 22 theory theory NOUN cana-1013 10 23 in in ADP cana-1013 10 24 recent recent ADJ cana-1013 10 25 years year NOUN cana-1013 10 26 . . PUNCT cana-1013 11 1 algebraic algebraic ADJ cana-1013 11 2 graph graph NOUN cana-1013 11 3 theory theory NOUN cana-1013 11 4 states state VERB cana-1013 11 5 that that SCONJ cana-1013 11 6 every every DET cana-1013 11 7 graph graph NOUN cana-1013 11 8 may may AUX cana-1013 11 9 be be AUX cana-1013 11 10 associated associate VERB cana-1013 11 11 with with ADP cana-1013 11 12 a a DET cana-1013 11 13 group group NOUN cana-1013 11 14 , , PUNCT cana-1013 11 15 ring ring NOUN cana-1013 11 16 , , PUNCT cana-1013 11 17 module module NOUN cana-1013 11 18 , , PUNCT cana-1013 11 19 or or CCONJ cana-1013 11 20 any any DET cana-1013 11 21 other other ADJ cana-1013 11 22 algebraic algebraic ADJ cana-1013 11 23 structure structure NOUN cana-1013 11 24 . . PUNCT cana-1013 12 1 an an DET cana-1013 12 2 algebraic algebraic ADJ cana-1013 12 3 graph graph NOUN cana-1013 12 4 that that PRON cana-1013 12 5 is be AUX cana-1013 12 6 particularly particularly ADV cana-1013 12 7 interesting interesting ADJ cana-1013 12 8 is be AUX cana-1013 12 9 the the DET cana-1013 12 10 cayley cayley ADJ cana-1013 12 11 graph graph NOUN cana-1013 12 12 for for ADP cana-1013 12 13 a a DET cana-1013 12 14 group group NOUN cana-1013 12 15 and and CCONJ cana-1013 12 16 related relate VERB cana-1013 12 17 subset subset NOUN cana-1013 12 18 . . PUNCT cana-1013 13 1 in in ADP cana-1013 13 2 1878 1878 NUM cana-1013 13 3 , , PUNCT cana-1013 13 4 arthur arthur PROPN cana-1013 13 5 cayley cayley PROPN cana-1013 13 6 created create VERB cana-1013 13 7 the the DET cana-1013 13 8 cayley cayley ADJ cana-1013 13 9 graph graph NOUN cana-1013 13 10 to to PART cana-1013 13 11 provide provide VERB cana-1013 13 12 clarification clarification NOUN cana-1013 13 13 on on ADP cana-1013 13 14 the the DET cana-1013 13 15 concept concept NOUN cana-1013 13 16 of of ADP cana-1013 13 17 abstract abstract ADJ cana-1013 13 18 groups group NOUN cana-1013 13 19 , , PUNCT cana-1013 13 20 which which PRON cana-1013 13 21 at at ADP cana-1013 13 22 the the DET cana-1013 13 23 time time NOUN cana-1013 13 24 were be AUX cana-1013 13 25 created create VERB cana-1013 13 26 by by ADP cana-1013 13 27 a a DET cana-1013 13 28 group group NOUN cana-1013 13 29 of of ADP cana-1013 13 30 generators generator NOUN cana-1013 13 31 . . PUNCT cana-1013 14 1 a a DET cana-1013 14 2 graph graph NOUN cana-1013 14 3 with with ADP cana-1013 14 4 a a DET cana-1013 14 5 group group NOUN cana-1013 14 6 encoded encode VERB cana-1013 14 7 is be AUX cana-1013 14 8 called call VERB cana-1013 14 9 a a DET cana-1013 14 10 cayley cayley ADJ cana-1013 14 11 graph graph NOUN cana-1013 14 12 . . PUNCT cana-1013 15 1 assuming assume VERB cana-1013 15 2 ฯˆ ฯˆ NOUN cana-1013 15 3 is be AUX cana-1013 15 4 a a DET cana-1013 15 5 group group NOUN cana-1013 15 6 and and CCONJ cana-1013 15 7 s s NOUN cana-1013 15 8 is be AUX cana-1013 15 9 its its PRON cana-1013 15 10 inverse inverse NOUN cana-1013 15 11 closed close VERB cana-1013 15 12 subset subset NOUN cana-1013 15 13 , , PUNCT cana-1013 15 14 we we PRON cana-1013 15 15 may may AUX cana-1013 15 16 conclude conclude VERB cana-1013 15 17 that that PRON cana-1013 15 18 eโˆ‰s eโˆ‰s PROPN cana-1013 15 19 . . PUNCT cana-1013 16 1 as as ADP cana-1013 16 2 a a DET cana-1013 16 3 result result NOUN cana-1013 16 4 , , PUNCT cana-1013 16 5 the the DET cana-1013 16 6 cayley cayley NOUN cana-1013 16 7 graph graph NOUN cana-1013 16 8 cay(ฯˆ cay(ฯˆ PROPN cana-1013 16 9 , , PUNCT cana-1013 16 10 s s PART cana-1013 16 11 ) ) PUNCT cana-1013 16 12 is be AUX cana-1013 16 13 an an DET cana-1013 16 14 undirected undirected ADJ cana-1013 16 15 simpl simpl NOUN cana-1013 16 16 - - NOUN cana-1013 16 17 graph graph NOUN cana-1013 16 18 whose whose DET cana-1013 16 19 vertex vertex NOUN cana-1013 16 20 set set NOUN cana-1013 16 21 is be AUX cana-1013 16 22 made make VERB cana-1013 16 23 up up ADP cana-1013 16 24 of of ADP cana-1013 16 25 all all PRON cana-1013 16 26 of of ADP cana-1013 16 27 ฯˆ ฯˆ NOUN cana-1013 16 28 's 's PART cana-1013 16 29 members member NOUN cana-1013 16 30 , , PUNCT cana-1013 16 31 and and CCONJ cana-1013 16 32 x x X cana-1013 16 33 is be AUX cana-1013 16 34 next next ADJ cana-1013 16 35 to to ADP cana-1013 16 36 y y NOUN cana-1013 16 37 only only ADV cana-1013 16 38 if if SCONJ cana-1013 16 39 ๐‘ฅ๐‘ฆโˆ’1 ๐‘ฅ๐‘ฆโˆ’1 PROPN cana-1013 16 40 โˆˆ โˆˆ PROPN cana-1013 16 41 ๐‘†. ๐‘†. NOUN cana-1013 16 42 we we PRON cana-1013 16 43 note note VERB cana-1013 16 44 that that SCONJ cana-1013 16 45 ๐ถ๐‘Ž๐‘ฆ ๐ถ๐‘Ž๐‘ฆ PROPN cana-1013 16 46 is be AUX cana-1013 16 47 a a DET cana-1013 16 48 simple simple ADJ cana-1013 16 49 ๐‘Ÿ ๐‘Ÿ NOUN cana-1013 16 50 โˆ’regular โˆ’regular ADJ cana-1013 16 51 graph graph NOUN cana-1013 16 52 and and CCONJ cana-1013 16 53 it it PRON cana-1013 16 54 depends depend VERB cana-1013 16 55 on on ADP cana-1013 16 56 to to PART cana-1013 16 57 set set VERB cana-1013 16 58 ๐‘† ๐‘† PROPN cana-1013 16 59 of of ADP cana-1013 16 60 the the DET cana-1013 16 61 group group NOUN cana-1013 16 62 . . PUNCT cana-1013 17 1 also also ADV cana-1013 17 2 , , PUNCT cana-1013 17 3 ๐ถ๐‘Ž๐‘ฆ ๐ถ๐‘Ž๐‘ฆ PROPN cana-1013 17 4 is be AUX cana-1013 17 5 connectedโ†” connectedโ†” ADJ cana-1013 17 6 ๐‘† ๐‘† PROPN cana-1013 17 7 is be AUX cana-1013 17 8 a a DET cana-1013 17 9 generating generate VERB cana-1013 17 10 set set NOUN cana-1013 17 11 of of ADP cana-1013 17 12 ๐บ. ๐บ. NOUN cana-1013 17 13 a a DET cana-1013 17 14 new new ADJ cana-1013 17 15 definition definition NOUN cana-1013 17 16 of of ADP cana-1013 17 17 the the DET cana-1013 17 18 generalized generalized ADJ cana-1013 17 19 cayley cayley ADJ cana-1013 17 20 graph graph NOUN cana-1013 17 21 , , PUNCT cana-1013 17 22 called call VERB cana-1013 17 23 caym caym NOUN cana-1013 17 24 ( ( PUNCT cana-1013 17 25 ฯˆ ฯˆ X cana-1013 17 26 , , PUNCT cana-1013 17 27 s s PART cana-1013 17 28 ) ) PUNCT cana-1013 17 29 , , PUNCT cana-1013 17 30 was be AUX cana-1013 17 31 recently recently ADV cana-1013 17 32 provided provide VERB cana-1013 17 33 by by ADP cana-1013 17 34 erfanian erfanian NOUN cana-1013 17 35 in in ADP cana-1013 17 36 [ [ X cana-1013 17 37 4 4 NUM cana-1013 17 38 ] ] PUNCT cana-1013 17 39 . . PUNCT cana-1013 18 1 this this DET cana-1013 18 2 new new ADJ cana-1013 18 3 definition definition NOUN cana-1013 18 4 uses use VERB cana-1013 18 5 column column NOUN cana-1013 18 6 mร—1 mร—1 NOUN cana-1013 18 7 matrices matrice VERB cana-1013 18 8 and and CCONJ cana-1013 18 9 is be AUX cana-1013 18 10 a a DET cana-1013 18 11 novel novel ADJ cana-1013 18 12 extension extension NOUN cana-1013 18 13 of of ADP cana-1013 18 14 the the DET cana-1013 18 15 standard standard ADJ cana-1013 18 16 cay cay PROPN cana-1013 18 17 ( ( PUNCT cana-1013 18 18 ) ) PUNCT cana-1013 18 19 . . PUNCT cana-1013 19 1 the the DET cana-1013 19 2 generalized generalized ADJ cana-1013 19 3 cayley cayley ADJ cana-1013 19 4 graph graph NOUN cana-1013 19 5 , , PUNCT cana-1013 19 6 represented represent VERB cana-1013 19 7 as as ADP cana-1013 19 8 cay_m cay_m NOUN cana-1013 19 9 , , PUNCT cana-1013 19 10 is be AUX cana-1013 19 11 an an DET cana-1013 19 12 undirected undirected ADJ cana-1013 19 13 simple simple ADJ cana-1013 19 14 graph graph NOUN cana-1013 19 15 with with ADP cana-1013 19 16 two two NUM cana-1013 19 17 vertices vertex NOUN cana-1013 19 18 and and CCONJ cana-1013 19 19 a a DET cana-1013 19 20 vertex vertex NOUN cana-1013 19 21 set set NOUN cana-1013 19 22 made make VERB cana-1013 19 23 up up ADP cana-1013 19 24 of of ADP cana-1013 19 25 all all DET cana-1013 19 26 mร—1 mร—1 NOUN cana-1013 19 27 matrices matrix NOUN cana-1013 19 28 , , PUNCT cana-1013 19 29 where where SCONJ cana-1013 19 30 x_iโˆˆg,1โ‰คiโ‰คm x_iโˆˆg,1โ‰คiโ‰คm PROPN cana-1013 19 31 , , PUNCT cana-1013 19 32 for for ADP cana-1013 19 33 each each DET cana-1013 19 34 positive positive ADJ cana-1013 19 35 integer integer NOUN cana-1013 19 36 mโ‰ฅ1 mโ‰ฅ1 PROPN cana-1013 19 37 ๐‘‹ ๐‘‹ PROPN cana-1013 19 38 = = SYM cana-1013 20 1 [ [ X cana-1013 20 2 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 20 3 , , PUNCT cana-1013 20 4 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 20 5 , , PUNCT cana-1013 20 6 โ€ฆ โ€ฆ PUNCT cana-1013 20 7 , , PUNCT cana-1013 20 8 ๐‘ฅ๐‘š]๐‘ก ๐‘ฅ๐‘š]๐‘ก NOUN cana-1013 20 9 and and CCONJ cana-1013 20 10 ๐‘Œ ๐‘Œ PROPN cana-1013 21 1 = = PUNCT cana-1013 22 1 [ [ X cana-1013 22 2 ๐‘ฆ1 ๐‘ฆ1 PROPN cana-1013 22 3 , , PUNCT cana-1013 22 4 ๐‘ฆ2 ๐‘ฆ2 NOUN cana-1013 22 5 , , PUNCT cana-1013 22 6 โ€ฆ โ€ฆ PUNCT cana-1013 22 7 , , PUNCT cana-1013 22 8 ๐‘ฆ๐‘š]๐‘ก ๐‘ฆ๐‘š]๐‘ก NOUN cana-1013 22 9 are be AUX cana-1013 22 10 contiguous contiguous ADJ cana-1013 22 11 only only ADV cana-1013 22 12 in in ADP cana-1013 22 13 the the DET cana-1013 22 14 event event NOUN cana-1013 22 15 that that SCONJ cana-1013 22 16 x(y x(y PROPN cana-1013 22 17 ) ) PUNCT cana-1013 22 18 communications communication NOUN cana-1013 22 19 on on ADP cana-1013 22 20 applied apply VERB cana-1013 22 21 nonlinear nonlinear ADJ cana-1013 22 22 analysis analysis NOUN cana-1013 22 23 issn issn NOUN cana-1013 22 24 : : PUNCT cana-1013 22 25 1074 1074 NUM cana-1013 22 26 - - PUNCT cana-1013 22 27 133x 133x NUM cana-1013 22 28 vol vol NOUN cana-1013 22 29 31 31 NUM cana-1013 22 30 no no NOUN cana-1013 22 31 . . PUNCT cana-1013 23 1 5s 5s NUM cana-1013 23 2 ( ( PUNCT cana-1013 23 3 2024 2024 NUM cana-1013 23 4 ) ) PUNCT cana-1013 23 5 198 198 NUM cana-1013 23 6 https://internationalpubls.com https://internationalpubls.com X cana-1013 23 7 ) ) PUNCT cana-1013 23 8 tโˆˆm(s tโˆˆm(s PROPN cana-1013 23 9 ) ) PUNCT cana-1013 23 10 . . PUNCT cana-1013 24 1 since since SCONJ cana-1013 24 2 it it PRON cana-1013 24 3 is be AUX cana-1013 24 4 obvious obvious ADJ cana-1013 24 5 that that SCONJ cana-1013 24 6 the the DET cana-1013 24 7 standard standard ADJ cana-1013 24 8 cayley cayley NOUN cana-1013 24 9 graph graph NOUN cana-1013 24 10 cay(ฯˆ cay(ฯˆ PROPN cana-1013 24 11 , , PUNCT cana-1013 24 12 s s PART cana-1013 24 13 ) ) PUNCT cana-1013 24 14 exists exist VERB cana-1013 24 15 if if SCONJ cana-1013 24 16 m=1 m=1 PROPN cana-1013 24 17 , , PUNCT cana-1013 24 18 we we PRON cana-1013 24 19 refer refer VERB cana-1013 24 20 to to ADP cana-1013 24 21 this this PRON cana-1013 24 22 as as ADP cana-1013 24 23 the the DET cana-1013 24 24 generalized generalized ADJ cana-1013 24 25 cayley cayley ADJ cana-1013 24 26 graph graph NOUN cana-1013 24 27 . . PUNCT cana-1013 25 1 in in ADP cana-1013 25 2 this this DET cana-1013 25 3 work work NOUN cana-1013 25 4 , , PUNCT cana-1013 25 5 we we PRON cana-1013 25 6 consistently consistently ADV cana-1013 25 7 assume assume VERB cana-1013 25 8 that that SCONJ cana-1013 25 9 s^(-1)โŠ†s s^(-1)โŠ†s PROPN cana-1013 25 10 , , PUNCT cana-1013 25 11 eโˆ‰s eโˆ‰s PROPN cana-1013 25 12 , , PUNCT cana-1013 25 13 and and CCONJ cana-1013 25 14 s s VERB cana-1013 25 15 is be AUX cana-1013 25 16 a a DET cana-1013 25 17 entertain entertain NOUN cana-1013 25 18 set set VERB cana-1013 25 19 of of ADP cana-1013 25 20 g. g. PROPN cana-1013 25 21 cay(g cay(g PROPN cana-1013 25 22 , , PUNCT cana-1013 25 23 s s PART cana-1013 25 24 ) ) PUNCT cana-1013 25 25 is be AUX cana-1013 25 26 therefore therefore ADV cana-1013 25 27 a a DET cana-1013 25 28 connected connected ADJ cana-1013 25 29 graph graph NOUN cana-1013 25 30 in in ADP cana-1013 25 31 this this DET cana-1013 25 32 case case NOUN cana-1013 25 33 . . PUNCT cana-1013 26 1 in in ADP cana-1013 26 2 this this DET cana-1013 26 3 paper paper NOUN cana-1013 26 4 , , PUNCT cana-1013 26 5 we we PRON cana-1013 26 6 focus focus VERB cana-1013 26 7 on on ADP cana-1013 26 8 the the DET cana-1013 26 9 cartesian cartesian ADJ cana-1013 26 10 product product NOUN cana-1013 26 11 of of ADP cana-1013 26 12 two two NUM cana-1013 26 13 graphs graph NOUN cana-1013 26 14 in in ADP cana-1013 26 15 order order NOUN cana-1013 26 16 to to PART cana-1013 26 17 determine determine VERB cana-1013 26 18 the the DET cana-1013 26 19 generalized generalized ADJ cana-1013 26 20 cayley cayley ADJ cana-1013 26 21 graph graph NOUN cana-1013 26 22 . . PUNCT cana-1013 27 1 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 27 2 , , PUNCT cana-1013 27 3 ๐‘† ๐‘† PROPN cana-1013 27 4 ) ) PUNCT cana-1013 27 5 = = SYM cana-1013 28 1 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 28 2 ร— ร— NOUN cana-1013 28 3 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 28 4 and and CCONJ cana-1013 28 5 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 28 6 , , PUNCT cana-1013 28 7 ๐‘† ๐‘† PROPN cana-1013 28 8 ) ) PUNCT cana-1013 28 9 = = SYM cana-1013 29 1 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 29 2 ร— ร— PROPN cana-1013 29 3 ๐ถ3 ๐ถ3 PROPN cana-1013 29 4 . . PUNCT cana-1013 30 1 binary binary ADJ cana-1013 30 2 operations operation NOUN cana-1013 30 3 create create VERB cana-1013 30 4 a a DET cana-1013 30 5 new new ADJ cana-1013 30 6 graph graph NOUN cana-1013 30 7 from from ADP cana-1013 30 8 two two NUM cana-1013 30 9 initial initial ADJ cana-1013 30 10 graphs graph NOUN cana-1013 30 11 ๐บ ๐บ PROPN cana-1013 30 12 , , PUNCT cana-1013 30 13 ๐ป ๐ป PROPN cana-1013 30 14 , , PUNCT cana-1013 30 15 such such ADJ cana-1013 30 16 as as ADP cana-1013 30 17 graph graph NOUN cana-1013 30 18 union union NOUN cana-1013 30 19 , , PUNCT cana-1013 30 20 cartesian cartesian ADJ cana-1013 30 21 graph graph NOUN cana-1013 30 22 product product NOUN cana-1013 30 23 , , PUNCT cana-1013 30 24 corona corona NOUN cana-1013 30 25 graph graph NOUN cana-1013 30 26 product product NOUN cana-1013 30 27 , , PUNCT cana-1013 30 28 and and CCONJ cana-1013 30 29 generalized generalized ADJ cana-1013 30 30 corona corona NOUN cana-1013 30 31 product product NOUN cana-1013 30 32 . . PUNCT cana-1013 31 1 here here ADV cana-1013 31 2 we we PRON cana-1013 31 3 define define VERB cana-1013 31 4 these these DET cana-1013 31 5 graph graph NOUN cana-1013 31 6 operations operation NOUN cana-1013 31 7 . . PUNCT cana-1013 32 1 definition definition NOUN cana-1013 32 2 1 1 NUM cana-1013 32 3 . . PUNCT cana-1013 33 1 assuming assume VERB cana-1013 33 2 ฯˆ ฯˆ X cana-1013 33 3 and and CCONJ cana-1013 33 4 h h NOUN cana-1013 33 5 represent represent VERB cana-1013 33 6 two two NUM cana-1013 33 7 graphs graph NOUN cana-1013 33 8 . . PUNCT cana-1013 34 1 following follow VERB cana-1013 34 2 that that PRON cana-1013 34 3 , , PUNCT cana-1013 34 4 the the DET cana-1013 34 5 graph graph NOUN cana-1013 34 6 represented represent VERB cana-1013 34 7 by by ADP cana-1013 34 8 ฯˆโˆชh ฯˆโˆชh PROPN cana-1013 34 9 , , PUNCT cana-1013 34 10 which which PRON cana-1013 34 11 is be AUX cana-1013 34 12 the the DET cana-1013 34 13 union union NOUN cana-1013 34 14 of of ADP cana-1013 34 15 ฯˆ ฯˆ PROPN cana-1013 34 16 and and CCONJ cana-1013 34 17 h h NOUN cana-1013 34 18 ๐‘‰(๐บ ๐‘‰(๐บ NOUN cana-1013 34 19 โˆช โˆช ADP cana-1013 34 20 ๐ป ๐ป NOUN cana-1013 34 21 ) ) PUNCT cana-1013 34 22 = = SYM cana-1013 34 23 ๐‘‰(๐บ ๐‘‰(๐บ NOUN cana-1013 34 24 ) ) PUNCT cana-1013 34 25 โˆช โˆช NOUN cana-1013 34 26 ๐‘‰(๐ป ๐‘‰(๐ป NOUN cana-1013 34 27 ) ) PUNCT cana-1013 34 28 and and CCONJ cana-1013 34 29 ๐ธ(๐บ ๐ธ(๐บ PROPN cana-1013 34 30 โˆช โˆช PROPN cana-1013 34 31 ๐ป ๐ป PROPN cana-1013 34 32 ) ) PUNCT cana-1013 34 33 = = SYM cana-1013 34 34 ๐ธ(๐บ ๐ธ(๐บ X cana-1013 34 35 ) ) PUNCT cana-1013 34 36 โˆช โˆช PROPN cana-1013 34 37 ๐ธ(๐ป ๐ธ(๐ป PROPN cana-1013 34 38 ) ) PUNCT cana-1013 34 39 . . PUNCT cana-1013 35 1 definition definition NOUN cana-1013 35 2 2 2 NUM cana-1013 35 3 . . PUNCT cana-1013 36 1 the the DET cana-1013 36 2 graph graph NOUN cana-1013 36 3 denoted denote VERB cana-1013 36 4 by by ADP cana-1013 36 5 ฯˆร—h ฯˆร—h PROPN cana-1013 36 6 is be AUX cana-1013 36 7 the the DET cana-1013 36 8 cartesian cartesian ADJ cana-1013 36 9 product product NOUN cana-1013 36 10 of of ADP cana-1013 36 11 ฯˆ ฯˆ PROPN cana-1013 36 12 and and CCONJ cana-1013 36 13 h h NOUN cana-1013 36 14 , , PUNCT cana-1013 36 15 with with ADP cana-1013 36 16 v(g)ร—v(h v(g)ร—v(h PROPN cana-1013 36 17 ) ) PUNCT cana-1013 36 18 as as ADP cana-1013 36 19 its its PRON cana-1013 36 20 vertex vertex NOUN cana-1013 36 21 set set NOUN cana-1013 36 22 . . PUNCT cana-1013 37 1 there there PRON cana-1013 37 2 are be VERB cana-1013 37 3 two two NUM cana-1013 37 4 vertices vertex NOUN cana-1013 37 5 ( ( PUNCT cana-1013 37 6 ๐‘” ๐‘” NOUN cana-1013 37 7 , , PUNCT cana-1013 37 8 โ„Ž โ„Ž PROPN cana-1013 37 9 ) ) PUNCT cana-1013 37 10 , , PUNCT cana-1013 37 11 ( ( PUNCT cana-1013 37 12 ๐‘”โ€ฒ ๐‘”โ€ฒ X cana-1013 37 13 , , PUNCT cana-1013 37 14 โ„Žโ€ฒ โ„Žโ€ฒ NOUN cana-1013 37 15 ) ) PUNCT cana-1013 37 16 are be AUX cana-1013 37 17 next next ADJ cana-1013 37 18 to to ADP cana-1013 37 19 each each DET cana-1013 37 20 other other ADJ cana-1013 37 21 if if SCONJ cana-1013 37 22 ( ( PUNCT cana-1013 37 23 gg'โˆˆ gg'โˆˆ NOUN cana-1013 37 24 g g ADP cana-1013 37 25 โ”ง โ”ง PROPN cana-1013 37 26 and and CCONJ cana-1013 37 27 โ”œ โ”œ NOUN cana-1013 37 28 h h NOUN cana-1013 37 29 = = NOUN cana-1013 37 30 h^'โˆˆe(h h^'โˆˆe(h NOUN cana-1013 37 31 ) ) PUNCT cana-1013 37 32 ) ) PUNCT cana-1013 37 33 or or CCONJ cana-1013 37 34 ( ( PUNCT cana-1013 37 35 g g NOUN cana-1013 37 36 = = SYM cana-1013 37 37 g g NOUN cana-1013 37 38 ' ' PUNCT cana-1013 37 39 โ”ง โ”ง PROPN cana-1013 37 40 and and CCONJ cana-1013 37 41 โ”œ โ”œ PROPN cana-1013 37 42 hh'โˆˆe(h hh'โˆˆe(h PROPN cana-1013 37 43 ) ) PUNCT cana-1013 37 44 ) ) PUNCT cana-1013 37 45 . . PUNCT cana-1013 38 1 therefore therefore ADV cana-1013 38 2 , , PUNCT cana-1013 38 3 e(gร—h)={(g e(gร—h)={(g PROPN cana-1013 38 4 , , PUNCT cana-1013 38 5 h)(g',h h)(g',h NUM cana-1013 38 6 ' ' PUNCT cana-1013 38 7 ) ) PUNCT cana-1013 39 1 โ”‚ โ”‚ VERB cana-1013 39 2 g g NOUN cana-1013 39 3 = = SYM cana-1013 39 4 g',hh'โˆˆe(h g',hh'โˆˆe(h NOUN cana-1013 39 5 ) ) PUNCT cana-1013 39 6 or or CCONJ cana-1013 39 7 gg'โˆˆ gg'โˆˆ PROPN cana-1013 39 8 e(g e(g PROPN cana-1013 39 9 ) ) PUNCT cana-1013 39 10 , , PUNCT cana-1013 40 1 โ”œ โ”œ PROPN cana-1013 40 2 h h NOUN cana-1013 40 3 = = NOUN cana-1013 40 4 h h NOUN cana-1013 40 5 ' ' PUNCT cana-1013 40 6 ) ) PUNCT cana-1013 40 7 } } PUNCT cana-1013 40 8 and and CCONJ cana-1013 40 9 v(gร—h)={(g v(gร—h)={(g PROPN cana-1013 40 10 , , PUNCT cana-1013 40 11 h) h) PROPN cana-1013 40 12 โ”œ โ”œ PROPN cana-1013 40 13 |gโˆˆ |gโˆˆ PROPN cana-1013 40 14 โ”ค โ”ค PROPN cana-1013 40 15 v(g),hโˆˆv v(g),hโˆˆv PROPN cana-1013 40 16 ( ( PUNCT cana-1013 40 17 h h NOUN cana-1013 40 18 ) ) PUNCT cana-1013 40 19 } } PUNCT cana-1013 40 20 . . PUNCT cana-1013 41 1 factors factor NOUN cana-1013 41 2 of of ADP cana-1013 41 3 gร—h gร—h PROPN cana-1013 41 4 are be AUX cana-1013 41 5 represented represent VERB cana-1013 41 6 by by ADP cana-1013 41 7 the the DET cana-1013 41 8 graph graph NOUN cana-1013 41 9 g g PROPN cana-1013 41 10 , , PUNCT cana-1013 41 11 h. h. PROPN cana-1013 41 12 definition definition NOUN cana-1013 41 13 3 3 X cana-1013 41 14 . . PUNCT cana-1013 41 15 assuming assume VERB cana-1013 41 16 ๐œ“ ๐œ“ NOUN cana-1013 41 17 and and CCONJ cana-1013 41 18 h h NOUN cana-1013 41 19 are be AUX cana-1013 41 20 graphs graph NOUN cana-1013 41 21 , , PUNCT cana-1013 41 22 one one PRON cana-1013 41 23 may may AUX cana-1013 41 24 derive derive VERB cana-1013 41 25 the the DET cana-1013 41 26 corona corona NOUN cana-1013 41 27 product product NOUN cana-1013 41 28 of of ADP cana-1013 41 29 ฯˆ ฯˆ PROPN cana-1013 41 30 and and CCONJ cana-1013 41 31 h h NOUN cana-1013 41 32 , , PUNCT cana-1013 41 33 represented represent VERB cana-1013 41 34 as as ADP cana-1013 41 35 ฯˆโˆ˜h ฯˆโˆ˜h NOUN cana-1013 41 36 , , PUNCT cana-1013 41 37 by by ADP cana-1013 41 38 linking link VERB cana-1013 41 39 each each DET cana-1013 41 40 vertex vertex NOUN cana-1013 41 41 of of ADP cana-1013 41 42 the the DET cana-1013 41 43 i i PROPN cana-1013 41 44 - - PUNCT cana-1013 41 45 th th PROPN cana-1013 41 46 copy copy NOUN cana-1013 41 47 of of ADP cana-1013 41 48 h h NOUN cana-1013 41 49 to to ADP cana-1013 41 50 the the DET cana-1013 41 51 i i PROPN cana-1013 41 52 - - PUNCT cana-1013 41 53 th th X cana-1013 41 54 vertex vertex NOUN cana-1013 41 55 of of ADP cana-1013 41 56 g g NOUN cana-1013 41 57 , , PUNCT cana-1013 41 58 where where SCONJ cana-1013 41 59 , , PUNCT cana-1013 41 60 using use VERB cana-1013 41 61 one one NUM cana-1013 41 62 copy copy NOUN cana-1013 41 63 of of ADP cana-1013 41 64 ฯˆ ฯˆ PROPN cana-1013 41 65 and and CCONJ cana-1013 41 66 |v(g)| |v(g)| NOUN cana-1013 41 67 the the DET cana-1013 41 68 functioning functioning NOUN cana-1013 41 69 of of ADP cana-1013 41 70 copies copy NOUN cana-1013 41 71 of of ADP cana-1013 41 72 h. h. PROPN cana-1013 41 73 corona corona PROPN cana-1013 41 74 product product NOUN cana-1013 41 75 is be AUX cana-1013 41 76 noncommutative noncommutative ADJ cana-1013 41 77 . . PUNCT cana-1013 41 78 . . PUNCT cana-1013 42 1 lemma lemma PROPN cana-1013 42 2 4 4 X cana-1013 42 3 . . PUNCT cana-1013 42 4 let let VERB cana-1013 42 5 ๐‘‹ ๐‘‹ PROPN cana-1013 42 6 = = PUNCT cana-1013 43 1 [ [ X cana-1013 43 2 ๐œ›1 ๐œ›1 NOUN cana-1013 43 3 , , PUNCT cana-1013 43 4 ๐œ›2 ๐œ›2 NOUN cana-1013 43 5 , , PUNCT cana-1013 43 6 โ€ฆ โ€ฆ PUNCT cana-1013 43 7 , , PUNCT cana-1013 43 8 ๐œ›๐‘š]๐‘ก ๐œ›๐‘š]๐‘ก PROPN cana-1013 43 9 and and CCONJ cana-1013 43 10 ๐‘Œ ๐‘Œ PROPN cana-1013 44 1 = = PUNCT cana-1013 45 1 [ [ X cana-1013 45 2 ๐‘ฆ1 ๐‘ฆ1 PROPN cana-1013 45 3 , , PUNCT cana-1013 45 4 ๐‘ฆ2 ๐‘ฆ2 NOUN cana-1013 45 5 , , PUNCT cana-1013 45 6 โ€ฆ โ€ฆ PUNCT cana-1013 45 7 , , PUNCT cana-1013 45 8 ๐‘ฆ๐‘š]๐‘ก ๐‘ฆ๐‘š]๐‘ก ADV cana-1013 45 9 be be VERB cana-1013 45 10 two two NUM cana-1013 45 11 arbitrary arbitrary ADJ cana-1013 45 12 vertices vertex NOUN cana-1013 45 13 of of ADP cana-1013 45 14 ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ NOUN cana-1013 45 15 , , PUNCT cana-1013 45 16 ๐‘† ๐‘† PROPN cana-1013 45 17 ) ) PUNCT cana-1013 45 18 where where SCONJ cana-1013 45 19 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PRON cana-1013 45 20 and and CCONJ cana-1013 45 21 ๐‘ฆ๐‘— ๐‘ฆ๐‘— PROPN cana-1013 45 22 are be AUX cana-1013 45 23 in in ADP cana-1013 45 24 ๐บ ๐บ PROPN cana-1013 45 25 for for ADP cana-1013 45 26 all all PRON cana-1013 45 27 ๐‘– ๐‘– ADP cana-1013 45 28 , , PUNCT cana-1013 45 29 ๐‘— ๐‘— PROPN cana-1013 45 30 โˆˆ โˆˆ NOUN cana-1013 45 31 { { PUNCT cana-1013 45 32 1,2 1,2 NUM cana-1013 45 33 , , PUNCT cana-1013 45 34 . . PUNCT cana-1013 45 35 . . PUNCT cana-1013 46 1 . . PUNCT cana-1013 47 1 , , PUNCT cana-1013 47 2 ๐‘š ๐‘š X cana-1013 47 3 } } PUNCT cana-1013 47 4 , , PUNCT cana-1013 47 5 then then ADV cana-1013 47 6 ๐‘‹ ๐‘‹ PROPN cana-1013 47 7 and and CCONJ cana-1013 47 8 ๐‘Œ ๐‘Œ PROPN cana-1013 47 9 are be AUX cana-1013 47 10 adjacent adjacent ADJ cana-1013 47 11 โ†” โ†” PROPN cana-1013 47 12 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 47 13 is be AUX cana-1013 47 14 adjacent adjacent ADJ cana-1013 47 15 to to ADP cana-1013 47 16 ๐‘ฆ๐‘— ๐‘ฆ๐‘— VERB cana-1013 47 17 in in ADP cana-1013 47 18 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 47 19 , , PUNCT cana-1013 47 20 ๐‘† ๐‘† PROPN cana-1013 47 21 ) ) PUNCT cana-1013 47 22 โˆ€l โˆ€l NOUN cana-1013 47 23 ๐‘– ๐‘– PROPN cana-1013 47 24 , , PUNCT cana-1013 47 25 ๐‘— ๐‘— PROPN cana-1013 47 26 โˆˆ โˆˆ NOUN cana-1013 47 27 { { PUNCT cana-1013 47 28 1,2 1,2 NUM cana-1013 47 29 , , PUNCT cana-1013 47 30 . . PUNCT cana-1013 47 31 . . PUNCT cana-1013 48 1 . . PUNCT cana-1013 49 1 , , PUNCT cana-1013 49 2 ๐‘š ๐‘š X cana-1013 49 3 } } PUNCT cana-1013 49 4 . . PUNCT cana-1013 50 1 here here ADV cana-1013 50 2 , , PUNCT cana-1013 50 3 we we PRON cana-1013 50 4 find find VERB cana-1013 50 5 the the DET cana-1013 50 6 generalized generalized ADJ cana-1013 50 7 cayley cayley NOUN cana-1013 50 8 graph graph NOUN cana-1013 50 9 when when SCONJ cana-1013 50 10 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 50 11 , , PUNCT cana-1013 50 12 ๐‘† ๐‘† PROPN cana-1013 50 13 ) ) PUNCT cana-1013 50 14 is be AUX cana-1013 50 15 the the DET cana-1013 50 16 cartesian cartesian ADJ cana-1013 50 17 products product NOUN cana-1013 50 18 ๐‘ƒ2 ๐‘ƒ2 PRON cana-1013 50 19 ร— ร— PROPN cana-1013 50 20 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 50 21 and and CCONJ cana-1013 50 22 ๐‘ƒ2 ๐‘ƒ2 DET cana-1013 50 23 ร— ร— PROPN cana-1013 50 24 ๐ถ3 ๐ถ3 PROPN cana-1013 50 25 . . PUNCT cana-1013 51 1 lemma lemma PROPN cana-1013 51 2 5 5 X cana-1013 51 3 . . PUNCT cana-1013 52 1 let let VERB cana-1013 52 2 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ NUM cana-1013 52 3 , , PUNCT cana-1013 52 4 ๐‘† ๐‘† PROPN cana-1013 52 5 ) ) PUNCT cana-1013 53 1 = = SYM cana-1013 53 2 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 53 3 ร— ร— PROPN cana-1013 53 4 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 53 5 , , PUNCT cana-1013 53 6 then then ADV cana-1013 53 7 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 53 8 , , PUNCT cana-1013 53 9 ๐‘† ๐‘† PROPN cana-1013 53 10 ) ) PUNCT cana-1013 53 11 = = SYM cana-1013 53 12 ๐พ4,4 ๐พ4,4 X cana-1013 53 13 โˆช โˆช ADP cana-1013 53 14 8๐‘ƒ1 8๐‘ƒ1 NUM cana-1013 53 15 . . PUNCT cana-1013 54 1 proof proof NOUN cana-1013 54 2 : : PUNCT cana-1013 54 3 suppose suppose VERB cana-1013 54 4 that that SCONJ cana-1013 54 5 ๐บ1 ๐บ1 PROPN cana-1013 54 6 = = SYM cana-1013 54 7 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 54 8 with with ADP cana-1013 54 9 vertex vertex NOUN cana-1013 54 10 set set NOUN cana-1013 54 11 { { PUNCT cana-1013 54 12 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 54 13 , , PUNCT cana-1013 54 14 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 54 15 } } PUNCT cana-1013 54 16 and and CCONJ cana-1013 54 17 ๐บ2 ๐บ2 ADJ cana-1013 54 18 = = SYM cana-1013 54 19 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 54 20 with with ADP cana-1013 54 21 vertex vertex NOUN cana-1013 54 22 set set NOUN cana-1013 54 23 { { PUNCT cana-1013 54 24 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 54 25 , , PUNCT cana-1013 54 26 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 54 27 } } PUNCT cana-1013 54 28 and and CCONJ cana-1013 54 29 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 54 30 , , PUNCT cana-1013 54 31 ๐‘† ๐‘† PROPN cana-1013 54 32 ) ) PUNCT cana-1013 54 33 = = SYM cana-1013 54 34 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 54 35 ร— ร— PROPN cana-1013 54 36 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 54 37 . . PUNCT cana-1013 55 1 so so ADV cana-1013 55 2 , , PUNCT cana-1013 55 3 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 55 4 , , PUNCT cana-1013 55 5 ๐‘† ๐‘† PROPN cana-1013 55 6 ) ) PUNCT cana-1013 55 7 is be AUX cana-1013 55 8 a a DET cana-1013 55 9 cycle cycle NOUN cana-1013 55 10 of of ADP cana-1013 55 11 length length NOUN cana-1013 55 12 4 4 NUM cana-1013 55 13 and and CCONJ cana-1013 55 14 its its PRON cana-1013 55 15 vertex vertex NOUN cana-1013 55 16 set set NOUN cana-1013 55 17 is be AUX cana-1013 55 18 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ NUM cana-1013 55 19 , , PUNCT cana-1013 55 20 ๐‘† ๐‘† PROPN cana-1013 55 21 ) ) PUNCT cana-1013 55 22 ) ) PUNCT cana-1013 56 1 = = PRON cana-1013 56 2 { { PUNCT cana-1013 56 3 ( ( PUNCT cana-1013 56 4 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 56 5 , , PUNCT cana-1013 56 6 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 56 7 ) ) PUNCT cana-1013 56 8 , , PUNCT cana-1013 56 9 ( ( PUNCT cana-1013 56 10 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 56 11 , , PUNCT cana-1013 56 12 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 56 13 ) ) PUNCT cana-1013 56 14 , , PUNCT cana-1013 56 15 ( ( PUNCT cana-1013 56 16 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 56 17 , , PUNCT cana-1013 56 18 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 56 19 ) ) PUNCT cana-1013 56 20 , , PUNCT cana-1013 56 21 ( ( PUNCT cana-1013 56 22 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 56 23 , , PUNCT cana-1013 56 24 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 56 25 ) ) PUNCT cana-1013 56 26 } } PUNCT cana-1013 56 27 and and CCONJ cana-1013 56 28 the the DET cana-1013 56 29 set set NOUN cana-1013 56 30 of of ADP cana-1013 56 31 four four NUM cana-1013 56 32 edges edge NOUN cana-1013 56 33 { { PUNCT cana-1013 56 34 ( ( PUNCT cana-1013 56 35 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 56 36 , , PUNCT cana-1013 56 37 ๐‘ฅ3)(๐‘ฅ1 ๐‘ฅ3)(๐‘ฅ1 ADV cana-1013 56 38 , , PUNCT cana-1013 56 39 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 56 40 ) ) PUNCT cana-1013 56 41 , , PUNCT cana-1013 56 42 ( ( PUNCT cana-1013 56 43 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 56 44 , , PUNCT cana-1013 56 45 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 56 46 , , PUNCT cana-1013 56 47 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 56 48 ) ) PUNCT cana-1013 56 49 , , PUNCT cana-1013 56 50 ( ( PUNCT cana-1013 56 51 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 56 52 , , PUNCT cana-1013 56 53 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 56 54 , , PUNCT cana-1013 56 55 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 56 56 ) ) PUNCT cana-1013 56 57 , , PUNCT cana-1013 56 58 ( ( PUNCT cana-1013 56 59 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 56 60 , , PUNCT cana-1013 56 61 ๐‘ฅ4)(๐‘ฅ1 ๐‘ฅ4)(๐‘ฅ1 ADJ cana-1013 56 62 , , PUNCT cana-1013 56 63 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 56 64 ) ) PUNCT cana-1013 56 65 } } PUNCT cana-1013 56 66 since since SCONJ cana-1013 56 67 the the DET cana-1013 56 68 cayley cayley ADJ cana-1013 56 69 graph graph NOUN cana-1013 56 70 is be AUX cana-1013 56 71 a a DET cana-1013 56 72 cycle cycle NOUN cana-1013 56 73 ( ( PUNCT cana-1013 56 74 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 56 75 , , PUNCT cana-1013 56 76 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 56 77 ) ) PUNCT cana-1013 56 78 โˆ’ โˆ’ PROPN cana-1013 56 79 ( ( PUNCT cana-1013 56 80 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 56 81 , , PUNCT cana-1013 56 82 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 56 83 ) ) PUNCT cana-1013 57 1 โˆ’ โˆ’ PROPN cana-1013 57 2 ( ( PUNCT cana-1013 57 3 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 57 4 , , PUNCT cana-1013 57 5 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 57 6 ) ) PUNCT cana-1013 57 7 โˆ’ โˆ’ PROPN cana-1013 57 8 ( ( PUNCT cana-1013 57 9 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 57 10 , , PUNCT cana-1013 57 11 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 57 12 ) ) PUNCT cana-1013 58 1 โˆ’ โˆ’ PROPN cana-1013 58 2 ( ( PUNCT cana-1013 58 3 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 58 4 , , PUNCT cana-1013 58 5 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 58 6 ) ) PUNCT cana-1013 58 7 . . PUNCT cana-1013 59 1 then then ADV cana-1013 59 2 , , PUNCT cana-1013 59 3 we we PRON cana-1013 59 4 have have VERB cana-1013 59 5 42 42 NUM cana-1013 59 6 = = SYM cana-1013 59 7 16 16 NUM cana-1013 59 8 vertices vertex NOUN cana-1013 59 9 in in ADP cana-1013 59 10 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 59 11 , , PUNCT cana-1013 59 12 ๐‘† ๐‘† PROPN cana-1013 59 13 ) ) PUNCT cana-1013 59 14 and and CCONJ cana-1013 59 15 ๐‘‰(๐ถ๐‘Ž๐‘ฆ2(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ2(๐บ NOUN cana-1013 59 16 , , PUNCT cana-1013 59 17 ๐‘† ๐‘† PROPN cana-1013 59 18 ) ) PUNCT cana-1013 59 19 ) ) PUNCT cana-1013 60 1 = = PRON cana-1013 60 2 { { PUNCT cana-1013 60 3 [ [ PUNCT cana-1013 60 4 ๐‘Ž ๐‘Ž PROPN cana-1013 60 5 ๐‘ ๐‘ NOUN cana-1013 60 6 ] ] PUNCT cana-1013 60 7 |๐‘Ž |๐‘Ž NOUN cana-1013 60 8 , , PUNCT cana-1013 60 9 ๐‘ ๐‘ PROPN cana-1013 60 10 โˆˆ โˆˆ PROPN cana-1013 60 11 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 60 12 , , PUNCT cana-1013 60 13 ๐‘† ๐‘† PROPN cana-1013 60 14 ) ) PUNCT cana-1013 60 15 ) ) PUNCT cana-1013 60 16 } } PUNCT cana-1013 60 17 communications communication NOUN cana-1013 60 18 on on ADP cana-1013 60 19 applied apply VERB cana-1013 60 20 nonlinear nonlinear ADJ cana-1013 60 21 analysis analysis NOUN cana-1013 60 22 issn issn NOUN cana-1013 60 23 : : PUNCT cana-1013 60 24 1074 1074 NUM cana-1013 60 25 - - PUNCT cana-1013 60 26 133x 133x NUM cana-1013 60 27 vol vol NOUN cana-1013 60 28 31 31 NUM cana-1013 60 29 no no NOUN cana-1013 60 30 . . PUNCT cana-1013 61 1 5s 5s NUM cana-1013 61 2 ( ( PUNCT cana-1013 61 3 2024 2024 NUM cana-1013 61 4 ) ) PUNCT cana-1013 61 5 199 199 NUM cana-1013 61 6 https://internationalpubls.com https://internationalpubls.com X cana-1013 61 7 consequently consequently ADV cana-1013 61 8 . . PUNCT cana-1013 62 1 every every DET cana-1013 62 2 vertex vertex NOUN cana-1013 62 3 in in ADP cana-1013 62 4 set set NOUN cana-1013 62 5 a a PRON cana-1013 62 6 is be AUX cana-1013 62 7 obviously obviously ADV cana-1013 62 8 adjacent adjacent ADJ cana-1013 62 9 to to ADP cana-1013 62 10 every every DET cana-1013 62 11 vertex vertex NOUN cana-1013 62 12 in in ADP cana-1013 62 13 set set PROPN cana-1013 62 14 b b NOUN cana-1013 62 15 , , PUNCT cana-1013 62 16 and and CCONJ cana-1013 62 17 vice vice ADV cana-1013 62 18 versa versa ADV cana-1013 62 19 . . PUNCT cana-1013 63 1 thus thus ADV cana-1013 63 2 , , PUNCT cana-1013 63 3 the the DET cana-1013 63 4 bipartite bipartite PROPN cana-1013 63 5 graph graph NOUN cana-1013 63 6 is be AUX cana-1013 63 7 obtained obtain VERB cana-1013 63 8 ๐พ4,4 ๐พ4,4 VERB cana-1013 63 9 . . PUNCT cana-1013 64 1 we we PRON cana-1013 64 2 demonstrate demonstrate VERB cana-1013 64 3 that that SCONJ cana-1013 64 4 every every DET cana-1013 64 5 other other ADJ cana-1013 64 6 vertex vertex NOUN cana-1013 64 7 is be AUX cana-1013 64 8 an an DET cana-1013 64 9 independent independent ADJ cana-1013 64 10 vertex vertex NOUN cana-1013 64 11 . . PUNCT cana-1013 65 1 assume assume VERB cana-1013 65 2 , , PUNCT cana-1013 65 3 without without ADP cana-1013 65 4 losing lose VERB cana-1013 65 5 generality generality NOUN cana-1013 65 6 , , PUNCT cana-1013 65 7 that that PRON cana-1013 65 8 is be AUX cana-1013 65 9 not not PART cana-1013 65 10 isolated isolate VERB cana-1013 65 11 . . PUNCT cana-1013 66 1 so so ADV cana-1013 66 2 , , PUNCT cana-1013 66 3 there there PRON cana-1013 66 4 is be VERB cana-1013 66 5 a a DET cana-1013 66 6 vertex vertex NOUN cana-1013 66 7 [ [ PUNCT cana-1013 66 8 ( ( PUNCT cana-1013 66 9 ๐‘Ž ๐‘Ž X cana-1013 66 10 , , PUNCT cana-1013 66 11 ๐‘ ๐‘ NOUN cana-1013 66 12 ) ) PUNCT cana-1013 66 13 ( ( PUNCT cana-1013 66 14 ๐‘ ๐‘ NOUN cana-1013 66 15 , , PUNCT cana-1013 66 16 ๐‘‘ ๐‘‘ NOUN cana-1013 66 17 ) ) PUNCT cana-1013 66 18 ] ] PUNCT cana-1013 66 19 โˆˆ โˆˆ PROPN cana-1013 66 20 ๐‘‰(๐ถ๐‘Ž๐‘ฆ2(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ2(๐บ NOUN cana-1013 66 21 , , PUNCT cana-1013 66 22 ๐‘† ๐‘† PROPN cana-1013 66 23 ) ) PUNCT cana-1013 66 24 ) ) PUNCT cana-1013 67 1 such such ADJ cana-1013 67 2 that that SCONJ cana-1013 67 3 ( ( PUNCT cana-1013 67 4 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 67 5 , , PUNCT cana-1013 67 6 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 67 7 ) ) PUNCT cana-1013 67 8 โˆ’ โˆ’ PROPN cana-1013 68 1 ( ( PUNCT cana-1013 68 2 ๐‘Ž ๐‘Ž X cana-1013 68 3 , , PUNCT cana-1013 68 4 ๐‘ ๐‘ NOUN cana-1013 68 5 ) ) PUNCT cana-1013 68 6 , , PUNCT cana-1013 68 7 ( ( PUNCT cana-1013 68 8 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 68 9 , , PUNCT cana-1013 68 10 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 68 11 ) ) PUNCT cana-1013 68 12 โˆ’ โˆ’ PROPN cana-1013 69 1 ( ( PUNCT cana-1013 69 2 ๐‘ ๐‘ NOUN cana-1013 69 3 , , PUNCT cana-1013 69 4 ๐‘‘ ๐‘‘ NOUN cana-1013 69 5 ) ) PUNCT cana-1013 69 6 , , PUNCT cana-1013 69 7 ( ( PUNCT cana-1013 69 8 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 69 9 , , PUNCT cana-1013 69 10 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 69 11 ) ) PUNCT cana-1013 69 12 โˆ’ โˆ’ PROPN cana-1013 69 13 ( ( PUNCT cana-1013 69 14 ๐‘Ž ๐‘Ž X cana-1013 69 15 , , PUNCT cana-1013 69 16 ๐‘ ๐‘ NOUN cana-1013 69 17 ) ) PUNCT cana-1013 69 18 and and CCONJ cana-1013 69 19 ( ( PUNCT cana-1013 69 20 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 69 21 , , PUNCT cana-1013 69 22 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 69 23 ) ) PUNCT cana-1013 70 1 โˆ’ โˆ’ PROPN cana-1013 70 2 ( ( PUNCT cana-1013 70 3 ๐‘ ๐‘ NOUN cana-1013 70 4 , , PUNCT cana-1013 70 5 ๐‘‘ ๐‘‘ NOUN cana-1013 70 6 ) ) PUNCT cana-1013 70 7 . . PUNCT cana-1013 71 1 so so ADV cana-1013 71 2 , , PUNCT cana-1013 71 3 ( ( PUNCT cana-1013 71 4 ๐‘Ž ๐‘Ž X cana-1013 71 5 , , PUNCT cana-1013 71 6 ๐‘ ๐‘ NOUN cana-1013 71 7 ) ) PUNCT cana-1013 71 8 = = SYM cana-1013 71 9 ( ( PUNCT cana-1013 71 10 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 71 11 , , PUNCT cana-1013 71 12 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 71 13 ) ) PUNCT cana-1013 71 14 or or CCONJ cana-1013 71 15 ( ( PUNCT cana-1013 71 16 ๐‘Ž ๐‘Ž X cana-1013 71 17 , , PUNCT cana-1013 71 18 ๐‘ ๐‘ NOUN cana-1013 71 19 ) ) PUNCT cana-1013 71 20 = = SYM cana-1013 71 21 ( ( PUNCT cana-1013 71 22 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 71 23 , , PUNCT cana-1013 71 24 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 71 25 ) ) PUNCT cana-1013 71 26 . . PUNCT cana-1013 72 1 if if SCONJ cana-1013 72 2 ( ( PUNCT cana-1013 72 3 ๐‘Ž ๐‘Ž X cana-1013 72 4 , , PUNCT cana-1013 72 5 ๐‘ ๐‘ NOUN cana-1013 72 6 ) ) PUNCT cana-1013 72 7 = = SYM cana-1013 72 8 ( ( PUNCT cana-1013 72 9 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 72 10 , , PUNCT cana-1013 72 11 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 72 12 ) ) PUNCT cana-1013 72 13 , , PUNCT cana-1013 72 14 then then ADV cana-1013 72 15 ( ( PUNCT cana-1013 72 16 ๐‘Ž ๐‘Ž X cana-1013 72 17 , , PUNCT cana-1013 72 18 ๐‘ ๐‘ NOUN cana-1013 72 19 ) ) PUNCT cana-1013 73 1 โˆ’ โˆ’ PROPN cana-1013 73 2 ( ( PUNCT cana-1013 73 3 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 73 4 , , PUNCT cana-1013 73 5 ๐‘ฅ4 ๐‘ฅ4 PROPN cana-1013 73 6 ) ) PUNCT cana-1013 73 7 then then ADV cana-1013 73 8 it it PRON cana-1013 73 9 implies imply VERB cana-1013 73 10 that that SCONJ cana-1013 73 11 ( ( PUNCT cana-1013 73 12 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 73 13 , , PUNCT cana-1013 73 14 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 73 15 ) ) PUNCT cana-1013 73 16 โˆ’ โˆ’ PROPN cana-1013 73 17 ( ( PUNCT cana-1013 73 18 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 73 19 , , PUNCT cana-1013 73 20 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 73 21 ) ) PUNCT cana-1013 73 22 which which PRON cana-1013 73 23 is be AUX cana-1013 73 24 a a DET cana-1013 73 25 contradiction contradiction NOUN cana-1013 73 26 . . PUNCT cana-1013 74 1 similarly similarly ADV cana-1013 74 2 , , PUNCT cana-1013 74 3 if if SCONJ cana-1013 74 4 ( ( PUNCT cana-1013 74 5 ๐‘Ž ๐‘Ž X cana-1013 74 6 , , PUNCT cana-1013 74 7 ๐‘ ๐‘ NOUN cana-1013 74 8 ) ) PUNCT cana-1013 74 9 = = SYM cana-1013 74 10 ( ( PUNCT cana-1013 74 11 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 74 12 , , PUNCT cana-1013 74 13 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 74 14 ) ) PUNCT cana-1013 74 15 , , PUNCT cana-1013 74 16 then then ADV cana-1013 74 17 ( ( PUNCT cana-1013 74 18 ๐‘Ž ๐‘Ž X cana-1013 74 19 , , PUNCT cana-1013 74 20 ๐‘ ๐‘ NOUN cana-1013 74 21 ) ) PUNCT cana-1013 74 22 โˆ’ โˆ’ PROPN cana-1013 74 23 ( ( PUNCT cana-1013 74 24 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 74 25 , , PUNCT cana-1013 74 26 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 74 27 ) ) PUNCT cana-1013 74 28 which which PRON cana-1013 74 29 implies imply VERB cana-1013 74 30 that that SCONJ cana-1013 74 31 ( ( PUNCT cana-1013 74 32 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 74 33 , , PUNCT cana-1013 74 34 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 74 35 ) ) PUNCT cana-1013 74 36 โˆ’ โˆ’ PROPN cana-1013 74 37 ( ( PUNCT cana-1013 74 38 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 74 39 , , PUNCT cana-1013 74 40 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 74 41 ) ) PUNCT cana-1013 74 42 and and CCONJ cana-1013 74 43 gain gain VERB cana-1013 74 44 it it PRON cana-1013 74 45 is be AUX cana-1013 74 46 a a DET cana-1013 74 47 contradiction contradiction NOUN cana-1013 74 48 . . PUNCT cana-1013 75 1 hence hence ADV cana-1013 75 2 , , PUNCT cana-1013 75 3 [ [ PUNCT cana-1013 75 4 ( ( PUNCT cana-1013 75 5 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 75 6 , , PUNCT cana-1013 75 7 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 75 8 ) ) PUNCT cana-1013 75 9 ( ( PUNCT cana-1013 75 10 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 75 11 , , PUNCT cana-1013 75 12 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 75 13 ) ) PUNCT cana-1013 75 14 ] ] PUNCT cana-1013 75 15 is be AUX cana-1013 75 16 an an DET cana-1013 75 17 isolated isolated ADJ cana-1013 75 18 vertex vertex NOUN cana-1013 75 19 . . PUNCT cana-1013 76 1 the the DET cana-1013 76 2 following follow VERB cana-1013 76 3 procedure procedure NOUN cana-1013 76 4 may may AUX cana-1013 76 5 be be AUX cana-1013 76 6 used use VERB cana-1013 76 7 to to ADP cana-1013 76 8 other other ADJ cana-1013 76 9 vertices vertex NOUN cana-1013 76 10 as as ADV cana-1013 76 11 well well ADV cana-1013 76 12 . . PUNCT cana-1013 77 1 there there PRON cana-1013 77 2 are be VERB cana-1013 77 3 these these DET cana-1013 77 4 solitary solitary ADJ cana-1013 77 5 vertices vertex NOUN cana-1013 77 6 in in ADP cana-1013 77 7 an an DET cana-1013 77 8 amount amount NOUN cana-1013 77 9 of of ADP cana-1013 77 10 42 42 NUM cana-1013 77 11 โˆ’ โˆ’ NOUN cana-1013 77 12 8 8 NUM cana-1013 77 13 = = SYM cana-1013 77 14 8 8 NUM cana-1013 77 15 , , PUNCT cana-1013 77 16 and and CCONJ cana-1013 77 17 hence hence ADV cana-1013 77 18 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 77 19 , , PUNCT cana-1013 77 20 ๐‘† ๐‘† PROPN cana-1013 77 21 ) ) PUNCT cana-1013 77 22 = = SYM cana-1013 77 23 ๐พ4,4 ๐พ4,4 X cana-1013 77 24 โˆช โˆช ADP cana-1013 77 25 8๐‘ƒ1 8๐‘ƒ1 NUM cana-1013 77 26 . . PUNCT cana-1013 78 1 the the DET cana-1013 78 2 graph graph NOUN cana-1013 78 3 of of ADP cana-1013 78 4 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 78 5 , , PUNCT cana-1013 78 6 ๐‘† ๐‘† PROPN cana-1013 78 7 ) ) PUNCT cana-1013 78 8 in in ADP cana-1013 78 9 this this DET cana-1013 78 10 case case NOUN cana-1013 78 11 , , PUNCT cana-1013 78 12 is be AUX cana-1013 78 13 shown show VERB cana-1013 78 14 below below ADP cana-1013 78 15 . . PUNCT cana-1013 79 1 ๏‚ก ๏‚ก PUNCT cana-1013 79 2 the the DET cana-1013 79 3 graph graph NOUN cana-1013 79 4 ๐‘ƒ2 ๐‘ƒ2 ADJ cana-1013 79 5 ร— ร— PROPN cana-1013 79 6 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 79 7 a a DET cana-1013 79 8 component component NOUN cana-1013 79 9 of of ADP cana-1013 79 10 graph graph NOUN cana-1013 79 11 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 79 12 , , PUNCT cana-1013 79 13 ๐‘† ๐‘† PROPN cana-1013 79 14 ) ) PUNCT cana-1013 79 15 of of ADP cana-1013 79 16 ๐‘ƒ2 ๐‘ƒ2 ADJ cana-1013 79 17 ร— ร— PROPN cana-1013 79 18 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 79 19 in in ADP cana-1013 79 20 the the DET cana-1013 79 21 next next ADJ cana-1013 79 22 theorem theorem NOUN cana-1013 79 23 , , PUNCT cana-1013 79 24 we we PRON cana-1013 79 25 generalized generalize VERB cana-1013 79 26 the the DET cana-1013 79 27 cayley cayley ADJ cana-1013 79 28 graph graph NOUN cana-1013 79 29 for for ADP cana-1013 79 30 each each PRON cana-1013 79 31 ๐‘š=3 ๐‘š=3 PUNCT cana-1013 79 32 when when SCONJ cana-1013 79 33 the the DET cana-1013 79 34 common common ADJ cana-1013 79 35 cayley cayley ADJ cana-1013 79 36 graph graph NOUN cana-1013 79 37 is be AUX cana-1013 79 38 ๐‘ƒ2 ๐‘ƒ2 ADJ cana-1013 79 39 ร— ร— PROPN cana-1013 79 40 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 79 41 . . PUNCT cana-1013 80 1 lemma lemma PROPN cana-1013 80 2 6 6 NUM cana-1013 80 3 . . PUNCT cana-1013 81 1 let let VERB cana-1013 81 2 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ NUM cana-1013 81 3 , , PUNCT cana-1013 81 4 ๐‘† ๐‘† PROPN cana-1013 81 5 ) ) PUNCT cana-1013 82 1 = = SYM cana-1013 82 2 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 82 3 ร— ร— PROPN cana-1013 82 4 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 82 5 , , PUNCT cana-1013 82 6 then then ADV cana-1013 82 7 ๐ถ๐‘Ž๐‘ฆ3(๐บ ๐ถ๐‘Ž๐‘ฆ3(๐บ PROPN cana-1013 82 8 , , PUNCT cana-1013 82 9 ๐‘† ๐‘† PROPN cana-1013 82 10 ) ) PUNCT cana-1013 82 11 = = PUNCT cana-1013 82 12 ๐พ8,8 ๐พ8,8 ADJ cana-1013 82 13 โˆช โˆช ADJ cana-1013 82 14 48๐‘ƒ1 48๐‘ƒ1 NUM cana-1013 82 15 . . PUNCT cana-1013 83 1 proof proof NOUN cana-1013 83 2 : : PUNCT cana-1013 83 3 suppose suppose VERB cana-1013 83 4 that that SCONJ cana-1013 83 5 ๐บ1 ๐บ1 PROPN cana-1013 83 6 = = SYM cana-1013 83 7 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 83 8 with with ADP cana-1013 83 9 vertex vertex NOUN cana-1013 83 10 set set NOUN cana-1013 83 11 { { PUNCT cana-1013 83 12 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 83 13 , , PUNCT cana-1013 83 14 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 83 15 } } PUNCT cana-1013 83 16 and and CCONJ cana-1013 83 17 ๐บ2 ๐บ2 ADJ cana-1013 83 18 = = SYM cana-1013 83 19 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 83 20 with with ADP cana-1013 83 21 vertex vertex NOUN cana-1013 83 22 set set NOUN cana-1013 83 23 { { PUNCT cana-1013 83 24 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 83 25 , , PUNCT cana-1013 83 26 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 83 27 } } PUNCT cana-1013 83 28 and and CCONJ cana-1013 83 29 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 83 30 , , PUNCT cana-1013 83 31 ๐‘† ๐‘† PROPN cana-1013 83 32 ) ) PUNCT cana-1013 83 33 = = SYM cana-1013 83 34 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 83 35 ร— ร— PROPN cana-1013 83 36 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 83 37 . . PUNCT cana-1013 84 1 so so ADV cana-1013 84 2 , , PUNCT cana-1013 84 3 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 84 4 , , PUNCT cana-1013 84 5 ๐‘† ๐‘† PROPN cana-1013 84 6 ) ) PUNCT cana-1013 84 7 is be AUX cana-1013 84 8 a a DET cana-1013 84 9 cycle cycle NOUN cana-1013 84 10 of of ADP cana-1013 84 11 lenฯˆth lenฯˆth NOUN cana-1013 84 12 4 4 NUM cana-1013 84 13 and and CCONJ cana-1013 84 14 its its PRON cana-1013 84 15 vertex vertex NOUN cana-1013 84 16 set set NOUN cana-1013 84 17 is be AUX cana-1013 84 18 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ NUM cana-1013 84 19 , , PUNCT cana-1013 84 20 ๐‘† ๐‘† PROPN cana-1013 84 21 ) ) PUNCT cana-1013 84 22 ) ) PUNCT cana-1013 85 1 = = PRON cana-1013 85 2 { { PUNCT cana-1013 85 3 ( ( PUNCT cana-1013 85 4 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 85 5 , , PUNCT cana-1013 85 6 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 85 7 ) ) PUNCT cana-1013 85 8 , , PUNCT cana-1013 85 9 ( ( PUNCT cana-1013 85 10 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 85 11 , , PUNCT cana-1013 85 12 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 85 13 ) ) PUNCT cana-1013 85 14 , , PUNCT cana-1013 85 15 ( ( PUNCT cana-1013 85 16 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 85 17 , , PUNCT cana-1013 85 18 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 85 19 ) ) PUNCT cana-1013 85 20 , , PUNCT cana-1013 85 21 ( ( PUNCT cana-1013 85 22 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 85 23 , , PUNCT cana-1013 85 24 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 85 25 ) ) PUNCT cana-1013 85 26 } } PUNCT cana-1013 85 27 and and CCONJ cana-1013 85 28 the the DET cana-1013 85 29 set set NOUN cana-1013 85 30 of of ADP cana-1013 85 31 four four NUM cana-1013 85 32 edฯˆes edฯˆes NOUN cana-1013 85 33 { { PUNCT cana-1013 85 34 ( ( PUNCT cana-1013 85 35 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 85 36 , , PUNCT cana-1013 85 37 ๐‘ฅ3)(๐‘ฅ1 ๐‘ฅ3)(๐‘ฅ1 ADV cana-1013 85 38 , , PUNCT cana-1013 85 39 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 85 40 ) ) PUNCT cana-1013 85 41 , , PUNCT cana-1013 85 42 ( ( PUNCT cana-1013 85 43 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 85 44 , , PUNCT cana-1013 85 45 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 85 46 , , PUNCT cana-1013 85 47 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 85 48 ) ) PUNCT cana-1013 85 49 , , PUNCT cana-1013 85 50 ( ( PUNCT cana-1013 85 51 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 85 52 , , PUNCT cana-1013 85 53 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 85 54 , , PUNCT cana-1013 85 55 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 85 56 ) ) PUNCT cana-1013 85 57 , , PUNCT cana-1013 85 58 ( ( PUNCT cana-1013 85 59 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 85 60 , , PUNCT cana-1013 85 61 ๐‘ฅ4)(๐‘ฅ1 ๐‘ฅ4)(๐‘ฅ1 ADJ cana-1013 85 62 , , PUNCT cana-1013 85 63 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 85 64 ) ) PUNCT cana-1013 85 65 } } PUNCT cana-1013 85 66 since since SCONJ cana-1013 85 67 the the DET cana-1013 85 68 cayley cayley ADJ cana-1013 85 69 graph graph NOUN cana-1013 85 70 is be AUX cana-1013 85 71 a a DET cana-1013 85 72 cycle cycle NOUN cana-1013 85 73 ( ( PUNCT cana-1013 85 74 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 85 75 , , PUNCT cana-1013 85 76 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 85 77 ) ) PUNCT cana-1013 85 78 โˆ’ โˆ’ PROPN cana-1013 85 79 ( ( PUNCT cana-1013 85 80 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 85 81 , , PUNCT cana-1013 85 82 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 85 83 ) ) PUNCT cana-1013 86 1 โˆ’ โˆ’ PROPN cana-1013 86 2 ( ( PUNCT cana-1013 86 3 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 86 4 , , PUNCT cana-1013 86 5 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 86 6 ) ) PUNCT cana-1013 86 7 โˆ’ โˆ’ PROPN cana-1013 86 8 ( ( PUNCT cana-1013 86 9 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 86 10 , , PUNCT cana-1013 86 11 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 86 12 ) ) PUNCT cana-1013 87 1 โˆ’ โˆ’ PROPN cana-1013 87 2 ( ( PUNCT cana-1013 87 3 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 87 4 , , PUNCT cana-1013 87 5 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 87 6 ) ) PUNCT cana-1013 87 7 . . PUNCT cana-1013 88 1 then then ADV cana-1013 88 2 , , PUNCT cana-1013 88 3 we we PRON cana-1013 88 4 have have VERB cana-1013 88 5 communications communication NOUN cana-1013 88 6 on on ADP cana-1013 88 7 applied apply VERB cana-1013 88 8 nonlinear nonlinear ADJ cana-1013 88 9 analysis analysis NOUN cana-1013 88 10 issn issn NOUN cana-1013 88 11 : : PUNCT cana-1013 88 12 1074 1074 NUM cana-1013 88 13 - - PUNCT cana-1013 88 14 133x 133x NUM cana-1013 88 15 vol vol NOUN cana-1013 88 16 31 31 NUM cana-1013 88 17 no no NOUN cana-1013 88 18 . . PUNCT cana-1013 89 1 5s 5s NUM cana-1013 89 2 ( ( PUNCT cana-1013 89 3 2024 2024 NUM cana-1013 89 4 ) ) PUNCT cana-1013 89 5 200 200 NUM cana-1013 89 6 https://internationalpubls.com https://internationalpubls.com X cana-1013 89 7 43 43 NUM cana-1013 89 8 = = SYM cana-1013 89 9 64 64 NUM cana-1013 89 10 vertices vertex NOUN cana-1013 89 11 in in ADP cana-1013 89 12 ๐ถ๐‘Ž๐‘ฆ3(๐บ ๐ถ๐‘Ž๐‘ฆ3(๐บ PROPN cana-1013 89 13 , , PUNCT cana-1013 89 14 ๐‘† ๐‘† PROPN cana-1013 89 15 ) ) PUNCT cana-1013 89 16 and and CCONJ cana-1013 89 17 ๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ ADV cana-1013 89 18 , , PUNCT cana-1013 89 19 ๐‘† ๐‘† PROPN cana-1013 89 20 ) ) PUNCT cana-1013 89 21 ) ) PUNCT cana-1013 90 1 = = PRON cana-1013 90 2 { { PUNCT cana-1013 90 3 [ [ PUNCT cana-1013 90 4 ๐‘Ž ๐‘Ž PROPN cana-1013 90 5 ๐‘ ๐‘ PROPN cana-1013 90 6 ๐‘ ๐‘ PROPN cana-1013 90 7 ] ] PUNCT cana-1013 90 8 |๐‘Ž |๐‘Ž PROPN cana-1013 90 9 , , PUNCT cana-1013 90 10 ๐‘ ๐‘ NOUN cana-1013 90 11 , , PUNCT cana-1013 90 12 ๐‘ ๐‘ PROPN cana-1013 90 13 โˆˆ โˆˆ PROPN cana-1013 90 14 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 90 15 , , PUNCT cana-1013 90 16 ๐‘† ๐‘† PROPN cana-1013 90 17 ) ) PUNCT cana-1013 90 18 ) ) PUNCT cana-1013 90 19 } } PUNCT cana-1013 90 20 . . PUNCT cana-1013 91 1 so so ADV cana-1013 91 2 , , PUNCT cana-1013 91 3 ๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ ADV cana-1013 91 4 , , PUNCT cana-1013 91 5 ๐‘† ๐‘† PROPN cana-1013 91 6 ) ) PUNCT cana-1013 91 7 ) ) PUNCT cana-1013 92 1 = = PRON cana-1013 92 2 { { PUNCT cana-1013 92 3 [ [ PUNCT cana-1013 92 4 ( ( PUNCT cana-1013 92 5 ๐‘ฅ๐‘– ๐‘ฅ๐‘– X cana-1013 92 6 , , PUNCT cana-1013 92 7 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 92 8 ) ) PUNCT cana-1013 92 9 ( ( PUNCT cana-1013 92 10 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1013 92 11 , , PUNCT cana-1013 92 12 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 92 13 ) ) PUNCT cana-1013 92 14 ( ( PUNCT cana-1013 92 15 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 92 16 , , PUNCT cana-1013 92 17 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 92 18 ) ) PUNCT cana-1013 92 19 ] ] PUNCT cana-1013 93 1 โˆถ โˆถ NOUN cana-1013 93 2 ๐‘– ๐‘– SYM cana-1013 93 3 , , PUNCT cana-1013 93 4 ๐‘— ๐‘— PROPN cana-1013 93 5 , , PUNCT cana-1013 93 6 ๐‘˜ ๐‘˜ PROPN cana-1013 93 7 , , PUNCT cana-1013 93 8 ๐‘™ ๐‘™ X cana-1013 93 9 , , PUNCT cana-1013 93 10 ๐‘Ÿ ๐‘Ÿ NOUN cana-1013 93 11 , , PUNCT cana-1013 93 12 ๐‘  ๐‘  PROPN cana-1013 93 13 = = SYM cana-1013 93 14 1,2,3,4 1,2,3,4 NUM cana-1013 93 15 } } PUNCT cana-1013 93 16 . . PUNCT cana-1013 94 1 therefore therefore ADV cana-1013 94 2 , , PUNCT cana-1013 94 3 we we PRON cana-1013 94 4 have have VERB cana-1013 94 5 two two NUM cana-1013 94 6 independent independent ADJ cana-1013 94 7 sets set NOUN cana-1013 94 8 it it PRON cana-1013 94 9 is be AUX cana-1013 94 10 clear clear ADJ cana-1013 94 11 that that SCONJ cana-1013 94 12 every every DET cana-1013 94 13 vertex vertex NOUN cana-1013 94 14 in in ADP cana-1013 94 15 set set NOUN cana-1013 94 16 a a PRON cana-1013 94 17 is be AUX cana-1013 94 18 adjacent adjacent ADJ cana-1013 94 19 to to ADP cana-1013 94 20 all all DET cana-1013 94 21 vertices vertex NOUN cana-1013 94 22 in in ADP cana-1013 94 23 set set PROPN cana-1013 94 24 b b PROPN cana-1013 94 25 and and CCONJ cana-1013 94 26 vice vice ADV cana-1013 94 27 versa versa ADV cana-1013 94 28 . . PUNCT cana-1013 95 1 thus thus ADV cana-1013 95 2 , , PUNCT cana-1013 95 3 we we PRON cana-1013 95 4 ฯˆet ฯˆet VERB cana-1013 95 5 the the DET cana-1013 95 6 bipartite bipartite PROPN cana-1013 95 7 graph graph NOUN cana-1013 95 8 ๐พ8,8 ๐พ8,8 PROPN cana-1013 95 9 . . PUNCT cana-1013 96 1 we we PRON cana-1013 96 2 demonstrate demonstrate VERB cana-1013 96 3 that that SCONJ cana-1013 96 4 every every DET cana-1013 96 5 other other ADJ cana-1013 96 6 vertex vertex NOUN cana-1013 96 7 is be AUX cana-1013 96 8 an an DET cana-1013 96 9 independent independent ADJ cana-1013 96 10 vertex vertex NOUN cana-1013 96 11 . . PUNCT cana-1013 97 1 absent absent ADJ cana-1013 97 2 loss loss NOUN cana-1013 97 3 of of ADP cana-1013 97 4 generality generality NOUN cana-1013 97 5 , , PUNCT cana-1013 97 6 suppose suppose VERB cana-1013 97 7 that that SCONJ cana-1013 97 8 [ [ PUNCT cana-1013 97 9 ( ( PUNCT cana-1013 97 10 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 97 11 , , PUNCT cana-1013 97 12 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 97 13 ) ) PUNCT cana-1013 97 14 ( ( PUNCT cana-1013 97 15 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1013 97 16 , , PUNCT cana-1013 97 17 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 97 18 ) ) PUNCT cana-1013 97 19 ( ( PUNCT cana-1013 97 20 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 97 21 , , PUNCT cana-1013 97 22 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 97 23 ) ) PUNCT cana-1013 97 24 ] ] PUNCT cana-1013 97 25 is be AUX cana-1013 97 26 not not PART cana-1013 97 27 isolated isolate VERB cana-1013 97 28 where where SCONJ cana-1013 97 29 ๐‘– ๐‘– X cana-1013 97 30 , , PUNCT cana-1013 97 31 ๐‘˜ ๐‘˜ PROPN cana-1013 97 32 , , PUNCT cana-1013 97 33 ๐‘Ÿ ๐‘Ÿ NOUN cana-1013 97 34 = = SYM cana-1013 97 35 1,2 1,2 NUM cana-1013 97 36 and and CCONJ cana-1013 97 37 ๐‘— ๐‘— NOUN cana-1013 97 38 , , PUNCT cana-1013 97 39 ๐‘™ ๐‘™ PROPN cana-1013 97 40 , , PUNCT cana-1013 97 41 ๐‘  ๐‘  NOUN cana-1013 97 42 = = PUNCT cana-1013 97 43 3,4 3,4 NUM cana-1013 97 44 . . PUNCT cana-1013 98 1 so so ADV cana-1013 98 2 , , PUNCT cana-1013 98 3 there there PRON cana-1013 98 4 is be VERB cana-1013 98 5 a a DET cana-1013 98 6 vertex vertex NOUN cana-1013 98 7 [ [ PUNCT cana-1013 98 8 ( ( PUNCT cana-1013 98 9 ๐‘Ž ๐‘Ž X cana-1013 98 10 , , PUNCT cana-1013 98 11 ๐‘ ๐‘ NOUN cana-1013 98 12 ) ) PUNCT cana-1013 98 13 ( ( PUNCT cana-1013 98 14 ๐‘ ๐‘ NOUN cana-1013 98 15 , , PUNCT cana-1013 98 16 ๐‘‘ ๐‘‘ NOUN cana-1013 98 17 ) ) PUNCT cana-1013 98 18 ( ( PUNCT cana-1013 98 19 ๐‘’ ๐‘’ PROPN cana-1013 98 20 , , PUNCT cana-1013 98 21 ๐‘“ ๐‘“ X cana-1013 98 22 ) ) PUNCT cana-1013 98 23 ] ] PUNCT cana-1013 99 1 โˆˆ โˆˆ PROPN cana-1013 99 2 ๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ ADV cana-1013 99 3 , , PUNCT cana-1013 99 4 ๐‘† ๐‘† PROPN cana-1013 99 5 ) ) PUNCT cana-1013 99 6 ) ) PUNCT cana-1013 100 1 such such ADJ cana-1013 100 2 that that SCONJ cana-1013 100 3 ( ( PUNCT cana-1013 100 4 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 100 5 , , PUNCT cana-1013 100 6 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 100 7 ) ) PUNCT cana-1013 100 8 โˆ’ โˆ’ PROPN cana-1013 101 1 ( ( PUNCT cana-1013 101 2 ๐‘Ž ๐‘Ž X cana-1013 101 3 , , PUNCT cana-1013 101 4 ๐‘ ๐‘ NOUN cana-1013 101 5 ) ) PUNCT cana-1013 101 6 , , PUNCT cana-1013 101 7 ( ( PUNCT cana-1013 101 8 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 101 9 , , PUNCT cana-1013 101 10 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 101 11 ) ) PUNCT cana-1013 101 12 โˆ’ โˆ’ PROPN cana-1013 102 1 ( ( PUNCT cana-1013 102 2 ๐‘ ๐‘ NOUN cana-1013 102 3 , , PUNCT cana-1013 102 4 ๐‘‘ ๐‘‘ NOUN cana-1013 102 5 ) ) PUNCT cana-1013 102 6 , , PUNCT cana-1013 102 7 ( ( PUNCT cana-1013 102 8 ๐‘ฅ๐‘– ๐‘ฅ๐‘– X cana-1013 102 9 , , PUNCT cana-1013 102 10 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 102 11 ) ) PUNCT cana-1013 102 12 โˆ’ โˆ’ PROPN cana-1013 103 1 ( ( PUNCT cana-1013 103 2 ๐‘’ ๐‘’ PROPN cana-1013 103 3 , , PUNCT cana-1013 103 4 ๐‘“ ๐‘“ X cana-1013 103 5 ) ) PUNCT cana-1013 103 6 and and CCONJ cana-1013 103 7 ( ( PUNCT cana-1013 103 8 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1013 103 9 , , PUNCT cana-1013 103 10 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 103 11 ) ) PUNCT cana-1013 103 12 โˆ’ โˆ’ PROPN cana-1013 104 1 ( ( PUNCT cana-1013 104 2 ๐‘Ž ๐‘Ž X cana-1013 104 3 , , PUNCT cana-1013 104 4 ๐‘ ๐‘ NOUN cana-1013 104 5 ) ) PUNCT cana-1013 104 6 , , PUNCT cana-1013 104 7 ( ( PUNCT cana-1013 104 8 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ INTJ cana-1013 104 9 , , PUNCT cana-1013 104 10 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 104 11 ) ) PUNCT cana-1013 104 12 โˆ’ โˆ’ PROPN cana-1013 105 1 ( ( PUNCT cana-1013 105 2 ๐‘ ๐‘ NOUN cana-1013 105 3 , , PUNCT cana-1013 105 4 ๐‘‘ ๐‘‘ NOUN cana-1013 105 5 ) ) PUNCT cana-1013 105 6 , , PUNCT cana-1013 105 7 ( ( PUNCT cana-1013 105 8 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1013 105 9 , , PUNCT cana-1013 105 10 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 105 11 ) ) PUNCT cana-1013 105 12 โˆ’ โˆ’ PROPN cana-1013 106 1 ( ( PUNCT cana-1013 106 2 ๐‘’ ๐‘’ PROPN cana-1013 106 3 , , PUNCT cana-1013 106 4 ๐‘“ ๐‘“ X cana-1013 106 5 ) ) PUNCT cana-1013 106 6 and and CCONJ cana-1013 106 7 ( ( PUNCT cana-1013 106 8 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 106 9 , , PUNCT cana-1013 106 10 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 106 11 ) ) PUNCT cana-1013 106 12 โˆ’ โˆ’ PROPN cana-1013 107 1 ( ( PUNCT cana-1013 107 2 ๐‘Ž ๐‘Ž X cana-1013 107 3 , , PUNCT cana-1013 107 4 ๐‘ ๐‘ NOUN cana-1013 107 5 ) ) PUNCT cana-1013 107 6 , , PUNCT cana-1013 107 7 ( ( PUNCT cana-1013 107 8 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 107 9 , , PUNCT cana-1013 107 10 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 107 11 ) ) PUNCT cana-1013 107 12 โˆ’ โˆ’ PROPN cana-1013 108 1 ( ( PUNCT cana-1013 108 2 ๐‘ ๐‘ NOUN cana-1013 108 3 , , PUNCT cana-1013 108 4 ๐‘‘ ๐‘‘ NOUN cana-1013 108 5 ) ) PUNCT cana-1013 108 6 , , PUNCT cana-1013 108 7 ( ( PUNCT cana-1013 108 8 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 108 9 , , PUNCT cana-1013 108 10 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 108 11 ) ) PUNCT cana-1013 108 12 โˆ’ โˆ’ PROPN cana-1013 109 1 ( ( PUNCT cana-1013 109 2 ๐‘’ ๐‘’ PROPN cana-1013 109 3 , , PUNCT cana-1013 109 4 ๐‘“ ๐‘“ PRON cana-1013 109 5 ) ) PUNCT cana-1013 109 6 . . PUNCT cana-1013 110 1 but but CCONJ cana-1013 110 2 ( ( PUNCT cana-1013 110 3 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 110 4 , , PUNCT cana-1013 110 5 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 110 6 ) ) PUNCT cana-1013 110 7 is be AUX cana-1013 110 8 of of ADP cana-1013 110 9 deฯˆree deฯˆree NOUN cana-1013 110 10 2 2 NUM cana-1013 110 11 . . PUNCT cana-1013 111 1 so so ADV cana-1013 111 2 , , PUNCT cana-1013 111 3 ( ( PUNCT cana-1013 111 4 ๐‘Ž ๐‘Ž X cana-1013 111 5 , , PUNCT cana-1013 111 6 ๐‘ ๐‘ NOUN cana-1013 111 7 ) ) PUNCT cana-1013 111 8 = = SYM cana-1013 111 9 ( ( PUNCT cana-1013 111 10 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 111 11 , , PUNCT cana-1013 111 12 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 111 13 ) ) PUNCT cana-1013 111 14 or or CCONJ cana-1013 111 15 ( ( PUNCT cana-1013 111 16 ๐‘Ž ๐‘Ž X cana-1013 111 17 , , PUNCT cana-1013 111 18 ๐‘ ๐‘ NOUN cana-1013 111 19 ) ) PUNCT cana-1013 111 20 = = SYM cana-1013 111 21 ( ( PUNCT cana-1013 111 22 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1013 111 23 , , PUNCT cana-1013 111 24 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 111 25 ) ) PUNCT cana-1013 111 26 or or CCONJ cana-1013 111 27 ( ( PUNCT cana-1013 111 28 ๐‘Ž ๐‘Ž X cana-1013 111 29 , , PUNCT cana-1013 111 30 ๐‘ ๐‘ NOUN cana-1013 111 31 ) ) PUNCT cana-1013 111 32 = = SYM cana-1013 111 33 ( ( PUNCT cana-1013 111 34 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 111 35 , , PUNCT cana-1013 111 36 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 111 37 ) ) PUNCT cana-1013 111 38 . . PUNCT cana-1013 112 1 if if SCONJ cana-1013 112 2 ( ( PUNCT cana-1013 112 3 ๐‘Ž ๐‘Ž X cana-1013 112 4 , , PUNCT cana-1013 112 5 ๐‘ ๐‘ NOUN cana-1013 112 6 ) ) PUNCT cana-1013 112 7 = = SYM cana-1013 112 8 ( ( PUNCT cana-1013 112 9 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 112 10 , , PUNCT cana-1013 112 11 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 112 12 ) ) PUNCT cana-1013 112 13 , , PUNCT cana-1013 112 14 then then ADV cana-1013 112 15 it it PRON cana-1013 112 16 implies imply VERB cana-1013 112 17 that that SCONJ cana-1013 112 18 ( ( PUNCT cana-1013 112 19 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 112 20 , , PUNCT cana-1013 112 21 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 112 22 ) ) PUNCT cana-1013 112 23 โˆ’ โˆ’ PROPN cana-1013 112 24 ( ( PUNCT cana-1013 112 25 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 112 26 , , PUNCT cana-1013 112 27 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 112 28 ) ) PUNCT cana-1013 112 29 which which PRON cana-1013 112 30 is be AUX cana-1013 112 31 a a DET cana-1013 112 32 contradiction contradiction NOUN cana-1013 112 33 . . PUNCT cana-1013 113 1 likewise likewise ADV cana-1013 113 2 , , PUNCT cana-1013 113 3 if if SCONJ cana-1013 113 4 ( ( PUNCT cana-1013 113 5 ๐‘Ž ๐‘Ž X cana-1013 113 6 , , PUNCT cana-1013 113 7 ๐‘ ๐‘ NOUN cana-1013 113 8 ) ) PUNCT cana-1013 113 9 = = SYM cana-1013 113 10 ( ( PUNCT cana-1013 113 11 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1013 113 12 , , PUNCT cana-1013 113 13 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 113 14 ) ) PUNCT cana-1013 113 15 and and CCONJ cana-1013 113 16 ( ( PUNCT cana-1013 113 17 ๐‘Ž ๐‘Ž X cana-1013 113 18 , , PUNCT cana-1013 113 19 ๐‘ ๐‘ NOUN cana-1013 113 20 ) ) PUNCT cana-1013 113 21 = = SYM cana-1013 113 22 ( ( PUNCT cana-1013 113 23 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 113 24 , , PUNCT cana-1013 113 25 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 113 26 ) ) PUNCT cana-1013 113 27 we we PRON cana-1013 113 28 ฯˆet ฯˆet VERB cana-1013 113 29 the the DET cana-1013 113 30 a a DET cana-1013 113 31 contradiction contradiction NOUN cana-1013 113 32 . . PUNCT cana-1013 114 1 hence hence ADV cana-1013 114 2 , , PUNCT cana-1013 114 3 the the DET cana-1013 114 4 rest rest NOUN cana-1013 114 5 vertices vertex NOUN cana-1013 114 6 [ [ PUNCT cana-1013 114 7 ( ( PUNCT cana-1013 114 8 ๐‘ฅ๐‘– ๐‘ฅ๐‘– X cana-1013 114 9 , , PUNCT cana-1013 114 10 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 114 11 ) ) PUNCT cana-1013 114 12 ( ( PUNCT cana-1013 114 13 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ PROPN cana-1013 114 14 , , PUNCT cana-1013 114 15 ๐‘ฅ๐‘™ ๐‘ฅ๐‘™ NOUN cana-1013 114 16 ) ) PUNCT cana-1013 114 17 ( ( PUNCT cana-1013 114 18 ๐‘ฅ๐‘Ÿ ๐‘ฅ๐‘Ÿ INTJ cana-1013 114 19 , , PUNCT cana-1013 114 20 ๐‘ฅ๐‘  ๐‘ฅ๐‘  PROPN cana-1013 114 21 ) ) PUNCT cana-1013 114 22 ] ] PUNCT cana-1013 114 23 are be AUX cana-1013 114 24 isolated isolated ADJ cana-1013 114 25 vertices vertex NOUN cana-1013 114 26 . . PUNCT cana-1013 115 1 we we PRON cana-1013 115 2 can can AUX cana-1013 115 3 prove prove VERB cana-1013 115 4 by by ADP cana-1013 115 5 the the DET cana-1013 115 6 same same ADJ cana-1013 115 7 method method NOUN cana-1013 115 8 as as ADP cana-1013 115 9 above above ADV cana-1013 115 10 for for ADP cana-1013 115 11 more more ADJ cana-1013 115 12 vertices vertex NOUN cana-1013 115 13 . . PUNCT cana-1013 116 1 there there PRON cana-1013 116 2 are be VERB cana-1013 116 3 these these DET cana-1013 116 4 solitary solitary ADJ cana-1013 116 5 vertices vertex NOUN cana-1013 116 6 in in ADP cana-1013 116 7 an an DET cana-1013 116 8 amount amount NOUN cana-1013 116 9 of of ADP cana-1013 116 10 |๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ |๐‘‰(๐ถ๐‘Ž๐‘ฆ3(๐บ NOUN cana-1013 116 11 , , PUNCT cana-1013 116 12 ๐‘†)| ๐‘†)| VERB cana-1013 116 13 โˆ’ โˆ’ PROPN cana-1013 116 14 ( ( PUNCT cana-1013 116 15 |๐ด| |๐ด| NOUN cana-1013 116 16 + + ADJ cana-1013 116 17 |๐ต| |๐ต| NOUN cana-1013 116 18 ) ) PUNCT cana-1013 116 19 = = SYM cana-1013 116 20 43 43 NUM cana-1013 116 21 โˆ’ โˆ’ NOUN cana-1013 116 22 ( ( PUNCT cana-1013 116 23 8 8 NUM cana-1013 116 24 + + SYM cana-1013 116 25 8) 8) NUM cana-1013 116 26 = = SYM cana-1013 116 27 48 48 NUM cana-1013 116 28 , , PUNCT cana-1013 116 29 and and CCONJ cana-1013 116 30 hence hence ADV cana-1013 116 31 ๐ถ๐‘Ž๐‘ฆ3(๐บ ๐ถ๐‘Ž๐‘ฆ3(๐บ PROPN cana-1013 116 32 , , PUNCT cana-1013 116 33 ๐‘† ๐‘† PROPN cana-1013 116 34 ) ) PUNCT cana-1013 116 35 = = PUNCT cana-1013 116 36 ๐พ8,8 ๐พ8,8 ADJ cana-1013 116 37 โˆช โˆช ADJ cana-1013 116 38 48๐‘ƒ1 48๐‘ƒ1 NUM cana-1013 116 39 . . PUNCT cana-1013 117 1 the the DET cana-1013 117 2 graph graph NOUN cana-1013 117 3 of of ADP cana-1013 117 4 ๐ถ๐‘Ž๐‘ฆ3(๐บ ๐ถ๐‘Ž๐‘ฆ3(๐บ PROPN cana-1013 117 5 , , PUNCT cana-1013 117 6 ๐‘† ๐‘† PROPN cana-1013 117 7 ) ) PUNCT cana-1013 117 8 is be AUX cana-1013 117 9 shown show VERB cana-1013 117 10 below below ADP cana-1013 117 11 . . PUNCT cana-1013 118 1 ๏‚ก ๏‚ก PUNCT cana-1013 118 2 communications communication NOUN cana-1013 118 3 on on ADP cana-1013 118 4 applied apply VERB cana-1013 118 5 nonlinear nonlinear ADJ cana-1013 118 6 analysis analysis NOUN cana-1013 118 7 issn issn NOUN cana-1013 118 8 : : PUNCT cana-1013 118 9 1074 1074 NUM cana-1013 118 10 - - PUNCT cana-1013 118 11 133x 133x NUM cana-1013 118 12 vol vol NOUN cana-1013 118 13 31 31 NUM cana-1013 118 14 no no NOUN cana-1013 118 15 . . PUNCT cana-1013 119 1 5s 5s NUM cana-1013 119 2 ( ( PUNCT cana-1013 119 3 2024 2024 NUM cana-1013 119 4 ) ) PUNCT cana-1013 119 5 201 201 NUM cana-1013 119 6 https://internationalpubls.com https://internationalpubls.com X cana-1013 119 7 a a DET cana-1013 119 8 component component NOUN cana-1013 119 9 of of ADP cana-1013 119 10 graph graph NOUN cana-1013 119 11 ๐ถ๐‘Ž๐‘ฆ3(๐บ ๐ถ๐‘Ž๐‘ฆ3(๐บ PROPN cana-1013 119 12 , , PUNCT cana-1013 119 13 ๐‘† ๐‘† PROPN cana-1013 119 14 ) ) PUNCT cana-1013 119 15 of of ADP cana-1013 119 16 ๐‘ƒ2 ๐‘ƒ2 ADJ cana-1013 119 17 ร— ร— PROPN cana-1013 119 18 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 119 19 in in ADP cana-1013 119 20 the the DET cana-1013 119 21 next next ADJ cana-1013 119 22 theorem theorem NOUN cana-1013 119 23 , , PUNCT cana-1013 119 24 we we PRON cana-1013 119 25 generalized generalize VERB cana-1013 119 26 the the DET cana-1013 119 27 cayley cayley ADJ cana-1013 119 28 graph graph NOUN cana-1013 119 29 for for ADP cana-1013 119 30 each each DET cana-1013 119 31 ๐‘š ๐‘š PROPN cana-1013 119 32 โ‰ฅ โ‰ฅ NUM cana-1013 119 33 2 2 NUM cana-1013 119 34 when when SCONJ cana-1013 119 35 the the DET cana-1013 119 36 common common ADJ cana-1013 119 37 cayley cayley ADJ cana-1013 119 38 graph graph NOUN cana-1013 119 39 is be AUX cana-1013 119 40 ๐‘ƒ2 ๐‘ƒ2 ADJ cana-1013 119 41 ร— ร— PROPN cana-1013 119 42 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 119 43 . . PUNCT cana-1013 120 1 theorem theorem PROPN cana-1013 120 2 7 7 NUM cana-1013 120 3 . . PUNCT cana-1013 121 1 let let VERB cana-1013 121 2 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ NUM cana-1013 121 3 , , PUNCT cana-1013 121 4 ๐‘† ๐‘† PROPN cana-1013 121 5 ) ) PUNCT cana-1013 122 1 = = SYM cana-1013 122 2 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 122 3 ร— ร— PROPN cana-1013 122 4 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 122 5 , , PUNCT cana-1013 122 6 then then ADV cana-1013 122 7 the the DET cana-1013 122 8 generalized generalized ADJ cana-1013 122 9 cayley cayley NOUN cana-1013 122 10 graph graph NOUN cana-1013 122 11 ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ NOUN cana-1013 122 12 , , PUNCT cana-1013 122 13 ๐‘† ๐‘† PROPN cana-1013 122 14 ) ) PUNCT cana-1013 122 15 is be AUX cana-1013 122 16 the the DET cana-1013 122 17 graph graph NOUN cana-1013 122 18 ๐พ2๐‘š,2๐‘š ๐พ2๐‘š,2๐‘š ADP cana-1013 122 19 โˆช โˆช ADJ cana-1013 122 20 ( ( PUNCT cana-1013 122 21 2๐‘š+1(2๐‘šโˆ’1 2๐‘š+1(2๐‘šโˆ’1 NUM cana-1013 122 22 โˆ’ โˆ’ NUM cana-1013 122 23 1))๐‘ƒ1 1))๐‘ƒ1 NUM cana-1013 122 24 for for ADP cana-1013 122 25 all all DET cana-1013 122 26 ๐‘š ๐‘š PRON cana-1013 122 27 โ‰ฅ โ‰ฅ NUM cana-1013 122 28 2 2 NUM cana-1013 122 29 . . PUNCT cana-1013 123 1 proof proof NOUN cana-1013 123 2 : : PUNCT cana-1013 123 3 suppose suppose VERB cana-1013 123 4 that that SCONJ cana-1013 123 5 ๐บ1 ๐บ1 PROPN cana-1013 123 6 = = SYM cana-1013 123 7 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 123 8 with with ADP cana-1013 123 9 vertex vertex NOUN cana-1013 123 10 set set NOUN cana-1013 123 11 { { PUNCT cana-1013 123 12 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 123 13 , , PUNCT cana-1013 123 14 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 123 15 } } PUNCT cana-1013 123 16 and and CCONJ cana-1013 123 17 ๐บ2 ๐บ2 ADJ cana-1013 123 18 = = SYM cana-1013 123 19 ๐‘ƒ2 ๐‘ƒ2 NOUN cana-1013 123 20 with with ADP cana-1013 123 21 vertex vertex NOUN cana-1013 123 22 set set NOUN cana-1013 123 23 { { PUNCT cana-1013 123 24 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 123 25 , , PUNCT cana-1013 123 26 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 123 27 } } PUNCT cana-1013 123 28 and and CCONJ cana-1013 123 29 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 123 30 , , PUNCT cana-1013 123 31 ๐‘† ๐‘† PROPN cana-1013 123 32 ) ) PUNCT cana-1013 123 33 = = SYM cana-1013 123 34 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 123 35 ร— ร— PROPN cana-1013 123 36 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 123 37 . . PUNCT cana-1013 124 1 so so ADV cana-1013 124 2 , , PUNCT cana-1013 124 3 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 124 4 , , PUNCT cana-1013 124 5 ๐‘† ๐‘† PROPN cana-1013 124 6 ) ) PUNCT cana-1013 124 7 is be AUX cana-1013 124 8 a a DET cana-1013 124 9 cycle cycle NOUN cana-1013 124 10 of of ADP cana-1013 124 11 lenฯˆth lenฯˆth NOUN cana-1013 124 12 4 4 NUM cana-1013 124 13 and and CCONJ cana-1013 124 14 its its PRON cana-1013 124 15 vertex vertex NOUN cana-1013 124 16 set set NOUN cana-1013 124 17 is be AUX cana-1013 124 18 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ NUM cana-1013 124 19 , , PUNCT cana-1013 124 20 ๐‘† ๐‘† PROPN cana-1013 124 21 ) ) PUNCT cana-1013 124 22 ) ) PUNCT cana-1013 125 1 = = PRON cana-1013 125 2 { { PUNCT cana-1013 125 3 ( ( PUNCT cana-1013 125 4 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 125 5 , , PUNCT cana-1013 125 6 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 125 7 ) ) PUNCT cana-1013 125 8 , , PUNCT cana-1013 125 9 ( ( PUNCT cana-1013 125 10 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 125 11 , , PUNCT cana-1013 125 12 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 125 13 ) ) PUNCT cana-1013 125 14 , , PUNCT cana-1013 125 15 ( ( PUNCT cana-1013 125 16 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 125 17 , , PUNCT cana-1013 125 18 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 125 19 ) ) PUNCT cana-1013 125 20 , , PUNCT cana-1013 125 21 ( ( PUNCT cana-1013 125 22 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 125 23 , , PUNCT cana-1013 125 24 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 125 25 ) ) PUNCT cana-1013 125 26 } } PUNCT cana-1013 125 27 and and CCONJ cana-1013 125 28 the the DET cana-1013 125 29 set set NOUN cana-1013 125 30 of of ADP cana-1013 125 31 edฯˆes edฯˆes PROPN cana-1013 125 32 is be AUX cana-1013 125 33 ( ( PUNCT cana-1013 125 34 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 125 35 , , PUNCT cana-1013 125 36 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 125 37 ) ) PUNCT cana-1013 125 38 โˆ’ โˆ’ PROPN cana-1013 126 1 ( ( PUNCT cana-1013 126 2 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 126 3 , , PUNCT cana-1013 126 4 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 126 5 ) ) PUNCT cana-1013 127 1 โˆ’ โˆ’ PROPN cana-1013 127 2 ( ( PUNCT cana-1013 127 3 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 127 4 , , PUNCT cana-1013 127 5 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 127 6 ) ) PUNCT cana-1013 127 7 โˆ’ โˆ’ PROPN cana-1013 127 8 ( ( PUNCT cana-1013 127 9 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 127 10 , , PUNCT cana-1013 127 11 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 127 12 ) ) PUNCT cana-1013 128 1 โˆ’ โˆ’ PROPN cana-1013 128 2 ( ( PUNCT cana-1013 128 3 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 128 4 , , PUNCT cana-1013 128 5 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 128 6 ) ) PUNCT cana-1013 128 7 . . PUNCT cana-1013 129 1 so so ADV cana-1013 129 2 , , PUNCT cana-1013 129 3 ๐‘‰ ๐‘‰ PROPN cana-1013 129 4 = = PUNCT cana-1013 129 5 ๐‘‰(๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ NOUN cana-1013 129 6 , , PUNCT cana-1013 129 7 ๐‘† ๐‘† PROPN cana-1013 129 8 ) ) PUNCT cana-1013 129 9 ) ) PUNCT cana-1013 130 1 = = PRON cana-1013 130 2 { { PUNCT cana-1013 131 1 [ [ X cana-1013 131 2 ๐‘Ž1 ๐‘Ž1 X cana-1013 131 3 , , PUNCT cana-1013 131 4 ๐‘Ž2 ๐‘Ž2 PROPN cana-1013 131 5 , , PUNCT cana-1013 131 6 โ€ฆ โ€ฆ PUNCT cana-1013 131 7 , , PUNCT cana-1013 131 8 ๐‘Ž๐‘š]๐‘ก ๐‘Ž๐‘š]๐‘ก NOUN cana-1013 131 9 |๐‘Ž1 |๐‘Ž1 NOUN cana-1013 131 10 , , PUNCT cana-1013 131 11 ๐‘Ž2 ๐‘Ž2 PROPN cana-1013 131 12 , , PUNCT cana-1013 131 13 โ‹ฏ โ‹ฏ PROPN cana-1013 131 14 , , PUNCT cana-1013 131 15 ๐‘Ž๐‘š ๐‘Ž๐‘š PROPN cana-1013 131 16 โˆˆ โˆˆ PROPN cana-1013 131 17 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 131 18 , , PUNCT cana-1013 131 19 ๐‘† ๐‘† PROPN cana-1013 131 20 ) ) PUNCT cana-1013 131 21 ) ) PUNCT cana-1013 131 22 } } PUNCT cana-1013 131 23 . . PUNCT cana-1013 132 1 therefore therefore ADV cana-1013 132 2 , , PUNCT cana-1013 132 3 |๐‘‰(๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ |๐‘‰(๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ PROPN cana-1013 132 4 , , PUNCT cana-1013 132 5 ๐‘†))| ๐‘†))| VERB cana-1013 132 6 = = SYM cana-1013 132 7 4๐‘š. 4๐‘š. PRON cana-1013 132 8 consider consider VERB cana-1013 132 9 the the DET cana-1013 132 10 subsets subset NOUN cana-1013 132 11 ๐ด ๐ด PROPN cana-1013 132 12 and and CCONJ cana-1013 132 13 ๐ต ๐ต NOUN cana-1013 132 14 of of ADP cana-1013 132 15 ๐‘‰ ๐‘‰ PROPN cana-1013 132 16 as as SCONJ cana-1013 132 17 follows follow VERB cana-1013 132 18 : : PUNCT cana-1013 132 19 ๐ด ๐ด PROPN cana-1013 132 20 = = PUNCT cana-1013 132 21 { { PUNCT cana-1013 132 22 [ [ X cana-1013 132 23 ๐‘Ž1 ๐‘Ž1 X cana-1013 132 24 , , PUNCT cana-1013 132 25 ๐‘Ž2 ๐‘Ž2 PROPN cana-1013 132 26 , , PUNCT cana-1013 132 27 โ€ฆ โ€ฆ PUNCT cana-1013 132 28 , , PUNCT cana-1013 132 29 ๐‘Ž๐‘š]๐‘ก ๐‘Ž๐‘š]๐‘ก ADJ cana-1013 132 30 โˆถ โˆถ NOUN cana-1013 132 31 ๐‘Ž๐‘– ๐‘Ž๐‘– PRON cana-1013 132 32 โˆˆ โˆˆ PROPN cana-1013 132 33 { { PUNCT cana-1013 132 34 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 132 35 , , PUNCT cana-1013 132 36 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 132 37 } } PUNCT cana-1013 132 38 , , PUNCT cana-1013 132 39 ๐‘– ๐‘– SYM cana-1013 132 40 = = SYM cana-1013 132 41 1,2 1,2 NUM cana-1013 132 42 , , PUNCT cana-1013 132 43 โ€ฆ โ€ฆ PUNCT cana-1013 132 44 , , PUNCT cana-1013 132 45 ๐‘š ๐‘š NOUN cana-1013 132 46 } } PUNCT cana-1013 132 47 and and CCONJ cana-1013 132 48 ๐ต ๐ต NOUN cana-1013 132 49 = = NOUN cana-1013 132 50 { { PUNCT cana-1013 133 1 [ [ X cana-1013 133 2 ๐‘Ž1 ๐‘Ž1 X cana-1013 133 3 , , PUNCT cana-1013 133 4 ๐‘Ž2 ๐‘Ž2 PROPN cana-1013 133 5 , , PUNCT cana-1013 133 6 โ€ฆ โ€ฆ PUNCT cana-1013 133 7 , , PUNCT cana-1013 133 8 ๐‘Ž๐‘š]๐‘ก ๐‘Ž๐‘š]๐‘ก ADJ cana-1013 133 9 โˆถ โˆถ NOUN cana-1013 133 10 ๐‘Ž๐‘– ๐‘Ž๐‘– PRON cana-1013 133 11 โˆˆ โˆˆ PROPN cana-1013 133 12 { { PUNCT cana-1013 133 13 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 133 14 , , PUNCT cana-1013 133 15 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 133 16 } } PUNCT cana-1013 133 17 , , PUNCT cana-1013 133 18 ๐‘– ๐‘– SYM cana-1013 133 19 = = SYM cana-1013 133 20 1,2 1,2 NUM cana-1013 133 21 , , PUNCT cana-1013 133 22 โ€ฆ โ€ฆ PUNCT cana-1013 133 23 , , PUNCT cana-1013 133 24 ๐‘š ๐‘š NOUN cana-1013 133 25 } } PUNCT cana-1013 133 26 . . PUNCT cana-1013 134 1 we we PRON cana-1013 134 2 can can AUX cana-1013 134 3 see see VERB cana-1013 134 4 that that SCONJ cana-1013 134 5 a a PRON cana-1013 134 6 and and CCONJ cana-1013 134 7 b b NOUN cana-1013 134 8 are be AUX cana-1013 134 9 independent independent ADJ cana-1013 134 10 sets set NOUN cana-1013 134 11 and and CCONJ cana-1013 134 12 that that SCONJ cana-1013 134 13 every every DET cana-1013 134 14 vertex vertex NOUN cana-1013 134 15 from from ADP cana-1013 134 16 one one NUM cana-1013 134 17 is be AUX cana-1013 134 18 adjacent adjacent ADJ cana-1013 134 19 to to ADP cana-1013 134 20 another another DET cana-1013 134 21 set set NOUN cana-1013 134 22 using use VERB cana-1013 134 23 the the DET cana-1013 134 24 same same ADJ cana-1013 134 25 technique technique NOUN cana-1013 134 26 used use VERB cana-1013 134 27 in in ADP cana-1013 134 28 the the DET cana-1013 134 29 demonstration demonstration NOUN cana-1013 134 30 of of ADP cana-1013 134 31 the the DET cana-1013 134 32 preceding precede VERB cana-1013 134 33 lemma lemma PROPN cana-1013 134 34 . . PUNCT cana-1013 135 1 as as ADP cana-1013 135 2 a a DET cana-1013 135 3 result result NOUN cana-1013 135 4 , , PUNCT cana-1013 135 5 the the DET cana-1013 135 6 entire entire ADJ cana-1013 135 7 bipartite bipartite NOUN cana-1013 135 8 network network NOUN cana-1013 135 9 is be AUX cana-1013 135 10 induced induce VERB cana-1013 135 11 by by ADP cana-1013 135 12 the the DET cana-1013 135 13 union union NOUN cana-1013 135 14 of of ADP cana-1013 135 15 disjoint disjoint NOUN cana-1013 135 16 sets set NOUN cana-1013 135 17 aโˆช aโˆช ADP cana-1013 135 18 ฬ‡b ฬ‡b ADJ cana-1013 135 19 , , PUNCT cana-1013 135 20 and and CCONJ cana-1013 135 21 the the DET cana-1013 135 22 remaining remain VERB cana-1013 135 23 vertices vertex NOUN cana-1013 135 24 are be AUX cana-1013 135 25 all all PRON cana-1013 135 26 isolated isolated ADJ cana-1013 135 27 vertices vertex NOUN cana-1013 135 28 . . PUNCT cana-1013 136 1 consequently consequently ADV cana-1013 136 2 , , PUNCT cana-1013 136 3 ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ ๐ถ๐‘Ž๐‘ฆ๐‘š(๐บ PROPN cana-1013 136 4 , , PUNCT cana-1013 136 5 ๐‘† ๐‘† PROPN cana-1013 136 6 ) ) PUNCT cana-1013 136 7 = = NOUN cana-1013 136 8 ๐พ2๐‘š,2๐‘š ๐พ2๐‘š,2๐‘š ADP cana-1013 136 9 โˆช โˆช X cana-1013 136 10 ( ( PUNCT cana-1013 136 11 2๐‘š+1(2๐‘šโˆ’1 2๐‘š+1(2๐‘šโˆ’1 NUM cana-1013 136 12 โˆ’ โˆ’ NUM cana-1013 136 13 1))๐‘ƒ1 1))๐‘ƒ1 NUM cana-1013 136 14 for for ADP cana-1013 136 15 all all DET cana-1013 136 16 ๐‘š ๐‘š PRON cana-1013 136 17 โ‰ฅ โ‰ฅ NUM cana-1013 136 18 2 2 NUM cana-1013 136 19 . . PUNCT cana-1013 136 20 ๏‚ก ๏‚ก PUNCT cana-1013 136 21 in in ADP cana-1013 136 22 the the DET cana-1013 136 23 next next ADJ cana-1013 136 24 lemma lemma PROPN cana-1013 136 25 , , PUNCT cana-1013 136 26 we we PRON cana-1013 136 27 find find VERB cana-1013 136 28 the the DET cana-1013 136 29 generalized generalized ADJ cana-1013 136 30 cayley cayley ADJ cana-1013 136 31 graph graph NOUN cana-1013 136 32 for for ADP cana-1013 136 33 the the DET cana-1013 136 34 special special ADJ cana-1013 136 35 case case NOUN cana-1013 136 36 ๐‘› ๐‘› PRON cana-1013 136 37 = = SYM cana-1013 136 38 2 2 NUM cana-1013 136 39 when when SCONJ cana-1013 136 40 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 136 41 , , PUNCT cana-1013 136 42 ๐‘† ๐‘† PROPN cana-1013 136 43 ) ) PUNCT cana-1013 136 44 = = SYM cana-1013 137 1 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 137 2 ร— ร— PROPN cana-1013 137 3 ๐ถ3 ๐ถ3 PROPN cana-1013 137 4 . . PUNCT cana-1013 138 1 theorem theorem VERB cana-1013 138 2 8 8 NUM cana-1013 138 3 . . PUNCT cana-1013 139 1 let let VERB cana-1013 139 2 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ NUM cana-1013 139 3 , , PUNCT cana-1013 139 4 ๐‘† ๐‘† PROPN cana-1013 139 5 ) ) PUNCT cana-1013 139 6 = = SYM cana-1013 140 1 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 140 2 ร— ร— PROPN cana-1013 140 3 ๐ถ3 ๐ถ3 NOUN cana-1013 140 4 , , PUNCT cana-1013 140 5 then then ADV cana-1013 140 6 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 140 7 , , PUNCT cana-1013 140 8 ๐‘† ๐‘† PROPN cana-1013 140 9 ) ) PUNCT cana-1013 140 10 has have VERB cana-1013 140 11 ( ( PUNCT cana-1013 140 12 ( ( PUNCT cana-1013 140 13 ๐‘ƒ2 ๐‘ƒ2 DET cana-1013 140 14 ร— ร— PROPN cana-1013 140 15 ๐ถ3 ๐ถ3 PROPN cana-1013 140 16 ) ) PUNCT cana-1013 140 17 โˆ˜ โˆ˜ NOUN cana-1013 140 18 2๐‘ƒ1 2๐‘ƒ1 NUM cana-1013 140 19 ) ) PUNCT cana-1013 140 20 โˆช โˆช X cana-1013 140 21 18๐‘ƒ1 18๐‘ƒ1 NUM cana-1013 140 22 as as ADP cana-1013 140 23 a a DET cana-1013 140 24 subgraph subgraph NOUN cana-1013 140 25 . . PUNCT cana-1013 141 1 proof proof NOUN cana-1013 141 2 : : PUNCT cana-1013 141 3 suppose suppose VERB cana-1013 141 4 that that SCONJ cana-1013 141 5 ๐บ1 ๐บ1 PROPN cana-1013 141 6 = = SYM cana-1013 141 7 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 141 8 with with ADP cana-1013 141 9 vertex vertex NOUN cana-1013 141 10 set set NOUN cana-1013 141 11 { { PUNCT cana-1013 141 12 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 141 13 , , PUNCT cana-1013 141 14 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 141 15 } } PUNCT cana-1013 141 16 and and CCONJ cana-1013 141 17 ๐บ2 ๐บ2 ADJ cana-1013 141 18 = = SYM cana-1013 141 19 ๐ถ3 ๐ถ3 NOUN cana-1013 141 20 with with ADP cana-1013 141 21 vertex vertex NOUN cana-1013 141 22 set set NOUN cana-1013 141 23 { { PUNCT cana-1013 141 24 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 141 25 , , PUNCT cana-1013 141 26 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 141 27 , , PUNCT cana-1013 141 28 ๐‘ฅ5 ๐‘ฅ5 PROPN cana-1013 141 29 } } PUNCT cana-1013 141 30 and and CCONJ cana-1013 141 31 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 141 32 , , PUNCT cana-1013 141 33 ๐‘† ๐‘† PROPN cana-1013 141 34 ) ) PUNCT cana-1013 141 35 = = SYM cana-1013 142 1 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 142 2 ร— ร— PROPN cana-1013 142 3 ๐ถ3 ๐ถ3 NOUN cana-1013 142 4 . . PUNCT cana-1013 143 1 so so ADV cana-1013 143 2 , , PUNCT cana-1013 143 3 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 143 4 , , PUNCT cana-1013 143 5 ๐‘† ๐‘† PROPN cana-1013 143 6 ) ) PUNCT cana-1013 143 7 ) ) PUNCT cana-1013 144 1 = = PRON cana-1013 144 2 { { PUNCT cana-1013 144 3 ( ( PUNCT cana-1013 144 4 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 144 5 , , PUNCT cana-1013 144 6 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 144 7 ) ) PUNCT cana-1013 144 8 , , PUNCT cana-1013 144 9 ( ( PUNCT cana-1013 144 10 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 144 11 , , PUNCT cana-1013 144 12 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 144 13 ) ) PUNCT cana-1013 144 14 , , PUNCT cana-1013 144 15 ( ( PUNCT cana-1013 144 16 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 144 17 , , PUNCT cana-1013 144 18 ๐‘ฅ5 ๐‘ฅ5 PROPN cana-1013 144 19 ) ) PUNCT cana-1013 144 20 , , PUNCT cana-1013 144 21 ( ( PUNCT cana-1013 144 22 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 144 23 , , PUNCT cana-1013 144 24 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 144 25 , , PUNCT cana-1013 144 26 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 144 27 ) ) PUNCT cana-1013 144 28 , , PUNCT cana-1013 144 29 ( ( PUNCT cana-1013 144 30 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 144 31 , , PUNCT cana-1013 144 32 ๐‘ฅ5 ๐‘ฅ5 PROPN cana-1013 144 33 ) ) PUNCT cana-1013 144 34 } } PUNCT cana-1013 144 35 and and CCONJ cana-1013 144 36 |๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ |๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 144 37 , , PUNCT cana-1013 144 38 ๐‘†)| ๐‘†)| VERB cana-1013 144 39 = = SYM cana-1013 144 40 6 6 NUM cana-1013 144 41 and and CCONJ cana-1013 144 42 ๐ธ(๐ถ๐‘Ž๐‘ฆ(๐บ ๐ธ(๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 144 43 , , PUNCT cana-1013 144 44 ๐‘† ๐‘† PROPN cana-1013 144 45 ) ) PUNCT cana-1013 144 46 ) ) PUNCT cana-1013 145 1 = = PRON cana-1013 145 2 { { PUNCT cana-1013 145 3 ( ( PUNCT cana-1013 145 4 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 145 5 , , PUNCT cana-1013 145 6 ๐‘ฅ3)(๐‘ฅ1 ๐‘ฅ3)(๐‘ฅ1 ADV cana-1013 145 7 , , PUNCT cana-1013 145 8 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 145 9 ) ) PUNCT cana-1013 145 10 , , PUNCT cana-1013 145 11 ( ( PUNCT cana-1013 145 12 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 145 13 , , PUNCT cana-1013 145 14 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 145 15 , , PUNCT cana-1013 145 16 ๐‘ฅ3 ๐‘ฅ3 NOUN cana-1013 145 17 ) ) PUNCT cana-1013 145 18 , , PUNCT cana-1013 145 19 ( ( PUNCT cana-1013 145 20 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 145 21 , , PUNCT cana-1013 145 22 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 145 23 , , PUNCT cana-1013 145 24 ๐‘ฅ4 ๐‘ฅ4 ADJ cana-1013 145 25 ) ) PUNCT cana-1013 145 26 , , PUNCT cana-1013 145 27 ( ( PUNCT cana-1013 145 28 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 145 29 , , PUNCT cana-1013 145 30 ๐‘ฅ4)(๐‘ฅ1 ๐‘ฅ4)(๐‘ฅ1 ADJ cana-1013 145 31 , , PUNCT cana-1013 145 32 ๐‘ฅ4 ๐‘ฅ4 NOUN cana-1013 145 33 ) ) PUNCT cana-1013 145 34 , , PUNCT cana-1013 145 35 ( ( PUNCT cana-1013 145 36 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 145 37 , , PUNCT cana-1013 145 38 ๐‘ฅ4)(๐‘ฅ2 ๐‘ฅ4)(๐‘ฅ2 NUM cana-1013 145 39 , , PUNCT cana-1013 145 40 ๐‘ฅ5 ๐‘ฅ5 PROPN cana-1013 145 41 ) ) PUNCT cana-1013 145 42 , , PUNCT cana-1013 145 43 ( ( PUNCT cana-1013 145 44 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 145 45 , , PUNCT cana-1013 145 46 ๐‘ฅ3)(๐‘ฅ2 ๐‘ฅ3)(๐‘ฅ2 NUM cana-1013 145 47 , , PUNCT cana-1013 145 48 ๐‘ฅ5 ๐‘ฅ5 PROPN cana-1013 145 49 ) ) PUNCT cana-1013 145 50 , , PUNCT cana-1013 145 51 ( ( PUNCT cana-1013 145 52 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 145 53 , , PUNCT cana-1013 145 54 ๐‘ฅ3)(๐‘ฅ1 ๐‘ฅ3)(๐‘ฅ1 ADV cana-1013 145 55 , , PUNCT cana-1013 145 56 ๐‘ฅ5 ๐‘ฅ5 PROPN cana-1013 145 57 ) ) PUNCT cana-1013 145 58 , , PUNCT cana-1013 145 59 , , PUNCT cana-1013 145 60 ( ( PUNCT cana-1013 145 61 ๐‘ฅ1 ๐‘ฅ1 NOUN cana-1013 145 62 , , PUNCT cana-1013 145 63 ๐‘ฅ4)(๐‘ฅ1 ๐‘ฅ4)(๐‘ฅ1 ADJ cana-1013 145 64 , , PUNCT cana-1013 145 65 ๐‘ฅ5 ๐‘ฅ5 PROPN cana-1013 145 66 ) ) PUNCT cana-1013 145 67 , , PUNCT cana-1013 145 68 ( ( PUNCT cana-1013 145 69 ๐‘ฅ2 ๐‘ฅ2 NOUN cana-1013 145 70 , , PUNCT cana-1013 145 71 ๐‘ฅ5)(๐‘ฅ1 ๐‘ฅ5)(๐‘ฅ1 NOUN cana-1013 145 72 , , PUNCT cana-1013 145 73 ๐‘ฅ5)t ๐‘ฅ5)t NOUN cana-1013 145 74 he he PRON cana-1013 145 75 graph graph VERB cana-1013 145 76 ๐ถ๐‘Ž๐‘ฆ(๐บ ๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 145 77 , , PUNCT cana-1013 145 78 ๐‘† ๐‘† PROPN cana-1013 145 79 ) ) PUNCT cana-1013 145 80 = = NOUN cana-1013 146 1 ๐‘ƒ2 ๐‘ƒ2 PROPN cana-1013 146 2 ร— ร— PROPN cana-1013 146 3 ๐ถ3 ๐ถ3 NOUN cana-1013 146 4 shown show VERB cana-1013 146 5 in in ADP cana-1013 146 6 below below ADV cana-1013 146 7 . . PUNCT cana-1013 147 1 communications communication NOUN cana-1013 147 2 on on ADP cana-1013 147 3 applied apply VERB cana-1013 147 4 nonlinear nonlinear ADJ cana-1013 147 5 analysis analysis NOUN cana-1013 147 6 issn issn NOUN cana-1013 147 7 : : PUNCT cana-1013 147 8 1074 1074 NUM cana-1013 147 9 - - PUNCT cana-1013 147 10 133x 133x NUM cana-1013 147 11 vol vol NOUN cana-1013 147 12 31 31 NUM cana-1013 147 13 no no NOUN cana-1013 147 14 . . PUNCT cana-1013 148 1 5s 5s NUM cana-1013 148 2 ( ( PUNCT cana-1013 148 3 2024 2024 NUM cana-1013 148 4 ) ) PUNCT cana-1013 148 5 202 202 NUM cana-1013 148 6 https://internationalpubls.com https://internationalpubls.com X cana-1013 149 1 then then ADV cana-1013 149 2 , , PUNCT cana-1013 149 3 we we PRON cana-1013 149 4 have have VERB cana-1013 149 5 62 62 NUM cana-1013 149 6 = = SYM cana-1013 149 7 36 36 NUM cana-1013 149 8 vertices vertex NOUN cana-1013 149 9 in in ADP cana-1013 149 10 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 149 11 , , PUNCT cana-1013 149 12 ๐‘† ๐‘† PROPN cana-1013 149 13 ) ) PUNCT cana-1013 149 14 and and CCONJ cana-1013 149 15 ๐‘‰(๐ถ๐‘Ž๐‘ฆ2(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ2(๐บ NOUN cana-1013 149 16 , , PUNCT cana-1013 149 17 ๐‘† ๐‘† PROPN cana-1013 149 18 ) ) PUNCT cana-1013 149 19 ) ) PUNCT cana-1013 150 1 = = PRON cana-1013 150 2 { { PUNCT cana-1013 150 3 [ [ PUNCT cana-1013 150 4 ๐‘Ž ๐‘Ž PROPN cana-1013 150 5 ๐‘ ๐‘ NOUN cana-1013 150 6 ] ] PUNCT cana-1013 150 7 |๐‘Ž |๐‘Ž NOUN cana-1013 150 8 , , PUNCT cana-1013 150 9 ๐‘ ๐‘ PROPN cana-1013 150 10 โˆˆ โˆˆ PROPN cana-1013 150 11 ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ ๐‘‰(๐ถ๐‘Ž๐‘ฆ(๐บ PROPN cana-1013 150 12 , , PUNCT cana-1013 150 13 ๐‘† ๐‘† PROPN cana-1013 150 14 ) ) PUNCT cana-1013 150 15 ) ) PUNCT cana-1013 150 16 } } PUNCT cana-1013 150 17 = = PUNCT cana-1013 150 18 therefore therefore ADV cana-1013 150 19 , , PUNCT cana-1013 150 20 each each DET cana-1013 150 21 vertex vertex NOUN cana-1013 150 22 [ [ PUNCT cana-1013 150 23 ( ( PUNCT cana-1013 150 24 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 150 25 , , PUNCT cana-1013 150 26 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 150 27 ) ) PUNCT cana-1013 150 28 ( ( PUNCT cana-1013 150 29 ๐‘ฅ๐‘– ๐‘ฅ๐‘– PROPN cana-1013 150 30 , , PUNCT cana-1013 150 31 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 150 32 ) ) PUNCT cana-1013 150 33 ] ] PUNCT cana-1013 150 34 has have VERB cana-1013 150 35 deฯˆree deฯˆree NOUN cana-1013 150 36 4 4 NUM cana-1013 150 37 and and CCONJ cana-1013 150 38 it it PRON cana-1013 150 39 is be AUX cana-1013 150 40 adjacent adjacent ADJ cana-1013 150 41 to to ADP cana-1013 150 42 the the DET cana-1013 150 43 vertices vertex NOUN cana-1013 150 44 [ [ PUNCT cana-1013 150 45 ( ( PUNCT cana-1013 150 46 ๐‘ฅ๐‘– ๐‘ฅ๐‘– X cana-1013 150 47 , , PUNCT cana-1013 150 48 ๐‘ฅ๐‘—+1 ๐‘ฅ๐‘—+1 NOUN cana-1013 150 49 ) ) PUNCT cana-1013 150 50 ( ( PUNCT cana-1013 150 51 ๐‘ฅ๐‘–+1 ๐‘ฅ๐‘–+1 PROPN cana-1013 150 52 , , PUNCT cana-1013 150 53 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 150 54 ) ) PUNCT cana-1013 150 55 ] ] PUNCT cana-1013 150 56 and and CCONJ cana-1013 150 57 [ [ PUNCT cana-1013 150 58 ( ( PUNCT cana-1013 150 59 ๐‘ฅ๐‘–+1 ๐‘ฅ๐‘–+1 ADJ cana-1013 150 60 , , PUNCT cana-1013 150 61 ๐‘ฅ๐‘— ๐‘ฅ๐‘— PROPN cana-1013 150 62 ) ) PUNCT cana-1013 150 63 ( ( PUNCT cana-1013 150 64 ๐‘ฅ๐‘– ๐‘ฅ๐‘– NOUN cana-1013 150 65 , , PUNCT cana-1013 150 66 ๐‘ฅ๐‘—+1 ๐‘ฅ๐‘—+1 NOUN cana-1013 150 67 ) ) PUNCT cana-1013 150 68 ] ] PUNCT cana-1013 150 69 and and CCONJ cana-1013 150 70 [ [ PUNCT cana-1013 150 71 ( ( PUNCT cana-1013 150 72 ๐‘ฅ๐‘– ๐‘ฅ๐‘– NOUN cana-1013 150 73 , , PUNCT cana-1013 150 74 ๐‘ฅ๐‘—+1 ๐‘ฅ๐‘—+1 NOUN cana-1013 150 75 ) ) PUNCT cana-1013 150 76 ( ( PUNCT cana-1013 150 77 ๐‘ฅ๐‘– ๐‘ฅ๐‘– NOUN cana-1013 150 78 , , PUNCT cana-1013 150 79 ๐‘ฅ๐‘—+1 ๐‘ฅ๐‘—+1 NOUN cana-1013 150 80 ) ) PUNCT cana-1013 150 81 ] ] PUNCT cana-1013 150 82 and and CCONJ cana-1013 150 83 [ [ PUNCT cana-1013 150 84 ( ( PUNCT cana-1013 150 85 ๐‘ฅ๐‘– ๐‘ฅ๐‘– X cana-1013 150 86 , , PUNCT cana-1013 150 87 ๐‘ฅ๐‘—+2 ๐‘ฅ๐‘—+2 NOUN cana-1013 150 88 ) ) PUNCT cana-1013 150 89 ( ( PUNCT cana-1013 150 90 ๐‘ฅ๐‘– ๐‘ฅ๐‘– X cana-1013 150 91 , , PUNCT cana-1013 150 92 ๐‘ฅ๐‘—+2 ๐‘ฅ๐‘—+2 X cana-1013 150 93 ) ) PUNCT cana-1013 150 94 ] ] PUNCT cana-1013 150 95 . . PUNCT cana-1013 151 1 the the DET cana-1013 151 2 other other ADJ cana-1013 151 3 vertices vertex NOUN cana-1013 151 4 are be AUX cana-1013 151 5 isolated isolate VERB cana-1013 151 6 . . PUNCT cana-1013 152 1 the the DET cana-1013 152 2 graph graph NOUN cana-1013 152 3 ๐ถ๐‘Ž๐‘ฆ2(๐บ ๐ถ๐‘Ž๐‘ฆ2(๐บ PROPN cana-1013 152 4 , , PUNCT cana-1013 152 5 ๐‘† ๐‘† PROPN cana-1013 152 6 ) ) PUNCT cana-1013 152 7 is be AUX cana-1013 152 8 shown show VERB cana-1013 152 9 in in ADP cana-1013 152 10 below below ADV cana-1013 152 11 . . PUNCT cana-1013 153 1 ๏‚ก ๏‚ก PUNCT cana-1013 153 2 communications communication NOUN cana-1013 153 3 on on ADP cana-1013 153 4 applied apply VERB cana-1013 153 5 nonlinear nonlinear ADJ cana-1013 153 6 analysis analysis NOUN cana-1013 153 7 issn issn NOUN cana-1013 153 8 : : PUNCT cana-1013 153 9 1074 1074 NUM cana-1013 153 10 - - PUNCT cana-1013 153 11 133x 133x NUM cana-1013 153 12 vol vol NOUN cana-1013 153 13 31 31 NUM cana-1013 153 14 no no NOUN cana-1013 153 15 . . PUNCT cana-1013 154 1 5s 5s NUM cana-1013 154 2 ( ( PUNCT cana-1013 154 3 2024 2024 NUM cana-1013 154 4 ) ) PUNCT cana-1013 154 5 203 203 NUM cana-1013 154 6 https://internationalpubls.com https://internationalpubls.com X cana-1013 154 7 references reference NOUN cana-1013 154 8 [ [ X cana-1013 154 9 1 1 NUM cana-1013 154 10 ] ] X cana-1013 154 11 farrokhi farrokhi PROPN cana-1013 154 12 dฯˆ dฯˆ PROPN cana-1013 154 13 , , PUNCT cana-1013 154 14 m. m. NOUN cana-1013 154 15 , , PUNCT cana-1013 154 16 rajabian rajabian NOUN cana-1013 154 17 , , PUNCT cana-1013 154 18 m. m. NOUN cana-1013 154 19 , , PUNCT cana-1013 154 20 & & CCONJ cana-1013 154 21 erfanian erfanian PROPN cana-1013 154 22 , , PUNCT cana-1013 154 23 a. a. NOUN cana-1013 154 24 ( ( PUNCT cana-1013 154 25 2019 2019 NUM cana-1013 154 26 ) ) PUNCT cana-1013 154 27 . . PUNCT cana-1013 155 1 relative relative ADJ cana-1013 155 2 cayley cayley ADJ cana-1013 155 3 graphs graph NOUN cana-1013 155 4 of of ADP cana-1013 155 5 finite finite ADJ cana-1013 155 6 ฯˆroups ฯˆroup NOUN cana-1013 155 7 . . PUNCT cana-1013 156 1 asian asian ADJ cana-1013 156 2 - - PUNCT cana-1013 156 3 european european ADJ cana-1013 156 4 journal journal NOUN cana-1013 156 5 of of ADP cana-1013 156 6 mathematics mathematic NOUN cana-1013 156 7 , , PUNCT cana-1013 156 8 12(07 12(07 NUM cana-1013 156 9 ) ) PUNCT cana-1013 156 10 , , PUNCT cana-1013 156 11 2050003 2050003 NUM cana-1013 156 12 . . PUNCT cana-1013 157 1 [ [ X cana-1013 157 2 2 2 NUM cana-1013 157 3 ] ] X cana-1013 157 4 frucht frucht NOUN cana-1013 157 5 , , PUNCT cana-1013 157 6 r. r. PROPN cana-1013 157 7 , , PUNCT cana-1013 157 8 harary harary PROPN cana-1013 157 9 , , PUNCT cana-1013 157 10 f. f. PROPN cana-1013 157 11 on on ADP cana-1013 157 12 the the DET cana-1013 157 13 corona corona NOUN cana-1013 157 14 of of ADP cana-1013 157 15 two two NUM cana-1013 157 16 graphs graph NOUN cana-1013 157 17 . . PUNCT cana-1013 158 1 aeq aeq PROPN cana-1013 158 2 . . PUNCT cana-1013 158 3 math math NOUN cana-1013 158 4 . . PUNCT cana-1013 159 1 4 4 NUM cana-1013 159 2 , , PUNCT cana-1013 159 3 322โ€“325 322โ€“325 NUM cana-1013 159 4 ( ( PUNCT cana-1013 159 5 1970 1970 NUM cana-1013 159 6 ) ) PUNCT cana-1013 159 7 . . PUNCT cana-1013 160 1 [ [ X cana-1013 160 2 3 3 X cana-1013 160 3 ] ] X cana-1013 160 4 s. s. PROPN cana-1013 160 5 mohammadi mohammadi PROPN cana-1013 160 6 , , PUNCT cana-1013 160 7 some some DET cana-1013 160 8 ฯˆeneralizations ฯˆeneralization NOUN cana-1013 160 9 of of ADP cana-1013 160 10 cayley cayley ADJ cana-1013 160 11 graph graph NOUN cana-1013 160 12 and and CCONJ cana-1013 160 13 intersection intersection NOUN cana-1013 160 14 graph graph NOUN cana-1013 160 15 , , PUNCT cana-1013 160 16 ph.d ph.d PROPN cana-1013 160 17 thesis thesis NOUN cana-1013 160 18 , , PUNCT cana-1013 160 19 ferdowsi ferdowsi NOUN cana-1013 160 20 university university PROPN cana-1013 160 21 of of ADP cana-1013 160 22 mashhad mashhad PROPN cana-1013 160 23 , , PUNCT cana-1013 160 24 mashhad mashhad PROPN cana-1013 160 25 , , PUNCT cana-1013 160 26 iran iran PROPN cana-1013 160 27 , , PUNCT cana-1013 160 28 2020 2020 NUM cana-1013 160 29 . . PUNCT cana-1013 161 1 [ [ X cana-1013 161 2 4 4 NUM cana-1013 161 3 ] ] X cana-1013 161 4 neamah neamah PROPN cana-1013 161 5 , , PUNCT cana-1013 161 6 a. a. NOUN cana-1013 161 7 a. a. PROPN cana-1013 161 8 , , PUNCT cana-1013 161 9 erfanian erfanian ADJ cana-1013 161 10 , , PUNCT cana-1013 161 11 a. a. PROPN cana-1013 161 12 , , PUNCT cana-1013 161 13 & & CCONJ cana-1013 161 14 majeed majeed PROPN cana-1013 161 15 , , PUNCT cana-1013 161 16 a. a. PROPN cana-1013 161 17 h. h. PROPN cana-1013 161 18 ( ( PUNCT cana-1013 161 19 2022 2022 NUM cana-1013 161 20 ) ) PUNCT cana-1013 161 21 . . PUNCT cana-1013 162 1 on on ADP cana-1013 162 2 a a DET cana-1013 162 3 generalized generalized ADJ cana-1013 162 4 cayley cayley ADJ cana-1013 162 5 graph graph NOUN cana-1013 162 6 of of ADP cana-1013 162 7 column column NOUN cana-1013 162 8 matrices matrix NOUN cana-1013 162 9 of of ADP cana-1013 162 10 elements element NOUN cana-1013 162 11 of of ADP cana-1013 162 12 a a DET cana-1013 162 13 finite finite ADJ cana-1013 162 14 ฯˆroup ฯˆroup NOUN cana-1013 162 15 . . PUNCT cana-1013 163 1 mathematica mathematica PROPN cana-1013 163 2 ( ( PUNCT cana-1013 163 3 1222 1222 NUM cana-1013 163 4 - - SYM cana-1013 163 5 9016 9016 NUM cana-1013 163 6 ) ) PUNCT cana-1013 163 7 , , PUNCT cana-1013 163 8 64(2 64(2 NUM cana-1013 163 9 ) ) PUNCT cana-1013 163 10 . . PUNCT cana-1013 164 1 [ [ X cana-1013 164 2 5 5 NUM cana-1013 164 3 ] ] X cana-1013 164 4 neamah neamah PROPN cana-1013 164 5 , , PUNCT cana-1013 164 6 a. a. NOUN cana-1013 164 7 a. a. PROPN cana-1013 164 8 , , PUNCT cana-1013 164 9 majeed majeed PROPN cana-1013 164 10 , , PUNCT cana-1013 164 11 a. a. PROPN cana-1013 164 12 h. h. PROPN cana-1013 164 13 , , PUNCT cana-1013 164 14 & & CCONJ cana-1013 164 15 erfanian erfanian PROPN cana-1013 164 16 , , PUNCT cana-1013 164 17 a. a. NOUN cana-1013 164 18 ( ( PUNCT cana-1013 164 19 2022 2022 NUM cana-1013 164 20 ) ) PUNCT cana-1013 164 21 . . PUNCT cana-1013 165 1 the the DET cana-1013 165 2 generalized generalized ADJ cana-1013 165 3 cayley cayley ADJ cana-1013 165 4 graph graph NOUN cana-1013 165 5 of of ADP cana-1013 165 6 complete complete ADJ cana-1013 165 7 graph graph NOUN cana-1013 165 8 k_nand k_nand NOUN cana-1013 165 9 complete complete ADJ cana-1013 165 10 multipartite multipartite ADJ cana-1013 165 11 graphs graph NOUN cana-1013 165 12 k_(n k_(n PROPN cana-1013 165 13 , , PUNCT cana-1013 165 14 n n CCONJ cana-1013 165 15 ) ) PUNCT cana-1013 165 16 and and CCONJ cana-1013 165 17 k_(n k_(n PROPN cana-1013 165 18 , , PUNCT cana-1013 165 19 n n CCONJ cana-1013 165 20 , , PUNCT cana-1013 165 21 n n CCONJ cana-1013 165 22 ) ) PUNCT cana-1013 165 23 . . PUNCT cana-1013 166 1 iraqi iraqi ADJ cana-1013 166 2 journal journal PROPN cana-1013 166 3 of of ADP cana-1013 166 4 science science NOUN cana-1013 166 5 , , PUNCT cana-1013 166 6 3103 3103 NUM cana-1013 166 7 - - SYM cana-1013 166 8 3110 3110 NUM cana-1013 166 9 . . PUNCT cana-1013 167 1 [ [ X cana-1013 167 2 6 6 NUM cana-1013 167 3 ] ] X cana-1013 167 4 neamah neamah PROPN cana-1013 167 5 , , PUNCT cana-1013 167 6 a. a. NOUN cana-1013 167 7 a. a. PROPN cana-1013 167 8 , , PUNCT cana-1013 167 9 erfanian erfanian ADJ cana-1013 167 10 , , PUNCT cana-1013 167 11 a. a. PROPN cana-1013 167 12 , , PUNCT cana-1013 167 13 & & CCONJ cana-1013 167 14 majeed majeed PROPN cana-1013 167 15 , , PUNCT cana-1013 167 16 a. a. PROPN cana-1013 167 17 h. h. PROPN cana-1013 167 18 ( ( PUNCT cana-1013 167 19 2023 2023 NUM cana-1013 167 20 , , PUNCT cana-1013 167 21 december december PROPN cana-1013 167 22 ) ) PUNCT cana-1013 167 23 . . PUNCT cana-1013 168 1 the the DET cana-1013 168 2 structure structure NOUN cana-1013 168 3 of of ADP cana-1013 168 4 generalized generalized ADJ cana-1013 168 5 cayley cayley ADJ cana-1013 168 6 graph graph NOUN cana-1013 168 7 when when SCONJ cana-1013 168 8 cay cay PROPN cana-1013 168 9 ( ( PUNCT cana-1013 168 10 g g NOUN cana-1013 168 11 , , PUNCT cana-1013 168 12 s)= s)= NOUN cana-1013 168 13 k_(n k_(n PROPN cana-1013 168 14 , , PUNCT cana-1013 168 15 n n CCONJ cana-1013 168 16 , , PUNCT cana-1013 168 17 n n CCONJ cana-1013 168 18 , , PUNCT cana-1013 168 19 n n CCONJ cana-1013 168 20 ) ) PUNCT cana-1013 168 21 . . PUNCT cana-1013 169 1 in in ADP cana-1013 169 2 aip aip PROPN cana-1013 169 3 conference conference NOUN cana-1013 169 4 proceedinฯˆs proceedinฯˆs PROPN cana-1013 169 5 ( ( PUNCT cana-1013 169 6 vol vol NOUN cana-1013 169 7 . . NOUN cana-1013 169 8 2834 2834 NUM cana-1013 169 9 , , PUNCT cana-1013 169 10 no no INTJ cana-1013 169 11 . . NOUN cana-1013 169 12 1 1 NUM cana-1013 169 13 ) ) PUNCT cana-1013 169 14 . . PUNCT cana-1013 170 1 aip aip PROPN cana-1013 170 2 publishinฯˆ publishinฯˆ PROPN cana-1013 170 3 . . PUNCT cana-1013 171 1 [ [ X cana-1013 171 2 7 7 NUM cana-1013 171 3 ] ] X cana-1013 171 4 neamah neamah PROPN cana-1013 171 5 , , PUNCT cana-1013 171 6 s. s. PROPN cana-1013 171 7 a. a. PROPN cana-1013 171 8 , , PUNCT cana-1013 171 9 & & CCONJ cana-1013 171 10 erfanian erfanian PROPN cana-1013 171 11 , , PUNCT cana-1013 171 12 a. a. NOUN cana-1013 171 13 ( ( PUNCT cana-1013 171 14 2023 2023 NUM cana-1013 171 15 ) ) PUNCT cana-1013 171 16 . . PUNCT cana-1013 172 1 some some DET cana-1013 172 2 results result NOUN cana-1013 172 3 on on ADP cana-1013 172 4 the the DET cana-1013 172 5 generalized generalized ADJ cana-1013 172 6 cayley cayley ADJ cana-1013 172 7 graph graph NOUN cana-1013 172 8 of of ADP cana-1013 172 9 complete complete ADJ cana-1013 172 10 graphs graph NOUN cana-1013 172 11 . . PUNCT cana-1013 173 1 iraqi iraqi ADJ cana-1013 173 2 journal journal PROPN cana-1013 173 3 of of ADP cana-1013 173 4 science science NOUN cana-1013 173 5 , , PUNCT cana-1013 173 6 3424 3424 NUM cana-1013 173 7 - - SYM cana-1013 173 8 3436 3436 NUM cana-1013 173 9 . . PUNCT cana-1013 174 1 [ [ X cana-1013 174 2 8 8 NUM cana-1013 174 3 ] ] SYM cana-1013 174 4 kelarev kelarev X cana-1013 174 5 , , PUNCT cana-1013 174 6 a. a. NOUN cana-1013 174 7 v. v. PROPN cana-1013 174 8 ( ( PUNCT cana-1013 174 9 2002 2002 NUM cana-1013 174 10 ) ) PUNCT cana-1013 174 11 . . PUNCT cana-1013 175 1 on on ADP cana-1013 175 2 undirected undirected ADJ cana-1013 175 3 cayley cayley ADJ cana-1013 175 4 graphs graph NOUN cana-1013 175 5 . . PUNCT cana-1013 176 1 australasian australasian ADJ cana-1013 176 2 journal journal NOUN cana-1013 176 3 of of ADP cana-1013 176 4 combinatorics combinatoric NOUN cana-1013 176 5 , , PUNCT cana-1013 176 6 25 25 NUM cana-1013 176 7 , , PUNCT cana-1013 176 8 73 73 NUM cana-1013 176 9 - - SYM cana-1013 176 10 78 78 NUM cana-1013 176 11 . . PUNCT cana-1013 177 1 [ [ X cana-1013 177 2 9 9 NUM cana-1013 177 3 ] ] X cana-1013 177 4 rajabian rajabian ADJ cana-1013 177 5 , , PUNCT cana-1013 177 6 m. m. NOUN cana-1013 177 7 , , PUNCT cana-1013 177 8 & & CCONJ cana-1013 177 9 erfanian erfanian PROPN cana-1013 177 10 , , PUNCT cana-1013 177 11 a. a. NOUN cana-1013 177 12 ( ( PUNCT cana-1013 177 13 2018 2018 NUM cana-1013 177 14 ) ) PUNCT cana-1013 177 15 . . PUNCT cana-1013 177 16 relative relative ADJ cana-1013 177 17 cayley cayley ADJ cana-1013 177 18 graphs graph NOUN cana-1013 177 19 of of ADP cana-1013 177 20 finite finite ADJ cana-1013 177 21 ฯˆroups ฯˆroup NOUN cana-1013 177 22 . . PUNCT cana-1013 178 1 asian asian ADJ cana-1013 178 2 - - PUNCT cana-1013 178 3 european european ADJ cana-1013 178 4 journal journal NOUN cana-1013 178 5 of of ADP cana-1013 178 6 mathematics mathematic NOUN cana-1013 178 7 , , PUNCT cana-1013 178 8 12 12 NUM cana-1013 178 9 . . PUNCT