id sid tid token lemma pos cana-5924 1 1 communications communication NOUN cana-5924 1 2 on on ADP cana-5924 1 3 applied apply VERB cana-5924 1 4 nonlinear nonlinear ADJ cana-5924 1 5 analysis analysis NOUN cana-5924 1 6 issn issn NOUN cana-5924 1 7 : : PUNCT cana-5924 1 8 1074 1074 NUM cana-5924 1 9 - - PUNCT cana-5924 1 10 133x 133x NUM cana-5924 1 11 vol vol NOUN cana-5924 1 12 31 31 NUM cana-5924 1 13 no no NOUN cana-5924 1 14 . . NOUN cana-5924 1 15 2 2 NUM cana-5924 1 16 ( ( PUNCT cana-5924 1 17 2024 2024 NUM cana-5924 1 18 ) ) PUNCT cana-5924 1 19 480 480 NUM cana-5924 1 20 https://internationalpubls.com https://internationalpubls.com X cana-5924 1 21 edge edge VERB cana-5924 1 22 binary binary ADJ cana-5924 1 23 coding coding NOUN cana-5924 1 24 of of ADP cana-5924 1 25 cayley cayley ADJ cana-5924 1 26 network network PROPN cana-5924 1 27 r. r. PROPN cana-5924 1 28 radha1*and radha1*and PROPN cana-5924 1 29 n. n. PROPN cana-5924 1 30 mohamed mohamed PROPN cana-5924 1 31 rilwan2 rilwan2 PROPN cana-5924 1 32 1*department 1*department NUM cana-5924 1 33 of of ADP cana-5924 1 34 mathematics mathematic NOUN cana-5924 1 35 , , PUNCT cana-5924 1 36 sri sri PROPN cana-5924 1 37 paramakalyani paramakalyani PROPN cana-5924 1 38 college college PROPN cana-5924 1 39 , , PUNCT cana-5924 1 40 affiliated affiliate VERB cana-5924 1 41 to to ADP cana-5924 1 42 manonmaniam manonmaniam PROPN cana-5924 1 43 sundaranar sundaranar PROPN cana-5924 1 44 university university PROPN cana-5924 1 45 , , PUNCT cana-5924 1 46 tenkasi tenkasi NOUN cana-5924 1 47 627 627 NUM cana-5924 1 48 412 412 NUM cana-5924 1 49 , , PUNCT cana-5924 1 50 tamil tamil PROPN cana-5924 1 51 nadu nadu PROPN cana-5924 1 52 , , PUNCT cana-5924 1 53 india india PROPN cana-5924 1 54 . . PUNCT cana-5924 2 1 2department 2department NUM cana-5924 2 2 of of ADP cana-5924 2 3 mathematics mathematic NOUN cana-5924 2 4 , , PUNCT cana-5924 2 5 sadakathullah sadakathullah ADJ cana-5924 2 6 appa appa PROPN cana-5924 2 7 college(autonomous college(autonomous PROPN cana-5924 2 8 ) ) PUNCT cana-5924 2 9 , , PUNCT cana-5924 2 10 affiliated affiliate VERB cana-5924 2 11 to to ADP cana-5924 2 12 manonmaniam manonmaniam PROPN cana-5924 2 13 sundaranar sundaranar PROPN cana-5924 2 14 university university PROPN cana-5924 2 15 , , PUNCT cana-5924 2 16 tirunelveli tirunelveli ADJ cana-5924 2 17 627 627 NUM cana-5924 2 18 011 011 NUM cana-5924 2 19 , , PUNCT cana-5924 2 20 tamil tamil PROPN cana-5924 2 21 nadu nadu PROPN cana-5924 2 22 , , PUNCT cana-5924 2 23 india india PROPN cana-5924 2 24 . . PUNCT cana-5924 3 1 email email NOUN cana-5924 3 2 address address NOUN cana-5924 3 3 : : PUNCT cana-5924 3 4 * * PUNCT cana-5924 3 5 kedarbaba20@gmail.com kedarbaba20@gmail.com X cana-5924 3 6 , , PUNCT cana-5924 3 7 rilwan2020@sadakath.ac.in rilwan2020@sadakath.ac.in PROPN cana-5924 3 8 _ _ PUNCT cana-5924 4 1 _ _ PUNCT cana-5924 5 1 _ _ PUNCT cana-5924 6 1 _ _ PUNCT cana-5924 7 1 _ _ PUNCT cana-5924 8 1 _ _ PUNCT cana-5924 9 1 _ _ PUNCT cana-5924 10 1 _ _ PUNCT cana-5924 11 1 _ _ PUNCT cana-5924 12 1 _ _ PUNCT cana-5924 13 1 _ _ PUNCT cana-5924 14 1 _ _ PUNCT cana-5924 15 1 _ _ PUNCT cana-5924 16 1 _ _ PUNCT cana-5924 17 1 _ _ PUNCT cana-5924 18 1 _ _ PUNCT cana-5924 19 1 _ _ PUNCT cana-5924 20 1 _ _ PUNCT cana-5924 21 1 _ _ PUNCT cana-5924 22 1 _ _ PUNCT cana-5924 23 1 _ _ PUNCT cana-5924 24 1 _ _ PUNCT cana-5924 25 1 _ _ PUNCT cana-5924 26 1 _ _ PUNCT cana-5924 27 1 _ _ PUNCT cana-5924 28 1 _ _ PUNCT cana-5924 29 1 _ _ PUNCT cana-5924 30 1 _ _ PUNCT cana-5924 31 1 _ _ PUNCT cana-5924 32 1 _ _ PUNCT cana-5924 33 1 _ _ PUNCT cana-5924 34 1 _ _ PUNCT cana-5924 35 1 _ _ PUNCT cana-5924 36 1 _ _ PUNCT cana-5924 37 1 _ _ PUNCT cana-5924 38 1 _ _ PUNCT cana-5924 39 1 _ _ PUNCT cana-5924 40 1 _ _ PUNCT cana-5924 41 1 _ _ PUNCT cana-5924 42 1 _ _ PUNCT cana-5924 43 1 _ _ PUNCT cana-5924 44 1 _ _ PUNCT cana-5924 45 1 _ _ PUNCT cana-5924 46 1 _ _ PUNCT cana-5924 47 1 _ _ PUNCT cana-5924 48 1 _ _ PUNCT cana-5924 49 1 _ _ PUNCT cana-5924 50 1 _ _ PUNCT cana-5924 51 1 _ _ PUNCT cana-5924 52 1 _ _ PUNCT cana-5924 53 1 _ _ PUNCT cana-5924 54 1 _ _ PUNCT cana-5924 55 1 _ _ PUNCT cana-5924 56 1 _ _ PUNCT cana-5924 57 1 _ _ PUNCT cana-5924 58 1 _ _ PUNCT cana-5924 59 1 _ _ PUNCT cana-5924 60 1 _ _ PUNCT cana-5924 61 1 _ _ PUNCT cana-5924 62 1 _ _ PUNCT cana-5924 63 1 _ _ PUNCT cana-5924 64 1 _ _ PUNCT cana-5924 65 1 _ _ PUNCT cana-5924 66 1 _ _ PUNCT cana-5924 67 1 _ _ PUNCT cana-5924 68 1 _ _ PUNCT cana-5924 69 1 _ _ PUNCT cana-5924 70 1 _ _ PUNCT cana-5924 71 1 _ _ PUNCT cana-5924 72 1 _ _ PUNCT cana-5924 73 1 _ _ PUNCT cana-5924 74 1 _ _ PUNCT cana-5924 75 1 _ _ PUNCT cana-5924 76 1 _ _ PUNCT cana-5924 77 1 _ _ PUNCT cana-5924 78 1 _ _ PUNCT cana-5924 79 1 _ _ PUNCT cana-5924 80 1 _ _ PUNCT cana-5924 81 1 _ _ PUNCT cana-5924 82 1 _ _ PUNCT cana-5924 83 1 _ _ PUNCT cana-5924 84 1 _ _ PUNCT cana-5924 85 1 _ _ PUNCT cana-5924 86 1 _ _ PUNCT cana-5924 87 1 _ _ PUNCT cana-5924 88 1 _ _ PUNCT cana-5924 89 1 _ _ PUNCT cana-5924 89 2 article article NOUN cana-5924 89 3 history history NOUN cana-5924 89 4 : : PUNCT cana-5924 89 5 received receive VERB cana-5924 89 6 : : PUNCT cana-5924 89 7 02 02 NUM cana-5924 89 8 - - SYM cana-5924 89 9 10 10 NUM cana-5924 89 10 - - PUNCT cana-5924 89 11 2024 2024 NUM cana-5924 89 12 revised revise VERB cana-5924 89 13 : : PUNCT cana-5924 89 14 25 25 NUM cana-5924 89 15 - - SYM cana-5924 89 16 11 11 NUM cana-5924 89 17 - - PUNCT cana-5924 89 18 2024 2024 NUM cana-5924 89 19 accepted accept VERB cana-5924 89 20 : : PUNCT cana-5924 89 21 26 26 NUM cana-5924 89 22 - - SYM cana-5924 89 23 12 12 NUM cana-5924 89 24 - - PUNCT cana-5924 89 25 2024 2024 NUM cana-5924 89 26 abstract abstract NOUN cana-5924 89 27 : : PUNCT cana-5924 89 28 let let VERB cana-5924 89 29 𝒒(𝑉 𝒒(𝑉 NOUN cana-5924 89 30 , , PUNCT cana-5924 89 31 𝐿 𝐿 PROPN cana-5924 89 32 ) ) PUNCT cana-5924 89 33 = = SYM cana-5924 90 1 cay(Ξ³ cay(Ξ³ PROPN cana-5924 90 2 , , PUNCT cana-5924 90 3 Ο‰ Ο‰ NOUN cana-5924 90 4 ) ) PUNCT cana-5924 90 5 is be AUX cana-5924 90 6 an an DET cana-5924 90 7 algebraic algebraic ADJ cana-5924 90 8 network network NOUN cana-5924 90 9 model model NOUN cana-5924 90 10 of of ADP cana-5924 90 11 the the DET cana-5924 90 12 cayley cayley ADJ cana-5924 90 13 graph graph NOUN cana-5924 90 14 . . PUNCT cana-5924 91 1 a a DET cana-5924 91 2 binary binary ADJ cana-5924 91 3 coding code VERB cana-5924 91 4 edge edge NOUN cana-5924 91 5 function function NOUN cana-5924 91 6 β„± β„± PROPN cana-5924 91 7 : : PUNCT cana-5924 91 8 l(𝒒 l(𝒒 PROPN cana-5924 91 9 ) ) PUNCT cana-5924 91 10 β†’ β†’ SYM cana-5924 91 11 { { PUNCT cana-5924 91 12 0 0 NUM cana-5924 91 13 , , PUNCT cana-5924 91 14 1 1 NUM cana-5924 91 15 } } PUNCT cana-5924 91 16 , , PUNCT cana-5924 91 17 it it PRON cana-5924 91 18 induces induce VERB cana-5924 91 19 𝑉(𝒒 𝑉(𝒒 ADV cana-5924 91 20 ) ) PUNCT cana-5924 91 21 as as ADP cana-5924 91 22 β„±(𝑣 β„±(𝑣 NUM cana-5924 91 23 ) ) PUNCT cana-5924 91 24 = = NOUN cana-5924 91 25 π›΄π‘’π‘£πœ–πΏ(𝒒 π›΄π‘’π‘£πœ–πΏ(𝒒 NOUN cana-5924 91 26 ) ) PUNCT cana-5924 91 27 β„±(𝑒𝑣)(π‘šπ‘œπ‘‘ β„±(𝑒𝑣)(π‘šπ‘œπ‘‘ SYM cana-5924 91 28 2 2 NUM cana-5924 91 29 ) ) PUNCT cana-5924 91 30 . . PUNCT cana-5924 92 1 the the DET cana-5924 92 2 function function NOUN cana-5924 92 3 β„± β„± PROPN cana-5924 92 4 is be AUX cana-5924 92 5 said say VERB cana-5924 92 6 to to PART cana-5924 92 7 be be AUX cana-5924 92 8 an an DET cana-5924 92 9 edge edge ADJ cana-5924 92 10 cordial cordial ADJ cana-5924 92 11 function function NOUN cana-5924 92 12 of of ADP cana-5924 92 13 𝒒 𝒒 PROPN cana-5924 92 14 , , PUNCT cana-5924 92 15 if if SCONJ cana-5924 92 16 the the DET cana-5924 92 17 difference difference NOUN cana-5924 92 18 between between ADP cana-5924 92 19 the the DET cana-5924 92 20 number number NOUN cana-5924 92 21 of of ADP cana-5924 92 22 vertices vertex NOUN cana-5924 92 23 ( ( PUNCT cana-5924 92 24 edges edge NOUN cana-5924 92 25 ) ) PUNCT cana-5924 92 26 labeled label VERB cana-5924 92 27 by by ADP cana-5924 92 28 zero zero NUM cana-5924 92 29 and and CCONJ cana-5924 92 30 the the DET cana-5924 92 31 number number NOUN cana-5924 92 32 of of ADP cana-5924 92 33 vertices vertex NOUN cana-5924 92 34 ( ( PUNCT cana-5924 92 35 edges edge NOUN cana-5924 92 36 ) ) PUNCT cana-5924 92 37 labeled label VERB cana-5924 92 38 by by ADP cana-5924 92 39 one one NUM cana-5924 92 40 is be AUX cana-5924 92 41 at at ADP cana-5924 92 42 most most ADJ cana-5924 92 43 one one NUM cana-5924 92 44 . . PUNCT cana-5924 93 1 in in ADP cana-5924 93 2 this this DET cana-5924 93 3 paper paper NOUN cana-5924 93 4 , , PUNCT cana-5924 93 5 we we PRON cana-5924 93 6 show show VERB cana-5924 93 7 the the DET cana-5924 93 8 edge edge NOUN cana-5924 93 9 binary binary ADJ cana-5924 93 10 coding coding NOUN cana-5924 93 11 of of ADP cana-5924 93 12 the the DET cana-5924 93 13 cayley cayley ADJ cana-5924 93 14 graph graph NOUN cana-5924 93 15 network network NOUN cana-5924 93 16 model model NOUN cana-5924 93 17 which which PRON cana-5924 93 18 satisfies satisfy VERB cana-5924 93 19 the the DET cana-5924 93 20 edge edge NOUN cana-5924 93 21 cordial cordial ADJ cana-5924 93 22 constraint constraint NOUN cana-5924 93 23 . . PUNCT cana-5924 94 1 keywords keyword NOUN cana-5924 94 2 and and CCONJ cana-5924 94 3 phrases phrase NOUN cana-5924 94 4 : : PUNCT cana-5924 94 5 cayley cayley ADJ cana-5924 94 6 network network NOUN cana-5924 94 7 , , PUNCT cana-5924 94 8 binary binary NOUN cana-5924 94 9 coding coding NOUN cana-5924 94 10 , , PUNCT cana-5924 94 11 edge edge NOUN cana-5924 94 12 cordiality cordiality NOUN cana-5924 94 13 , , PUNCT cana-5924 94 14 labeling labeling NOUN cana-5924 94 15 event event NOUN cana-5924 94 16 , , PUNCT cana-5924 94 17 congruence congruence PROPN cana-5924 94 18 classes class NOUN cana-5924 94 19 . . PUNCT cana-5924 95 1 2000 2000 NUM cana-5924 95 2 a.m.s a.m.s PROPN cana-5924 95 3 . . PUNCT cana-5924 96 1 subject subject ADJ cana-5924 96 2 classification classification NOUN cana-5924 96 3 : : PUNCT cana-5924 96 4 05b10 05b10 NUM cana-5924 96 5 , , PUNCT cana-5924 96 6 05b30 05b30 PRON cana-5924 96 7 , , PUNCT cana-5924 96 8 05c78 05c78 NUM cana-5924 96 9 1 1 X cana-5924 96 10 . . X cana-5924 96 11 introduction introduction NOUN cana-5924 96 12 and and CCONJ cana-5924 96 13 notations notation NOUN cana-5924 96 14 in in ADP cana-5924 96 15 mathematical mathematical ADJ cana-5924 96 16 modelling modelling NOUN cana-5924 96 17 , , PUNCT cana-5924 96 18 algebraic algebraic ADJ cana-5924 96 19 graphs graph NOUN cana-5924 96 20 are be AUX cana-5924 96 21 the the DET cana-5924 96 22 evergreen evergreen NOUN cana-5924 96 23 trending trending NOUN cana-5924 96 24 solution solution NOUN cana-5924 96 25 domain domain NOUN cana-5924 96 26 for for ADP cana-5924 96 27 many many ADJ cana-5924 96 28 practical practical ADJ cana-5924 96 29 problems problem NOUN cana-5924 96 30 . . PUNCT cana-5924 97 1 it it PRON cana-5924 97 2 describes describe VERB cana-5924 97 3 the the DET cana-5924 97 4 concept concept NOUN cana-5924 97 5 and and CCONJ cana-5924 97 6 make make VERB cana-5924 97 7 clarity clarity NOUN cana-5924 97 8 on on ADP cana-5924 97 9 a a DET cana-5924 97 10 concrete concrete ADJ cana-5924 97 11 solution solution NOUN cana-5924 97 12 to to ADP cana-5924 97 13 the the DET cana-5924 97 14 lot lot NOUN cana-5924 97 15 of of ADP cana-5924 97 16 abstract abstract ADJ cana-5924 97 17 problems problem NOUN cana-5924 97 18 through through ADP cana-5924 97 19 graphically graphically ADV cana-5924 97 20 , , PUNCT cana-5924 97 21 that that PRON cana-5924 97 22 ’s ’ VERB cana-5924 97 23 the the DET cana-5924 97 24 reason reason NOUN cana-5924 97 25 new new ADJ cana-5924 97 26 graphical graphical ADJ cana-5924 97 27 techniques technique NOUN cana-5924 97 28 and and CCONJ cana-5924 97 29 terminologies terminology NOUN cana-5924 97 30 are be AUX cana-5924 97 31 emerging emerge VERB cana-5924 97 32 in in ADP cana-5924 97 33 inter inter NOUN cana-5924 97 34 and and CCONJ cana-5924 97 35 under under ADP cana-5924 97 36 disciplined discipline VERB cana-5924 97 37 with with ADP cana-5924 97 38 the the DET cana-5924 97 39 algebraic algebraic ADJ cana-5924 97 40 base base NOUN cana-5924 97 41 . . PUNCT cana-5924 98 1 labelling labelling NOUN cana-5924 98 2 is be AUX cana-5924 98 3 one one NUM cana-5924 98 4 of of ADP cana-5924 98 5 such such ADJ cana-5924 98 6 encoding encoding NOUN cana-5924 98 7 technique technique NOUN cana-5924 98 8 , , PUNCT cana-5924 98 9 contributed contribute VERB cana-5924 98 10 by by ADP cana-5924 98 11 alex alex PROPN cana-5924 98 12 rosa[4]in rosa[4]in PROPN cana-5924 98 13 1967 1967 NUM cana-5924 98 14 . . PUNCT cana-5924 99 1 nowadays nowadays ADV cana-5924 99 2 different different ADJ cana-5924 99 3 types type NOUN cana-5924 99 4 of of ADP cana-5924 99 5 labelling labelling NOUN cana-5924 99 6 techniques technique NOUN cana-5924 99 7 are be AUX cana-5924 99 8 utilized utilize VERB cana-5924 99 9 in in ADP cana-5924 99 10 various various ADJ cana-5924 99 11 fields field NOUN cana-5924 99 12 of of ADP cana-5924 99 13 sciences science NOUN cana-5924 99 14 such such ADJ cana-5924 99 15 as as ADP cana-5924 99 16 coding code VERB cana-5924 99 17 theory theory NOUN cana-5924 99 18 , , PUNCT cana-5924 99 19 x x ADJ cana-5924 99 20 - - NOUN cana-5924 99 21 ray ray NOUN cana-5924 99 22 diffraction diffraction NOUN cana-5924 99 23 , , PUNCT cana-5924 99 24 crystallography crystallography NOUN cana-5924 99 25 , , PUNCT cana-5924 99 26 missile missile NOUN cana-5924 99 27 guidance guidance NOUN cana-5924 99 28 , , PUNCT cana-5924 99 29 astronomy astronomy NOUN cana-5924 99 30 , , PUNCT cana-5924 99 31 circuit circuit NOUN cana-5924 99 32 designing designing NOUN cana-5924 99 33 , , PUNCT cana-5924 99 34 communication communication NOUN cana-5924 99 35 network network NOUN cana-5924 99 36 addressing[3 addressing[3 PROPN cana-5924 99 37 ] ] X cana-5924 99 38 etc etc X cana-5924 99 39 . . X cana-5924 99 40 various various ADJ cana-5924 99 41 labelling labelling NOUN cana-5924 99 42 terminologies terminology NOUN cana-5924 99 43 are be AUX cana-5924 99 44 proven prove VERB cana-5924 99 45 on on ADP cana-5924 99 46 the the DET cana-5924 99 47 cayley cayley ADJ cana-5924 99 48 graphs graph NOUN cana-5924 99 49 however however SCONJ cana-5924 99 50 we we PRON cana-5924 99 51 mainly mainly ADV cana-5924 99 52 focus focus VERB cana-5924 99 53 the the DET cana-5924 99 54 classical classical ADJ cana-5924 99 55 binary binary ADJ cana-5924 99 56 encoding encoding NOUN cana-5924 99 57 technique technique NOUN cana-5924 99 58 , , PUNCT cana-5924 99 59 which which PRON cana-5924 99 60 is be AUX cana-5924 99 61 easy easy ADJ cana-5924 99 62 to to PART cana-5924 99 63 decode decode VERB cana-5924 99 64 than than ADP cana-5924 99 65 the the DET cana-5924 99 66 other other ADJ cana-5924 99 67 . . PUNCT cana-5924 100 1 we we PRON cana-5924 100 2 may may AUX cana-5924 100 3 say say VERB cana-5924 100 4 , , PUNCT cana-5924 100 5 binary binary ADJ cana-5924 100 6 encoding encoding NOUN cana-5924 100 7 is be AUX cana-5924 100 8 a a DET cana-5924 100 9 labelling labelling NOUN cana-5924 100 10 where where SCONJ cana-5924 100 11 as as SCONJ cana-5924 100 12 all all DET cana-5924 100 13 the the DET cana-5924 100 14 labelling labelling NOUN cana-5924 100 15 may may AUX cana-5924 100 16 not not PART cana-5924 100 17 be be AUX cana-5924 100 18 a a DET cana-5924 100 19 binary binary NOUN cana-5924 100 20 coding coding NOUN cana-5924 100 21 because because SCONJ cana-5924 100 22 every every DET cana-5924 100 23 labelling labelling NOUN cana-5924 100 24 has have VERB cana-5924 100 25 its its PRON cana-5924 100 26 own own ADJ cana-5924 100 27 well well ADV cana-5924 100 28 defined define VERB cana-5924 100 29 terminology terminology NOUN cana-5924 100 30 . . PUNCT cana-5924 101 1 when when SCONJ cana-5924 101 2 a a DET cana-5924 101 3 binary binary NOUN cana-5924 101 4 encoding encoding NOUN cana-5924 101 5 inherits inherit VERB cana-5924 101 6 the the DET cana-5924 101 7 ( ( PUNCT cana-5924 101 8 vertex vertex NOUN cana-5924 101 9 or or CCONJ cana-5924 101 10 edge edge NOUN cana-5924 101 11 or or CCONJ cana-5924 101 12 some some DET cana-5924 101 13 ) ) PUNCT cana-5924 101 14 cordial cordial ADJ cana-5924 101 15 labelling labelling NOUN cana-5924 101 16 terminology terminology NOUN cana-5924 101 17 is be AUX cana-5924 101 18 known know VERB cana-5924 101 19 as as ADP cana-5924 101 20 a a DET cana-5924 101 21 cordial cordial ADJ cana-5924 101 22 labelling labelling NOUN cana-5924 101 23 , , PUNCT cana-5924 101 24 therefore therefore ADV cana-5924 101 25 cordial cordial ADJ cana-5924 101 26 functions function NOUN cana-5924 101 27 are be AUX cana-5924 101 28 the the DET cana-5924 101 29 massive massive ADJ cana-5924 101 30 operator operator NOUN cana-5924 101 31 in in ADP cana-5924 101 32 logical logical ADJ cana-5924 101 33 and and CCONJ cana-5924 101 34 decision decision NOUN cana-5924 101 35 making make VERB cana-5924 101 36 algorithms algorithm NOUN cana-5924 101 37 . . PUNCT cana-5924 102 1 this this DET cana-5924 102 2 npcomplete npcomplete ADJ cana-5924 102 3 problems problem NOUN cana-5924 102 4 can can AUX cana-5924 102 5 be be AUX cana-5924 102 6 executed execute VERB cana-5924 102 7 in in ADP cana-5924 102 8 polynomial polynomial ADJ cana-5924 102 9 time time NOUN cana-5924 102 10 . . PUNCT cana-5924 103 1 β€œ " PUNCT cana-5924 103 2 cordial cordial ADJ cana-5924 103 3 labelling labelling NOUN cana-5924 103 4 may may AUX cana-5924 103 5 be be AUX cana-5924 103 6 considered consider VERB cana-5924 103 7 as as ADP cana-5924 103 8 a a DET cana-5924 103 9 weakened weakened ADJ cana-5924 103 10 version version NOUN cana-5924 103 11 of of ADP cana-5924 103 12 harmonious harmonious ADJ cana-5924 103 13 and and CCONJ cana-5924 103 14 graceful graceful ADJ cana-5924 103 15 labelling labelling NOUN cana-5924 103 16 ” " PUNCT cana-5924 103 17 is be AUX cana-5924 103 18 said say VERB cana-5924 103 19 by by ADP cana-5924 103 20 i. i. PROPN cana-5924 103 21 cahit[1 cahit[1 NUM cana-5924 103 22 ] ] PUNCT cana-5924 103 23 and and CCONJ cana-5924 103 24 introduced introduce VERB cana-5924 103 25 the the DET cana-5924 103 26 concept concept NOUN cana-5924 103 27 of of ADP cana-5924 103 28 cordial cordial ADJ cana-5924 103 29 labelling labelling NOUN cana-5924 103 30 in in ADP cana-5924 103 31 1987 1987 NUM cana-5924 103 32 with with ADP cana-5924 103 33 the the DET cana-5924 103 34 necessary necessary ADJ cana-5924 103 35 and and CCONJ cana-5924 103 36 sufficient sufficient ADJ cana-5924 103 37 condition condition NOUN cana-5924 103 38 for for ADP cana-5924 103 39 that that PRON cana-5924 103 40 and and CCONJ cana-5924 103 41 r. r. PROPN cana-5924 103 42 yilmaz yilmaz PROPN cana-5924 103 43 and and CCONJ cana-5924 103 44 i. i. PROPN cana-5924 103 45 cahit[7 cahit[7 PROPN cana-5924 103 46 ] ] PUNCT cana-5924 103 47 investigated investigate VERB cana-5924 103 48 the the DET cana-5924 103 49 edge edge NOUN cana-5924 103 50 cordiality cordiality NOUN cana-5924 103 51 of of ADP cana-5924 103 52 some some DET cana-5924 103 53 special special ADJ cana-5924 103 54 featured feature VERB cana-5924 103 55 graphs graph NOUN cana-5924 103 56 such such ADJ cana-5924 103 57 as as ADP cana-5924 103 58 complete complete ADJ cana-5924 103 59 bipartite bipartite PROPN cana-5924 103 60 , , PUNCT cana-5924 103 61 wheel wheel NOUN cana-5924 103 62 , , PUNCT cana-5924 103 63 cycles cycle NOUN cana-5924 103 64 etc etc X cana-5924 103 65 . . X cana-5924 103 66 , , PUNCT cana-5924 103 67 in in ADP cana-5924 103 68 1997 1997 NUM cana-5924 103 69 . . PUNCT cana-5924 104 1 any any DET cana-5924 104 2 graph graph NOUN cana-5924 104 3 of of ADP cana-5924 104 4 order order NOUN cana-5924 104 5 is be AUX cana-5924 104 6 congruent congruent ADJ cana-5924 104 7 to to ADP cana-5924 104 8 2(mod 2(mod NUM cana-5924 104 9 4 4 NUM cana-5924 104 10 ) ) PUNCT cana-5924 104 11 will will AUX cana-5924 104 12 not not PART cana-5924 104 13 admit admit VERB cana-5924 104 14 binary binary ADJ cana-5924 104 15 edge edge NOUN cana-5924 104 16 labelling labelling NOUN cana-5924 104 17 and and CCONJ cana-5924 104 18 suppose suppose VERB cana-5924 104 19 a a DET cana-5924 104 20 graph graph NOUN cana-5924 104 21 satisfies satisfy VERB cana-5924 104 22 the the DET cana-5924 104 23 binary binary NOUN cana-5924 104 24 coding coding NOUN cana-5924 104 25 under under ADP cana-5924 104 26 any any DET cana-5924 104 27 cordial cordial ADJ cana-5924 104 28 constraints constraint NOUN cana-5924 104 29 , , PUNCT cana-5924 104 30 from from ADP cana-5924 104 31 that that DET cana-5924 104 32 expected expect VERB cana-5924 104 33 optimal optimal ADJ cana-5924 104 34 solution solution NOUN cana-5924 104 35 derived derive VERB cana-5924 104 36 by by ADP cana-5924 104 37 decoding decode VERB cana-5924 104 38 . . PUNCT cana-5924 105 1 given give VERB cana-5924 105 2 network network NOUN cana-5924 105 3 model model NOUN cana-5924 105 4 is be AUX cana-5924 105 5 the the DET cana-5924 105 6 cayley cayley ADJ cana-5924 105 7 graph graph NOUN cana-5924 105 8 , , PUNCT cana-5924 106 1 𝒒 𝒒 PROPN cana-5924 106 2 = = SYM cana-5924 106 3 cay(Ξ³ cay(Ξ³ PROPN cana-5924 106 4 , , PUNCT cana-5924 106 5 Ο‰ Ο‰ NOUN cana-5924 106 6 ) ) PUNCT cana-5924 106 7 whose whose DET cana-5924 106 8 vertices vertex NOUN cana-5924 106 9 are be AUX cana-5924 106 10 the the DET cana-5924 106 11 elements element NOUN cana-5924 106 12 of of ADP cana-5924 106 13 group group NOUN cana-5924 106 14 𝛀 𝛀 PROPN cana-5924 106 15 and and CCONJ cana-5924 106 16 the the DET cana-5924 106 17 adjacency adjacency PROPN cana-5924 106 18 relation relation NOUN cana-5924 106 19 corresponds correspond VERB cana-5924 106 20 to to ADP cana-5924 106 21 the the DET cana-5924 106 22 operation operation NOUN cana-5924 106 23 on on ADP cana-5924 106 24 every every DET cana-5924 106 25 element element NOUN cana-5924 106 26 of of ADP cana-5924 106 27 Ξ³ Ξ³ NOUN cana-5924 106 28 with with ADP cana-5924 106 29 generators generator NOUN cana-5924 106 30 such such ADJ cana-5924 106 31 that that DET cana-5924 106 32 l(𝒒 l(𝒒 PROPN cana-5924 106 33 ) ) PUNCT cana-5924 106 34 = = PRON cana-5924 106 35 { { PUNCT cana-5924 106 36 ( ( PUNCT cana-5924 106 37 𝛾 𝛾 PROPN cana-5924 106 38 , , PUNCT cana-5924 106 39 π›Ύπœ” π›Ύπœ” VERB cana-5924 106 40 ): ): PUNCT cana-5924 106 41 π›Ύπœ– π›Ύπœ– PROPN cana-5924 106 42 Ξ³ Ξ³ PROPN cana-5924 106 43 , , PUNCT cana-5924 106 44 πœ”πœ– πœ”πœ– PROPN cana-5924 106 45 Ο‰ Ο‰ NOUN cana-5924 106 46 } } PUNCT cana-5924 106 47 . . PUNCT cana-5924 107 1 identity identity NOUN cana-5924 107 2 free free ADJ cana-5924 107 3 set set NOUN cana-5924 107 4 Ο‰ Ο‰ PROPN cana-5924 107 5 which which PRON cana-5924 107 6 is be AUX cana-5924 107 7 closed close VERB cana-5924 107 8 under under ADP cana-5924 107 9 inverse inverse NOUN cana-5924 107 10 , , PUNCT cana-5924 107 11 leads lead VERB cana-5924 107 12 to to ADP cana-5924 107 13 the the DET cana-5924 107 14 loop loop NOUN cana-5924 107 15 free free ADJ cana-5924 107 16 and and CCONJ cana-5924 107 17 symmetric symmetric ADJ cana-5924 107 18 structure structure NOUN cana-5924 107 19 of of ADP cana-5924 107 20 the the DET cana-5924 107 21 graph graph NOUN cana-5924 107 22 . . PUNCT cana-5924 108 1 its its PRON cana-5924 108 2 vertex vertex NOUN cana-5924 108 3 and and CCONJ cana-5924 108 4 edge edge NOUN cana-5924 108 5 transitivity transitivity NOUN cana-5924 108 6 feature feature NOUN cana-5924 108 7 will will AUX cana-5924 108 8 facilitate facilitate VERB cana-5924 108 9 the the DET cana-5924 108 10 lot lot NOUN cana-5924 108 11 of of ADP cana-5924 108 12 applications application NOUN cana-5924 108 13 in in ADP cana-5924 108 14 the the DET cana-5924 108 15 network network NOUN cana-5924 108 16 models model NOUN cana-5924 108 17 with with ADP cana-5924 108 18 fault fault NOUN cana-5924 108 19 handling handle VERB cana-5924 108 20 sensor sensor NOUN cana-5924 108 21 , , PUNCT cana-5924 108 22 in in ADP cana-5924 108 23 human human ADJ cana-5924 108 24 resource resource NOUN cana-5924 108 25 mailto:*kedarbaba20@gmail.com mailto:*kedarbaba20@gmail.com X cana-5924 108 26 rilwan2020@sadakath.ac.in rilwan2020@sadakath.ac.in PROPN cana-5924 108 27 communications communication NOUN cana-5924 108 28 on on ADP cana-5924 108 29 applied apply VERB cana-5924 108 30 nonlinear nonlinear ADJ cana-5924 108 31 analysis analysis NOUN cana-5924 108 32 issn issn NOUN cana-5924 108 33 : : PUNCT cana-5924 108 34 1074 1074 NUM cana-5924 108 35 - - PUNCT cana-5924 108 36 133x 133x NUM cana-5924 108 37 vol vol NOUN cana-5924 108 38 31 31 NUM cana-5924 108 39 no no NOUN cana-5924 108 40 . . NOUN cana-5924 108 41 2 2 NUM cana-5924 108 42 ( ( PUNCT cana-5924 108 43 2024 2024 NUM cana-5924 108 44 ) ) PUNCT cana-5924 108 45 481 481 NUM cana-5924 108 46 https://internationalpubls.com https://internationalpubls.com X cana-5924 108 47 development development NOUN cana-5924 108 48 as as ADP cana-5924 108 49 a a DET cana-5924 108 50 sociometry sociometry NOUN cana-5924 108 51 diagrams diagram NOUN cana-5924 108 52 and and CCONJ cana-5924 108 53 fixation fixation NOUN cana-5924 108 54 of of ADP cana-5924 108 55 radio radio NOUN cana-5924 108 56 frequencies frequency NOUN cana-5924 108 57 etc etc X cana-5924 108 58 . . X cana-5924 108 59 let let VERB cana-5924 108 60 β„± β„± PRON cana-5924 108 61 be be AUX cana-5924 108 62 a a DET cana-5924 108 63 binary binary ADJ cana-5924 108 64 function function NOUN cana-5924 108 65 is be AUX cana-5924 108 66 defined define VERB cana-5924 108 67 from from ADP cana-5924 108 68 the the DET cana-5924 108 69 edge edge NOUN cana-5924 108 70 set set VERB cana-5924 108 71 or or CCONJ cana-5924 108 72 the the DET cana-5924 108 73 vertex vertex NOUN cana-5924 108 74 set set NOUN cana-5924 108 75 of of ADP cana-5924 108 76 a a DET cana-5924 108 77 graph graph NOUN cana-5924 108 78 𝒒 𝒒 NOUN cana-5924 108 79 to to ADP cana-5924 108 80 a a DET cana-5924 108 81 set set NOUN cana-5924 108 82 { { PUNCT cana-5924 108 83 0 0 NUM cana-5924 108 84 , , PUNCT cana-5924 108 85 1 1 NUM cana-5924 108 86 } } PUNCT cana-5924 108 87 said say VERB cana-5924 108 88 to to PART cana-5924 108 89 be be AUX cana-5924 108 90 a a DET cana-5924 108 91 cordial cordial ADJ cana-5924 108 92 function function NOUN cana-5924 108 93 if if SCONJ cana-5924 108 94 β„± β„± PROPN cana-5924 108 95 satisfies satisfy VERB cana-5924 108 96 any any DET cana-5924 108 97 one one NUM cana-5924 108 98 of of ADP cana-5924 108 99 cordial cordial ADJ cana-5924 108 100 constrain constrain NOUN cana-5924 108 101 . . PUNCT cana-5924 109 1 suppose suppose VERB cana-5924 109 2 β„± β„± PROPN cana-5924 109 3 is be AUX cana-5924 109 4 said say VERB cana-5924 109 5 to to PART cana-5924 109 6 be be AUX cana-5924 109 7 an an DET cana-5924 109 8 edge edge ADJ cana-5924 109 9 cordial cordial ADJ cana-5924 109 10 function function NOUN cana-5924 109 11 , , PUNCT cana-5924 109 12 it it PRON cana-5924 109 13 is be AUX cana-5924 109 14 defined define VERB cana-5924 109 15 from from ADP cana-5924 109 16 the the DET cana-5924 109 17 edge edge NOUN cana-5924 109 18 set set NOUN cana-5924 109 19 of of ADP cana-5924 109 20 a a DET cana-5924 109 21 graph graph NOUN cana-5924 109 22 to to ADP cana-5924 109 23 a a DET cana-5924 109 24 cordial cordial ADJ cana-5924 109 25 set set NOUN cana-5924 109 26 { { PUNCT cana-5924 109 27 0 0 NUM cana-5924 109 28 , , PUNCT cana-5924 109 29 1 1 NUM cana-5924 109 30 } } PUNCT cana-5924 109 31 and and CCONJ cana-5924 109 32 it it PRON cana-5924 109 33 will will AUX cana-5924 109 34 satisfy satisfy VERB cana-5924 109 35 the the DET cana-5924 109 36 difference difference NOUN cana-5924 109 37 between between ADP cana-5924 109 38 the the DET cana-5924 109 39 number number NOUN cana-5924 109 40 of of ADP cana-5924 109 41 edges edge NOUN cana-5924 109 42 labelled label VERB cana-5924 109 43 by by ADP cana-5924 109 44 zero zero NUM cana-5924 109 45 and and CCONJ cana-5924 109 46 number number NOUN cana-5924 109 47 of of ADP cana-5924 109 48 edges edge NOUN cana-5924 109 49 labelled label VERB cana-5924 109 50 by by ADP cana-5924 109 51 one one NUM cana-5924 109 52 is be AUX cana-5924 109 53 at at ADP cana-5924 109 54 most most ADJ cana-5924 109 55 one one NUM cana-5924 109 56 , , PUNCT cana-5924 109 57 at at ADP cana-5924 109 58 the the DET cana-5924 109 59 same same ADJ cana-5924 109 60 time time NOUN cana-5924 109 61 it it PRON cana-5924 109 62 will will AUX cana-5924 109 63 induce induce VERB cana-5924 109 64 the the DET cana-5924 109 65 vertex vertex NOUN cana-5924 109 66 labels label NOUN cana-5924 109 67 ( ( PUNCT cana-5924 109 68 sum sum NOUN cana-5924 109 69 of of ADP cana-5924 109 70 incident incident NOUN cana-5924 109 71 edge edge NOUN cana-5924 109 72 labels label NOUN cana-5924 109 73 ) ) PUNCT cana-5924 109 74 and and CCONJ cana-5924 109 75 will will AUX cana-5924 109 76 meet meet VERB cana-5924 109 77 the the DET cana-5924 109 78 similar similar ADJ cana-5924 109 79 difference difference NOUN cana-5924 109 80 as as SCONJ cana-5924 109 81 we we PRON cana-5924 109 82 discussed discuss VERB cana-5924 109 83 for for ADP cana-5924 109 84 edges edge NOUN cana-5924 109 85 of of ADP cana-5924 109 86 𝒒. 𝒒. PROPN cana-5924 109 87 it it PRON cana-5924 109 88 is be AUX cana-5924 109 89 denoted denote VERB cana-5924 109 90 |𝑣ℱ(0 |𝑣ℱ(0 ADV cana-5924 109 91 ) ) PUNCT cana-5924 109 92 βˆ’ βˆ’ ADP cana-5924 109 93 𝑣ℱ(1 𝑣ℱ(1 NOUN cana-5924 109 94 ) ) PUNCT cana-5924 109 95 | | ADV cana-5924 109 96 ≀ ≀ NUM cana-5924 109 97 1 1 NUM cana-5924 109 98 and and CCONJ cana-5924 109 99 |𝑒ℱ(0 |𝑒ℱ(0 ADJ cana-5924 109 100 ) ) PUNCT cana-5924 110 1 βˆ’ βˆ’ PROPN cana-5924 111 1 𝑒ℱ(1)| 𝑒ℱ(1)| PROPN cana-5924 111 2 ≀ ≀ NOUN cana-5924 111 3 1 1 NUM cana-5924 111 4 . . PUNCT cana-5924 112 1 in in ADP cana-5924 112 2 this this DET cana-5924 112 3 paper paper NOUN cana-5924 112 4 , , PUNCT cana-5924 112 5 highly highly ADV cana-5924 112 6 symmetric symmetric ADJ cana-5924 112 7 network network NOUN cana-5924 112 8 model model NOUN cana-5924 112 9 cayley cayley PROPN cana-5924 112 10 graph graph NOUN cana-5924 112 11 is be AUX cana-5924 112 12 binary binary NOUN cana-5924 112 13 coded code VERB cana-5924 112 14 through through ADP cana-5924 112 15 the the DET cana-5924 112 16 edge edge NOUN cana-5924 112 17 cordial cordial ADJ cana-5924 112 18 constraint constraint NOUN cana-5924 112 19 , , PUNCT cana-5924 112 20 is be AUX cana-5924 112 21 said say VERB cana-5924 112 22 to to PART cana-5924 112 23 be be AUX cana-5924 112 24 an an DET cana-5924 112 25 edge edge NOUN cana-5924 112 26 cordiality cordiality NOUN cana-5924 112 27 of of ADP cana-5924 112 28 the the DET cana-5924 112 29 given give VERB cana-5924 112 30 network network NOUN cana-5924 112 31 . . PUNCT cana-5924 113 1 the the DET cana-5924 113 2 notations notation NOUN cana-5924 113 3 used use VERB cana-5924 113 4 in in ADP cana-5924 113 5 the the DET cana-5924 113 6 manuscript manuscript NOUN cana-5924 113 7 are be AUX cana-5924 113 8 given give VERB cana-5924 113 9 below below ADV cana-5924 113 10 . . PUNCT cana-5924 114 1 2 2 X cana-5924 114 2 . . X cana-5924 114 3 edge edge VERB cana-5924 114 4 binary binary NOUN cana-5924 114 5 coding coding NOUN cana-5924 114 6 on on ADP cana-5924 114 7 cayley cayley ADJ cana-5924 114 8 graph graph NOUN cana-5924 114 9 for for ADP cana-5924 114 10 convenience convenience NOUN cana-5924 114 11 , , PUNCT cana-5924 114 12 the the DET cana-5924 114 13 vertex vertex NOUN cana-5924 114 14 set set VERB cana-5924 114 15 𝑉(𝒒 𝑉(𝒒 ADV cana-5924 114 16 ) ) PUNCT cana-5924 114 17 contains contain VERB cana-5924 114 18 the the DET cana-5924 114 19 elements element NOUN cana-5924 114 20 of of ADP cana-5924 114 21 a a DET cana-5924 114 22 finite finite ADJ cana-5924 114 23 group group NOUN cana-5924 114 24 𝛀 𝛀 PROPN cana-5924 114 25 and and CCONJ cana-5924 114 26 the the DET cana-5924 114 27 edge edge NOUN cana-5924 114 28 set set VERB cana-5924 114 29 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 114 30 ) ) PUNCT cana-5924 114 31 can can AUX cana-5924 114 32 be be AUX cana-5924 114 33 split split VERB cana-5924 114 34 into into ADP cana-5924 114 35 the the DET cana-5924 114 36 sets set NOUN cana-5924 114 37 n n PRON cana-5924 114 38 and and CCONJ cana-5924 114 39 𝑆 𝑆 PROPN cana-5924 114 40 whose whose DET cana-5924 114 41 edges edge NOUN cana-5924 114 42 are be AUX cana-5924 114 43 generated generate VERB cana-5924 114 44 by by ADP cana-5924 114 45 the the DET cana-5924 114 46 non non ADJ cana-5924 114 47 - - ADJ cana-5924 114 48 self self ADJ cana-5924 114 49 inverse inverse NOUN cana-5924 114 50 elements element NOUN cana-5924 114 51 and and CCONJ cana-5924 114 52 self self NOUN cana-5924 114 53 inverse inverse NOUN cana-5924 114 54 elements element NOUN cana-5924 114 55 of of ADP cana-5924 114 56 𝛀 𝛀 PROPN cana-5924 114 57 respectively respectively ADV cana-5924 114 58 . . PUNCT cana-5924 115 1 suppose suppose VERB cana-5924 115 2 , , PUNCT cana-5924 115 3 the the DET cana-5924 115 4 set set NOUN cana-5924 115 5 of of ADP cana-5924 115 6 l(𝒒 l(𝒒 PROPN cana-5924 115 7 ) ) PUNCT cana-5924 116 1 = = PRON cana-5924 116 2 { { PUNCT cana-5924 116 3 n1 n1 PROPN cana-5924 116 4 , , PUNCT cana-5924 116 5 n2, n2, ADV cana-5924 116 6 … … SYM cana-5924 116 7 ,nΞ· ,nΞ· NOUN cana-5924 116 8 , , PUNCT cana-5924 116 9 𝑆1 𝑆1 NOUN cana-5924 116 10 , , PUNCT cana-5924 116 11 𝑆2 𝑆2 NOUN cana-5924 116 12 , , PUNCT cana-5924 116 13 … … PUNCT cana-5924 116 14 , , PUNCT cana-5924 116 15 𝑆𝛿 𝑆𝛿 PROPN cana-5924 116 16 } } PUNCT cana-5924 116 17 . . PUNCT cana-5924 117 1 assume assume VERB cana-5924 117 2 that that SCONJ cana-5924 117 3 Ο‰ Ο‰ PROPN cana-5924 117 4 = = PRON cana-5924 117 5 { { PUNCT cana-5924 117 6 πœƒ1 πœƒ1 NOUN cana-5924 117 7 , , PUNCT cana-5924 117 8 πœƒ2 πœƒ2 PROPN cana-5924 117 9 , , PUNCT cana-5924 117 10 . . PUNCT cana-5924 117 11 . . PUNCT cana-5924 117 12 . . PUNCT cana-5924 118 1 , , PUNCT cana-5924 118 2 πœƒ2πœ‚ πœƒ2πœ‚ NOUN cana-5924 118 3 , , PUNCT cana-5924 118 4 𝛼1 𝛼1 NOUN cana-5924 118 5 , , PUNCT cana-5924 118 6 𝛼2 𝛼2 ADJ cana-5924 118 7 , , PUNCT cana-5924 118 8 . . PUNCT cana-5924 118 9 . . PUNCT cana-5924 119 1 . . PUNCT cana-5924 120 1 , , PUNCT cana-5924 120 2 𝛼𝛿 𝛼𝛿 X cana-5924 120 3 } } PUNCT cana-5924 120 4 , , PUNCT cana-5924 120 5 where where SCONJ cana-5924 120 6 πœƒπ‘– πœƒπ‘– X cana-5924 120 7 , , PUNCT cana-5924 120 8 1 1 NUM cana-5924 120 9 ≀ ≀ NUM cana-5924 120 10 𝑖 𝑖 SYM cana-5924 120 11 ≀ ≀ NOUN cana-5924 120 12 2πœ‚ 2πœ‚ NOUN cana-5924 120 13 be be AUX cana-5924 120 14 the the DET cana-5924 120 15 generators generator NOUN cana-5924 120 16 which which PRON cana-5924 120 17 produces produce VERB cana-5924 120 18 a a DET cana-5924 120 19 cycles cycle NOUN cana-5924 120 20 and and CCONJ cana-5924 120 21 𝛼𝑗 𝛼𝑗 NOUN cana-5924 120 22 , , PUNCT cana-5924 120 23 1 1 NUM cana-5924 120 24 ≀ ≀ NUM cana-5924 120 25 𝑗 𝑗 PRON cana-5924 120 26 ≀ ≀ ADJ cana-5924 120 27 𝛿 𝛿 PROPN cana-5924 120 28 be be AUX cana-5924 120 29 the the DET cana-5924 120 30 generator generator NOUN cana-5924 120 31 which which PRON cana-5924 120 32 produces produce VERB cana-5924 120 33 matchings matching NOUN cana-5924 120 34 in in ADP cana-5924 120 35 𝒒. 𝒒. PROPN cana-5924 120 36 let let VERB cana-5924 120 37 πœ‘ πœ‘ PRON cana-5924 120 38 be be AUX cana-5924 120 39 the the DET cana-5924 120 40 number number NOUN cana-5924 120 41 of of ADP cana-5924 120 42 cycles cycle NOUN cana-5924 120 43 produced produce VERB cana-5924 120 44 by by ADP cana-5924 120 45 a a DET cana-5924 120 46 generator generator NOUN cana-5924 120 47 of of ADP cana-5924 120 48 the the DET cana-5924 120 49 set set NOUN cana-5924 120 50 n n PRON cana-5924 120 51 which which PRON cana-5924 120 52 is be AUX cana-5924 120 53 a a DET cana-5924 120 54 fraction fraction NOUN cana-5924 120 55 of of ADP cana-5924 120 56 the the DET cana-5924 120 57 group group NOUN cana-5924 120 58 order order NOUN cana-5924 120 59 ( ( PUNCT cana-5924 120 60 𝜌 𝜌 ADP cana-5924 120 61 ) ) PUNCT cana-5924 120 62 and and CCONJ cana-5924 120 63 the the DET cana-5924 120 64 order order NOUN cana-5924 120 65 of of ADP cana-5924 120 66 corresponding corresponding ADJ cana-5924 120 67 generator generator NOUN cana-5924 120 68 ( ( PUNCT cana-5924 120 69 β„“ β„“ NOUN cana-5924 120 70 ) ) PUNCT cana-5924 120 71 , , PUNCT cana-5924 120 72 where where SCONJ cana-5924 120 73 β„“ β„“ PROPN cana-5924 120 74 is be AUX cana-5924 120 75 known know VERB cana-5924 120 76 as as ADP cana-5924 120 77 a a DET cana-5924 120 78 length length NOUN cana-5924 120 79 of of ADP cana-5924 120 80 that that DET cana-5924 120 81 cycle cycle NOUN cana-5924 120 82 . . PUNCT cana-5924 121 1 as as ADP cana-5924 121 2 per per ADP cana-5924 121 3 the the DET cana-5924 121 4 requirement requirement NOUN cana-5924 121 5 , , PUNCT cana-5924 121 6 algebraic algebraic ADJ cana-5924 121 7 equations equation NOUN cana-5924 121 8 and and CCONJ cana-5924 121 9 congruence congruence PROPN cana-5924 121 10 classes class NOUN cana-5924 121 11 are be AUX cana-5924 121 12 defined define VERB cana-5924 121 13 and and CCONJ cana-5924 121 14 executed execute VERB cana-5924 121 15 in in ADP cana-5924 121 16 every every DET cana-5924 121 17 labeling labeling NOUN cana-5924 121 18 event event NOUN cana-5924 121 19 . . PUNCT cana-5924 122 1 let let VERB cana-5924 122 2 us we PRON cana-5924 122 3 consider consider VERB cana-5924 122 4 the the DET cana-5924 122 5 sets set NOUN cana-5924 122 6 𝐢1 𝐢1 PRON cana-5924 122 7 , , PUNCT cana-5924 122 8 𝐢2 𝐢2 ADJ cana-5924 122 9 , , PUNCT cana-5924 122 10 𝐢3 𝐢3 NOUN cana-5924 122 11 and and CCONJ cana-5924 122 12 𝐢4 𝐢4 PROPN cana-5924 122 13 consisting consist VERB cana-5924 122 14 the the DET cana-5924 122 15 elements element NOUN cana-5924 122 16 of of ADP cana-5924 122 17 congruence congruence NOUN cana-5924 122 18 classes class NOUN cana-5924 122 19 of of ADP cana-5924 122 20 [ [ X cana-5924 122 21 1]4 1]4 PROPN cana-5924 122 22 βˆͺ βˆͺ ADV cana-5924 122 23 [ [ X cana-5924 122 24 3]4 3]4 NOUN cana-5924 122 25 , , PUNCT cana-5924 122 26 [ [ X cana-5924 122 27 0]4 0]4 NOUN cana-5924 122 28 βˆͺ βˆͺ ADP cana-5924 122 29 [ [ X cana-5924 122 30 2]4 2]4 NOUN cana-5924 122 31 , , PUNCT cana-5924 122 32 [ [ X cana-5924 122 33 1]4 1]4 NOUN cana-5924 122 34 βˆͺ βˆͺ ADP cana-5924 122 35 [ [ X cana-5924 122 36 2]4 2]4 NOUN cana-5924 122 37 and and CCONJ cana-5924 122 38 [ [ X cana-5924 122 39 0]4 0]4 NOUN cana-5924 122 40 βˆͺ βˆͺ VERB cana-5924 122 41 [ [ X cana-5924 122 42 3]4 3]4 NUM cana-5924 122 43 respectively respectively ADV cana-5924 122 44 . . PUNCT cana-5924 123 1 here here ADV cana-5924 123 2 , , PUNCT cana-5924 123 3 [ [ X cana-5924 123 4 π‘Ž]𝑛 π‘Ž]𝑛 ADV cana-5924 123 5 is be AUX cana-5924 123 6 the the DET cana-5924 123 7 notation notation NOUN cana-5924 123 8 for for ADP cana-5924 123 9 the the DET cana-5924 123 10 congruence congruence PROPN cana-5924 123 11 classes class NOUN cana-5924 123 12 of of ADP cana-5924 123 13 a a DET cana-5924 123 14 modulo modulo NOUN cana-5924 123 15 𝑛 𝑛 ADP cana-5924 123 16 such such ADJ cana-5924 123 17 that that PRON cana-5924 123 18 for for ADP cana-5924 123 19 a a DET cana-5924 123 20 fixed fix VERB cana-5924 123 21 non non ADJ cana-5924 123 22 zero zero NUM cana-5924 123 23 integer integer PROPN cana-5924 123 24 𝑛 𝑛 PROPN cana-5924 123 25 , , PUNCT cana-5924 123 26 [ [ X cana-5924 123 27 π‘Ž]𝑛 π‘Ž]𝑛 NOUN cana-5924 123 28 = = SYM cana-5924 123 29 { { PUNCT cana-5924 123 30 𝑧 𝑧 X cana-5924 123 31 ∈ ∈ PROPN cana-5924 123 32 𝒁/ 𝒁/ PROPN cana-5924 123 33 𝑧 𝑧 DET cana-5924 123 34 ≑ ≑ PROPN cana-5924 123 35 π‘Ž(π‘šπ‘œπ‘‘ π‘Ž(π‘šπ‘œπ‘‘ PROPN cana-5924 123 36 𝑛 𝑛 PROPN cana-5924 123 37 ) ) PUNCT cana-5924 123 38 } } PUNCT cana-5924 123 39 . . PUNCT cana-5924 124 1 proposition proposition NOUN cana-5924 124 2 2.1 2.1 NUM cana-5924 124 3 . . PUNCT cana-5924 125 1 for for ADP cana-5924 125 2 a a DET cana-5924 125 3 group group NOUN cana-5924 125 4 𝛀 𝛀 PROPN cana-5924 125 5 of of ADP cana-5924 125 6 order order NOUN cana-5924 125 7 𝜌 𝜌 X cana-5924 125 8 = = SYM cana-5924 125 9 [ [ X cana-5924 125 10 0]4 0]4 X cana-5924 125 11 and and CCONJ cana-5924 125 12 Ο‰ Ο‰ NUM cana-5924 125 13 βŠ† βŠ† NUM cana-5924 125 14 𝛀 𝛀 PROPN cana-5924 125 15 , , PUNCT cana-5924 125 16 where where SCONJ cana-5924 125 17 Ο‰ Ο‰ NOUN cana-5924 125 18 is be AUX cana-5924 125 19 free free ADJ cana-5924 125 20 from from ADP cana-5924 125 21 non non ADJ cana-5924 125 22 - - ADJ cana-5924 125 23 self self ADJ cana-5924 125 24 inverse inverse ADJ cana-5924 125 25 elements element NOUN cana-5924 125 26 of of ADP cana-5924 125 27 Ξ³ Ξ³ PROPN cana-5924 125 28 . . PROPN cana-5924 126 1 then then ADV cana-5924 126 2 the the DET cana-5924 126 3 cayley cayley ADJ cana-5924 126 4 graph graph NOUN cana-5924 126 5 𝒒 𝒒 PROPN cana-5924 126 6 admits admit VERB cana-5924 126 7 edge edge NOUN cana-5924 126 8 cordiality cordiality NOUN cana-5924 126 9 . . PUNCT cana-5924 127 1 proof proof NOUN cana-5924 127 2 . . PUNCT cana-5924 128 1 if if SCONJ cana-5924 128 2 Ο‰ Ο‰ NUM cana-5924 128 3 = = SYM cana-5924 128 4 { { PUNCT cana-5924 128 5 𝛼1 𝛼1 NOUN cana-5924 128 6 , , PUNCT cana-5924 128 7 𝛼2 𝛼2 ADJ cana-5924 128 8 , , PUNCT cana-5924 128 9 . . PUNCT cana-5924 128 10 . . PUNCT cana-5924 128 11 . . PUNCT cana-5924 129 1 , , PUNCT cana-5924 129 2 𝛼𝛿 𝛼𝛿 X cana-5924 129 3 } } PUNCT cana-5924 129 4 . . PUNCT cana-5924 130 1 then then ADV cana-5924 130 2 the the DET cana-5924 130 3 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 130 4 ) ) PUNCT cana-5924 131 1 = = VERB cana-5924 131 2 𝑆1 𝑆1 NOUN cana-5924 131 3 βˆͺ βˆͺ ADJ cana-5924 131 4 𝑆2 𝑆2 PROPN cana-5924 131 5 βˆͺ βˆͺ X cana-5924 131 6 Β· Β· PUNCT cana-5924 131 7 Β· Β· PUNCT cana-5924 131 8 Β· Β· PUNCT cana-5924 131 9 βˆͺ βˆͺ ADP cana-5924 131 10 𝑆𝛿 𝑆𝛿 PROPN cana-5924 131 11 and and CCONJ cana-5924 131 12 without without ADP cana-5924 131 13 loss loss NOUN cana-5924 131 14 of of ADP cana-5924 131 15 generality generality NOUN cana-5924 131 16 , , PUNCT cana-5924 131 17 𝛼1 𝛼1 PROPN cana-5924 131 18 ∈ ∈ PROPN cana-5924 131 19 Ο‰ Ο‰ PROPN cana-5924 131 20 , , PUNCT cana-5924 131 21 𝑂(𝛼1 𝑂(𝛼1 PROPN cana-5924 131 22 ) ) PUNCT cana-5924 131 23 = = SYM cana-5924 131 24 2 2 NUM cana-5924 131 25 which which PRON cana-5924 131 26 generates generate VERB cana-5924 131 27 the the DET cana-5924 131 28 𝜌 𝜌 ADP cana-5924 131 29 2 2 NUM cana-5924 131 30 matchings matching NOUN cana-5924 131 31 in in ADP cana-5924 131 32 𝑆1 𝑆1 NOUN cana-5924 131 33 . . PUNCT cana-5924 132 1 i.e i.e NOUN cana-5924 132 2 , , PUNCT cana-5924 132 3 𝑆1 𝑆1 NOUN cana-5924 132 4 = = SYM cana-5924 132 5 π‘†πœ‰1 π‘†πœ‰1 NOUN cana-5924 132 6 1 1 NUM cana-5924 132 7 βˆͺ βˆͺ ADP cana-5924 132 8 π‘†πœ‰2 π‘†πœ‰2 NOUN cana-5924 132 9 βˆͺ βˆͺ NOUN cana-5924 132 10 . . PUNCT cana-5924 132 11 . . PUNCT cana-5924 133 1 .βˆͺ .βˆͺ PROPN cana-5924 134 1 π‘†πœ‰πœŒ π‘†πœ‰πœŒ PROPN cana-5924 134 2 2 2 NUM cana-5924 134 3 where where SCONJ cana-5924 134 4 π‘†πœ‰π‘– π‘†πœ‰π‘– PROPN cana-5924 134 5 = = PROPN cana-5924 134 6 𝑣𝑖 𝑣𝑖 ADV cana-5924 134 7 . . PUNCT cana-5924 135 1 𝛼1 𝛼1 NOUN cana-5924 135 2 with with ADP cana-5924 135 3 each each DET cana-5924 135 4 matching matching NOUN cana-5924 135 5 of of ADP cana-5924 135 6 length length NOUN cana-5924 135 7 exactly exactly ADV cana-5924 135 8 2 2 NUM cana-5924 135 9 . . X cana-5924 135 10 communications communication NOUN cana-5924 135 11 on on ADP cana-5924 135 12 applied apply VERB cana-5924 135 13 nonlinear nonlinear ADJ cana-5924 135 14 analysis analysis NOUN cana-5924 135 15 issn issn NOUN cana-5924 135 16 : : PUNCT cana-5924 135 17 1074 1074 NUM cana-5924 135 18 - - PUNCT cana-5924 135 19 133x 133x NUM cana-5924 135 20 vol vol NOUN cana-5924 135 21 31 31 NUM cana-5924 135 22 no no NOUN cana-5924 135 23 . . NOUN cana-5924 135 24 2 2 NUM cana-5924 135 25 ( ( PUNCT cana-5924 135 26 2024 2024 NUM cana-5924 135 27 ) ) PUNCT cana-5924 135 28 482 482 NUM cana-5924 135 29 https://internationalpubls.com https://internationalpubls.com X cana-5924 135 30 event event NOUN cana-5924 135 31 1 1 NUM cana-5924 135 32 . . PUNCT cana-5924 136 1 if if SCONJ cana-5924 136 2 |Ο‰| |Ο‰| PROPN cana-5924 136 3 is be AUX cana-5924 136 4 odd odd ADJ cana-5924 136 5 , , PUNCT cana-5924 136 6 then then ADV cana-5924 136 7 𝛿 𝛿 PRON cana-5924 136 8 must must AUX cana-5924 136 9 be be AUX cana-5924 136 10 odd odd ADJ cana-5924 136 11 . . PUNCT cana-5924 137 1 executing execute VERB cana-5924 137 2 a a DET cana-5924 137 3 cordial cordial ADJ cana-5924 137 4 function function NOUN cana-5924 137 5 β„± β„± PROPN cana-5924 137 6 from from ADP cana-5924 137 7 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 137 8 ) ) PUNCT cana-5924 137 9 to to ADP cana-5924 137 10 the the DET cana-5924 137 11 set set NOUN cana-5924 137 12 { { PUNCT cana-5924 137 13 0 0 NUM cana-5924 137 14 , , PUNCT cana-5924 137 15 1 1 NUM cana-5924 137 16 } } PUNCT cana-5924 137 17 such such ADJ cana-5924 137 18 that that SCONJ cana-5924 137 19 the the DET cana-5924 137 20 above above ADJ cana-5924 137 21 labelled label VERB cana-5924 137 22 assignment assignment NOUN cana-5924 137 23 persuading persuade VERB cana-5924 137 24 that that SCONJ cana-5924 137 25 , , PUNCT cana-5924 137 26 the the DET cana-5924 137 27 sum sum NOUN cana-5924 137 28 of of ADP cana-5924 137 29 the the DET cana-5924 137 30 labels label NOUN cana-5924 137 31 of of ADP cana-5924 137 32 incident incident NOUN cana-5924 137 33 edges edge NOUN cana-5924 137 34 on on ADP cana-5924 137 35 every every DET cana-5924 137 36 vertex vertex NOUN cana-5924 137 37 such such ADJ cana-5924 137 38 that that PRON cana-5924 137 39 for for ADP cana-5924 137 40 all all PRON cana-5924 137 41 1 1 NUM cana-5924 137 42 ≀ ≀ NUM cana-5924 137 43 𝑖 𝑖 SYM cana-5924 137 44 ≀ ≀ NOUN cana-5924 137 45 𝜌 𝜌 ADP cana-5924 137 46 , , PUNCT cana-5924 137 47 thus thus ADV cana-5924 137 48 , , PUNCT cana-5924 137 49 𝑣ℱ(0 𝑣ℱ(0 NUM cana-5924 137 50 ) ) PUNCT cana-5924 137 51 = = SYM cana-5924 137 52 𝑣ℱ(1 𝑣ℱ(1 NOUN cana-5924 137 53 ) ) PUNCT cana-5924 137 54 = = PUNCT cana-5924 137 55 𝜌 𝜌 ADP cana-5924 137 56 2 2 NUM cana-5924 137 57 and and CCONJ cana-5924 137 58 𝑒ℱ(0 𝑒ℱ(0 NUM cana-5924 137 59 ) ) PUNCT cana-5924 137 60 = = NOUN cana-5924 137 61 𝑒ℱ(1 𝑒ℱ(1 NOUN cana-5924 137 62 ) ) PUNCT cana-5924 137 63 = = PUNCT cana-5924 137 64 𝜌|Ο‰| 𝜌|Ο‰| VERB cana-5924 137 65 4 4 NUM cana-5924 137 66 . . PUNCT cana-5924 138 1 event event NOUN cana-5924 138 2 2 2 NUM cana-5924 138 3 . . PUNCT cana-5924 139 1 if if SCONJ cana-5924 139 2 |Ο‰| |Ο‰| PROPN cana-5924 139 3 is be AUX cana-5924 139 4 even even ADV cana-5924 139 5 , , PUNCT cana-5924 139 6 then then ADV cana-5924 139 7 𝛿 𝛿 PRON cana-5924 139 8 must must AUX cana-5924 139 9 be be AUX cana-5924 139 10 even even ADV cana-5924 139 11 . . PUNCT cana-5924 140 1 executing execute VERB cana-5924 140 2 a a DET cana-5924 140 3 cordial cordial ADJ cana-5924 140 4 function function NOUN cana-5924 140 5 β„± β„± PROPN cana-5924 140 6 on on ADP cana-5924 140 7 the the DET cana-5924 140 8 edge edge NOUN cana-5924 140 9 set set NOUN cana-5924 140 10 of of ADP cana-5924 140 11 ( ( PUNCT cana-5924 140 12 𝒒 𝒒 NOUN cana-5924 140 13 ) ) PUNCT cana-5924 140 14 such such ADJ cana-5924 140 15 that that SCONJ cana-5924 140 16 , , PUNCT cana-5924 140 17 the the DET cana-5924 140 18 above above ADJ cana-5924 140 19 labeled label VERB cana-5924 140 20 assignment assignment NOUN cana-5924 140 21 persuading persuade VERB cana-5924 140 22 that that SCONJ cana-5924 140 23 , , PUNCT cana-5924 140 24 the the DET cana-5924 140 25 sum sum NOUN cana-5924 140 26 of of ADP cana-5924 140 27 the the DET cana-5924 140 28 labels label NOUN cana-5924 140 29 of of ADP cana-5924 140 30 incident incident NOUN cana-5924 140 31 edges edge NOUN cana-5924 140 32 on on ADP cana-5924 140 33 every every DET cana-5924 140 34 vertex vertex NOUN cana-5924 140 35 for for ADP cana-5924 140 36 all all PRON cana-5924 140 37 1 1 NUM cana-5924 140 38 ≀ ≀ NUM cana-5924 140 39 𝑖 𝑖 SYM cana-5924 140 40 ≀ ≀ NUM cana-5924 140 41 𝜌 𝜌 ADP cana-5924 140 42 , , PUNCT cana-5924 140 43 theorem theorem VERB cana-5924 140 44 2.2 2.2 NUM cana-5924 140 45 . . PUNCT cana-5924 141 1 let let VERB cana-5924 141 2 Ο‰ Ο‰ NOUN cana-5924 141 3 βŠ† βŠ† NUM cana-5924 141 4 𝛀 𝛀 PRON cana-5924 141 5 be be VERB cana-5924 141 6 a a DET cana-5924 141 7 generating generate VERB cana-5924 141 8 set set NOUN cana-5924 141 9 , , PUNCT cana-5924 141 10 of of ADP cana-5924 141 11 a a DET cana-5924 141 12 group group NOUN cana-5924 141 13 Ξ³ Ξ³ X cana-5924 141 14 whose whose DET cana-5924 141 15 order order NOUN cana-5924 141 16 is be AUX cana-5924 141 17 𝜌 𝜌 ADP cana-5924 141 18 = = X cana-5924 142 1 [ [ X cana-5924 142 2 0]4 0]4 X cana-5924 142 3 . . PUNCT cana-5924 143 1 if if SCONJ cana-5924 143 2 |Ο‰| |Ο‰| PROPN cana-5924 143 3 is be AUX cana-5924 143 4 odd odd ADJ cana-5924 143 5 and and CCONJ cana-5924 143 6 Ο‰ Ο‰ PROPN cana-5924 143 7 contains contain VERB cana-5924 143 8 an an DET cana-5924 143 9 element element NOUN cana-5924 143 10 of of ADP cana-5924 143 11 order order NOUN cana-5924 143 12 at at ADP cana-5924 143 13 least least ADJ cana-5924 143 14 4β„“ 4β„“ NUM cana-5924 143 15 , , PUNCT cana-5924 143 16 β„“ β„“ PROPN cana-5924 143 17 β‰  β‰  PROPN cana-5924 143 18 0 0 NUM cana-5924 143 19 . . PUNCT cana-5924 144 1 then then ADV cana-5924 144 2 the the DET cana-5924 144 3 cayley cayley ADJ cana-5924 144 4 graph graph NOUN cana-5924 144 5 g g PROPN cana-5924 144 6 admits admit VERB cana-5924 144 7 edge edge NOUN cana-5924 144 8 binary binary ADJ cana-5924 144 9 coding coding NOUN cana-5924 144 10 . . PUNCT cana-5924 145 1 proof proof NOUN cana-5924 145 2 . . PUNCT cana-5924 146 1 suppose suppose VERB cana-5924 146 2 , , PUNCT cana-5924 146 3 Ο‰ Ο‰ PROPN cana-5924 146 4 = = SYM cana-5924 146 5 { { PUNCT cana-5924 146 6 πœƒ1 πœƒ1 NOUN cana-5924 146 7 , , PUNCT cana-5924 146 8 πœƒ2 πœƒ2 PROPN cana-5924 146 9 , , PUNCT cana-5924 146 10 . . PUNCT cana-5924 146 11 . . PUNCT cana-5924 146 12 . . PUNCT cana-5924 147 1 , , PUNCT cana-5924 147 2 πœƒ2πœ‚ πœƒ2πœ‚ NOUN cana-5924 147 3 , , PUNCT cana-5924 147 4 𝛼1 𝛼1 NOUN cana-5924 147 5 , , PUNCT cana-5924 147 6 𝛼2 𝛼2 ADJ cana-5924 147 7 , , PUNCT cana-5924 147 8 . . PUNCT cana-5924 147 9 . . PUNCT cana-5924 148 1 . . PUNCT cana-5924 149 1 , , PUNCT cana-5924 149 2 𝛼𝛿 𝛼𝛿 X cana-5924 149 3 } } PUNCT cana-5924 149 4 . . PUNCT cana-5924 150 1 note note VERB cana-5924 150 2 that that SCONJ cana-5924 150 3 |Ο‰| |Ο‰| PROPN cana-5924 150 4 is be AUX cana-5924 150 5 odd odd ADJ cana-5924 150 6 which which PRON cana-5924 150 7 implies imply VERB cana-5924 150 8 𝛿 𝛿 PROPN cana-5924 150 9 must must AUX cana-5924 150 10 be be AUX cana-5924 150 11 odd odd ADJ cana-5924 150 12 . . PUNCT cana-5924 151 1 let let VERB cana-5924 151 2 the the DET cana-5924 151 3 edge edge NOUN cana-5924 151 4 set set VERB cana-5924 151 5 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 151 6 ) ) PUNCT cana-5924 151 7 has have VERB cana-5924 151 8 a a DET cana-5924 151 9 partition partition NOUN cana-5924 151 10 { { PUNCT cana-5924 151 11 n1 n1 NOUN cana-5924 151 12 , , PUNCT cana-5924 151 13 n2, n2, ADV cana-5924 151 14 … … SYM cana-5924 151 15 ,nΞ· ,nΞ· NOUN cana-5924 151 16 , , PUNCT cana-5924 151 17 𝑆1 𝑆1 NOUN cana-5924 151 18 , , PUNCT cana-5924 151 19 𝑆2 𝑆2 NOUN cana-5924 151 20 , , PUNCT cana-5924 151 21 … … PUNCT cana-5924 151 22 , , PUNCT cana-5924 151 23 𝑆𝛿 𝑆𝛿 PROPN cana-5924 151 24 } } PUNCT cana-5924 151 25 . . PUNCT cana-5924 152 1 arbitrarily arbitrarily ADV cana-5924 152 2 , , PUNCT cana-5924 152 3 we we PRON cana-5924 152 4 assume assume VERB cana-5924 152 5 that that SCONJ cana-5924 152 6 Ο‰ Ο‰ PROPN cana-5924 152 7 is be AUX cana-5924 152 8 arranged arrange VERB cana-5924 152 9 so so SCONJ cana-5924 152 10 that that SCONJ cana-5924 152 11 πœƒπœ‚ πœƒπœ‚ PRON cana-5924 152 12 is be AUX cana-5924 152 13 an an DET cana-5924 152 14 element element NOUN cana-5924 152 15 , , PUNCT cana-5924 152 16 whose whose DET cana-5924 152 17 order order NOUN cana-5924 152 18 is be AUX cana-5924 152 19 at at ADP cana-5924 152 20 least least ADJ cana-5924 152 21 4β„“ 4β„“ NUM cana-5924 152 22 , , PUNCT cana-5924 152 23 β„“ β„“ PROPN cana-5924 152 24 β‰  β‰  PROPN cana-5924 152 25 0 0 NUM cana-5924 152 26 . . PUNCT cana-5924 153 1 if if SCONJ cana-5924 153 2 nΞ· nΞ· PROPN cana-5924 153 3 contains contain VERB cana-5924 153 4 communications communication NOUN cana-5924 153 5 on on ADP cana-5924 153 6 applied apply VERB cana-5924 153 7 nonlinear nonlinear ADJ cana-5924 153 8 analysis analysis NOUN cana-5924 153 9 issn issn NOUN cana-5924 153 10 : : PUNCT cana-5924 153 11 1074 1074 NUM cana-5924 153 12 - - PUNCT cana-5924 153 13 133x 133x NUM cana-5924 153 14 vol vol NOUN cana-5924 153 15 31 31 NUM cana-5924 153 16 no no NOUN cana-5924 153 17 . . NOUN cana-5924 153 18 2 2 NUM cana-5924 153 19 ( ( PUNCT cana-5924 153 20 2024 2024 NUM cana-5924 153 21 ) ) PUNCT cana-5924 153 22 483 483 NUM cana-5924 153 23 https://internationalpubls.com https://internationalpubls.com X cana-5924 153 24 πœ‘ πœ‘ PROPN cana-5924 153 25 cycles cycle NOUN cana-5924 153 26 and and CCONJ cana-5924 153 27 the the DET cana-5924 153 28 edges edge NOUN cana-5924 153 29 of of ADP cana-5924 153 30 πœ‘ πœ‘ PROPN cana-5924 153 31 cycles cycle NOUN cana-5924 153 32 are be AUX cana-5924 153 33 generated generate VERB cana-5924 153 34 by by ADP cana-5924 153 35 πœ‚ πœ‚ PROPN cana-5924 153 36 , , PUNCT cana-5924 153 37 can can AUX cana-5924 153 38 be be AUX cana-5924 153 39 represented represent VERB cana-5924 153 40 by by ADP cana-5924 153 41 a a DET cana-5924 153 42 set set ADJ cana-5924 153 43 nΞ·= nΞ·= NOUN cana-5924 153 44 nΞΆ1 nΞΆ1 ADP cana-5924 153 45 βˆͺnΞΆ2βˆͺ βˆͺnΞΆ2βˆͺ NOUN cana-5924 153 46 … … PUNCT cana-5924 153 47 βˆͺnΞΆΟ† βˆͺnΞΆΟ† X cana-5924 153 48 with with ADP cana-5924 153 49 each each DET cana-5924 153 50 cycle cycle NOUN cana-5924 153 51 of of ADP cana-5924 153 52 length length NOUN cana-5924 153 53 β„“. β„“. NOUN cana-5924 153 54 event event NOUN cana-5924 153 55 1 1 NUM cana-5924 153 56 . . PUNCT cana-5924 154 1 if if SCONJ cana-5924 154 2 πœ‚ πœ‚ PROPN cana-5924 154 3 β‰₯ β‰₯ NUM cana-5924 154 4 1 1 NUM cana-5924 154 5 , , PUNCT cana-5924 154 6 πœ‚ πœ‚ NOUN cana-5924 154 7 is be AUX cana-5924 154 8 even even ADV cana-5924 154 9 and and CCONJ cana-5924 154 10 𝛿 𝛿 ADJ cana-5924 154 11 is be AUX cana-5924 154 12 odd odd ADJ cana-5924 154 13 . . PUNCT cana-5924 155 1 executing execute VERB cana-5924 155 2 a a DET cana-5924 155 3 cordial cordial ADJ cana-5924 155 4 function function NOUN cana-5924 155 5 β„± β„± PROPN cana-5924 155 6 from from ADP cana-5924 155 7 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 155 8 ) ) PUNCT cana-5924 155 9 to to ADP cana-5924 155 10 the the DET cana-5924 155 11 set set NOUN cana-5924 155 12 { { PUNCT cana-5924 155 13 0 0 NUM cana-5924 155 14 , , PUNCT cana-5924 155 15 1 1 NUM cana-5924 155 16 } } PUNCT cana-5924 155 17 such such ADJ cana-5924 155 18 that that SCONJ cana-5924 155 19 , , PUNCT cana-5924 155 20 event event NOUN cana-5924 155 21 2 2 NUM cana-5924 155 22 . . PUNCT cana-5924 156 1 if if SCONJ cana-5924 156 2 πœ‚ πœ‚ PROPN cana-5924 156 3 β‰₯ β‰₯ NUM cana-5924 156 4 1 1 NUM cana-5924 156 5 , , PUNCT cana-5924 156 6 Ξ· Ξ· PROPN cana-5924 156 7 is be AUX cana-5924 156 8 odd odd ADJ cana-5924 156 9 which which PRON cana-5924 156 10 implies imply VERB cana-5924 156 11 Ξ΄ Ξ΄ PROPN cana-5924 156 12 is be AUX cana-5924 156 13 odd odd ADJ cana-5924 156 14 . . PUNCT cana-5924 157 1 executing execute VERB cana-5924 157 2 a a DET cana-5924 157 3 be be NOUN cana-5924 157 4 a a DET cana-5924 157 5 cordial cordial ADJ cana-5924 157 6 function function NOUN cana-5924 157 7 β„± β„± PROPN cana-5924 157 8 from from ADP cana-5924 157 9 l(𝒒 l(𝒒 PROPN cana-5924 157 10 ) ) PUNCT cana-5924 157 11 to to ADP cana-5924 157 12 the the DET cana-5924 157 13 set set NOUN cana-5924 157 14 { { PUNCT cana-5924 157 15 0 0 NUM cana-5924 157 16 , , PUNCT cana-5924 157 17 1 1 NUM cana-5924 157 18 } } PUNCT cana-5924 157 19 such such ADJ cana-5924 157 20 that that SCONJ cana-5924 157 21 the the DET cana-5924 157 22 above above ADJ cana-5924 157 23 two two NUM cana-5924 157 24 events event NOUN cana-5924 157 25 will will AUX cana-5924 157 26 ensure ensure VERB cana-5924 157 27 that that SCONJ cana-5924 157 28 the the DET cana-5924 157 29 persuaded persuade VERB cana-5924 157 30 vertex vertex NOUN cana-5924 157 31 sum sum NOUN cana-5924 157 32 of of ADP cana-5924 157 33 𝑉 𝑉 PROPN cana-5924 157 34 ( ( PUNCT cana-5924 157 35 𝒒 𝒒 PROPN cana-5924 157 36 ) ) PUNCT cana-5924 157 37 is be AUX cana-5924 157 38 , , PUNCT cana-5924 157 39 for for ADP cana-5924 157 40 all all PRON cana-5924 157 41 1 1 NUM cana-5924 157 42 ≀ ≀ NUM cana-5924 157 43 𝑖 𝑖 SYM cana-5924 157 44 ≀ ≀ NUM cana-5924 157 45 𝜌. 𝜌. NOUN cana-5924 157 46 event event NOUN cana-5924 157 47 3 3 NUM cana-5924 157 48 . . PUNCT cana-5924 157 49 suppose suppose VERB cana-5924 157 50 πœ‚ πœ‚ NOUN cana-5924 157 51 = = SYM cana-5924 157 52 0 0 NUM cana-5924 157 53 and and CCONJ cana-5924 157 54 odd odd ADJ cana-5924 157 55 Ξ΄ Ξ΄ PROPN cana-5924 157 56 . . PUNCT cana-5924 158 1 at at ADP cana-5924 158 2 this this DET cana-5924 158 3 instance instance NOUN cana-5924 158 4 the the DET cana-5924 158 5 generating generating NOUN cana-5924 158 6 subset subset NOUN cana-5924 158 7 Ο‰ Ο‰ PROPN cana-5924 158 8 contains contain VERB cana-5924 158 9 only only ADV cana-5924 158 10 the the DET cana-5924 158 11 self self NOUN cana-5924 158 12 inverse(order inverse(order PROPN cana-5924 158 13 two two NUM cana-5924 158 14 ) ) PUNCT cana-5924 158 15 elements element NOUN cana-5924 158 16 of of ADP cana-5924 158 17 𝛀. 𝛀. PROPN cana-5924 158 18 in in ADP cana-5924 158 19 this this DET cana-5924 158 20 case case NOUN cana-5924 158 21 , , PUNCT cana-5924 158 22 by by ADP cana-5924 158 23 proposition proposition NOUN cana-5924 158 24 2.1 2.1 NUM cana-5924 158 25 the the DET cana-5924 158 26 proof proof NOUN cana-5924 158 27 is be AUX cana-5924 158 28 immediate immediate ADJ cana-5924 158 29 . . PUNCT cana-5924 159 1 thus thus ADV cana-5924 159 2 the the DET cana-5924 159 3 network network NOUN cana-5924 159 4 model(𝒒 model(𝒒 PROPN cana-5924 159 5 ) ) PUNCT cana-5924 159 6 can can AUX cana-5924 159 7 be be AUX cana-5924 159 8 encoded encode VERB cana-5924 159 9 with with ADP cana-5924 159 10 the the DET cana-5924 159 11 binary binary ADJ cana-5924 159 12 labels label NOUN cana-5924 159 13 . . PUNCT cana-5924 160 1 example example NOUN cana-5924 160 2 2.3 2.3 NUM cana-5924 160 3 . . PUNCT cana-5924 161 1 the the DET cana-5924 161 2 following follow VERB cana-5924 161 3 figure figure NOUN cana-5924 161 4 shows show VERB cana-5924 161 5 the the DET cana-5924 161 6 e e NOUN cana-5924 161 7 - - NOUN cana-5924 161 8 cordiality cordiality NOUN cana-5924 161 9 of of ADP cana-5924 161 10 the the DET cana-5924 161 11 cayley cayley ADJ cana-5924 161 12 graph graph NOUN cana-5924 161 13 𝒒 𝒒 NOUN cana-5924 161 14 with with ADP cana-5924 161 15 respect respect NOUN cana-5924 161 16 to to ADP cana-5924 161 17 the the DET cana-5924 161 18 group group NOUN cana-5924 161 19 𝑆4 𝑆4 PROPN cana-5924 161 20 whose whose DET cana-5924 161 21 order order NOUN cana-5924 161 22 is be AUX cana-5924 161 23 24 24 NUM cana-5924 161 24 ≑ ≑ PROPN cana-5924 161 25 0 0 PUNCT cana-5924 162 1 ( ( PUNCT cana-5924 162 2 mod mod PROPN cana-5924 162 3 4 4 NUM cana-5924 162 4 ) ) PUNCT cana-5924 162 5 . . PUNCT cana-5924 163 1 this this DET cana-5924 163 2 network network NOUN cana-5924 163 3 is be AUX cana-5924 163 4 binary binary NOUN cana-5924 163 5 encoded encode VERB cana-5924 163 6 by by ADP cana-5924 163 7 applying apply VERB cana-5924 163 8 the the DET cana-5924 163 9 binary binary NOUN cana-5924 163 10 encoding encoding NOUN cana-5924 163 11 function function NOUN cana-5924 163 12 of of ADP cana-5924 163 13 theorem theorem NOUN cana-5924 163 14 2.2 2.2 NUM cana-5924 163 15 . . PUNCT cana-5924 164 1 the the DET cana-5924 164 2 dotted dotted ADJ cana-5924 164 3 lines line NOUN cana-5924 164 4 and and CCONJ cana-5924 164 5 non non ADJ cana-5924 164 6 - - ADJ cana-5924 164 7 darkened darken VERB cana-5924 164 8 nodes node NOUN cana-5924 164 9 indicating indicate VERB cana-5924 164 10 the the DET cana-5924 164 11 code code NOUN cana-5924 164 12 zero zero NUM cana-5924 164 13 and and CCONJ cana-5924 164 14 the the DET cana-5924 164 15 non non ADJ cana-5924 164 16 - - ADJ cana-5924 164 17 dotted dotted ADJ cana-5924 164 18 lines line NOUN cana-5924 164 19 and and CCONJ cana-5924 164 20 darkened darken VERB cana-5924 164 21 nodes node NOUN cana-5924 164 22 indicating indicate VERB cana-5924 164 23 code code NOUN cana-5924 164 24 is be AUX cana-5924 164 25 one one NUM cana-5924 164 26 . . PUNCT cana-5924 165 1 communications communication NOUN cana-5924 165 2 on on ADP cana-5924 165 3 applied apply VERB cana-5924 165 4 nonlinear nonlinear ADJ cana-5924 165 5 analysis analysis NOUN cana-5924 165 6 issn issn NOUN cana-5924 165 7 : : PUNCT cana-5924 165 8 1074 1074 NUM cana-5924 165 9 - - PUNCT cana-5924 165 10 133x 133x NUM cana-5924 165 11 vol vol NOUN cana-5924 165 12 31 31 NUM cana-5924 165 13 no no NOUN cana-5924 165 14 . . NOUN cana-5924 165 15 2 2 NUM cana-5924 165 16 ( ( PUNCT cana-5924 165 17 2024 2024 NUM cana-5924 165 18 ) ) PUNCT cana-5924 165 19 484 484 NUM cana-5924 165 20 https://internationalpubls.com https://internationalpubls.com X cana-5924 165 21 figure figure NOUN cana-5924 165 22 1 1 NUM cana-5924 165 23 : : PUNCT cana-5924 165 24 cay(𝑆4 cay(𝑆4 NOUN cana-5924 165 25 , , PUNCT cana-5924 165 26 Ο‰ Ο‰ NOUN cana-5924 165 27 ) ) PUNCT cana-5924 165 28 where where SCONJ cana-5924 165 29 Ο‰ Ο‰ NOUN cana-5924 165 30 = = SYM cana-5924 165 31 { { PUNCT cana-5924 165 32 ( ( PUNCT cana-5924 165 33 24 24 NUM cana-5924 165 34 ) ) PUNCT cana-5924 165 35 , , PUNCT cana-5924 165 36 ( ( PUNCT cana-5924 165 37 12)(34 12)(34 NUM cana-5924 165 38 ) ) PUNCT cana-5924 165 39 , , PUNCT cana-5924 165 40 ( ( PUNCT cana-5924 165 41 23)(14 23)(14 NUM cana-5924 165 42 ) ) PUNCT cana-5924 165 43 } } PUNCT cana-5924 165 44 theorem theorem VERB cana-5924 165 45 2.4 2.4 NUM cana-5924 165 46 . . PUNCT cana-5924 166 1 let let VERB cana-5924 166 2 Ο‰ Ο‰ NOUN cana-5924 166 3 βŠ† βŠ† NUM cana-5924 166 4 𝛀 𝛀 PRON cana-5924 166 5 be be VERB cana-5924 166 6 a a DET cana-5924 166 7 generating generating NOUN cana-5924 166 8 subset subset NOUN cana-5924 166 9 of of ADP cana-5924 166 10 a a DET cana-5924 166 11 group group NOUN cana-5924 166 12 𝛀 𝛀 PROPN cana-5924 166 13 whose whose DET cana-5924 166 14 order order NOUN cana-5924 166 15 is be AUX cana-5924 166 16 𝜌 𝜌 ADP cana-5924 166 17 = = X cana-5924 167 1 [ [ X cana-5924 167 2 0]4 0]4 X cana-5924 167 3 . . PUNCT cana-5924 168 1 if if SCONJ cana-5924 168 2 |Ο‰| |Ο‰| NUM cana-5924 168 3 is be AUX cana-5924 168 4 even even ADV cana-5924 168 5 and and CCONJ cana-5924 168 6 Ο‰ Ο‰ PROPN cana-5924 168 7 contains contain VERB cana-5924 168 8 an an DET cana-5924 168 9 element element NOUN cana-5924 168 10 of of ADP cana-5924 168 11 order order NOUN cana-5924 168 12 at at ADP cana-5924 168 13 least least ADJ cana-5924 168 14 4β„“ 4β„“ NUM cana-5924 168 15 , , PUNCT cana-5924 168 16 β„“ β„“ PROPN cana-5924 168 17 β‰  β‰  PROPN cana-5924 168 18 0 0 NUM cana-5924 168 19 . . PUNCT cana-5924 169 1 then then ADV cana-5924 169 2 the the DET cana-5924 169 3 network network NOUN cana-5924 169 4 model model NOUN cana-5924 169 5 cayley cayley PROPN cana-5924 169 6 graph graph NOUN cana-5924 169 7 𝒒 𝒒 PROPN cana-5924 169 8 is be AUX cana-5924 169 9 binary binary ADJ cana-5924 169 10 codable codable NOUN cana-5924 169 11 . . PUNCT cana-5924 170 1 proof proof NOUN cana-5924 170 2 . . PUNCT cana-5924 171 1 suppose suppose VERB cana-5924 171 2 , , PUNCT cana-5924 171 3 Ο‰ Ο‰ PROPN cana-5924 171 4 = = SYM cana-5924 171 5 { { PUNCT cana-5924 171 6 πœƒ1 πœƒ1 NOUN cana-5924 171 7 , , PUNCT cana-5924 171 8 πœƒ2 πœƒ2 PROPN cana-5924 171 9 , , PUNCT cana-5924 171 10 . . PUNCT cana-5924 171 11 . . PUNCT cana-5924 171 12 . . PUNCT cana-5924 172 1 , , PUNCT cana-5924 172 2 πœƒ2πœ‚ πœƒ2πœ‚ NOUN cana-5924 172 3 , , PUNCT cana-5924 172 4 𝛼1 𝛼1 NOUN cana-5924 172 5 , , PUNCT cana-5924 172 6 𝛼2 𝛼2 ADJ cana-5924 172 7 , , PUNCT cana-5924 172 8 . . PUNCT cana-5924 172 9 . . PUNCT cana-5924 173 1 . . PUNCT cana-5924 174 1 , , PUNCT cana-5924 174 2 𝛼𝛿 𝛼𝛿 X cana-5924 174 3 } } PUNCT cana-5924 174 4 and and CCONJ cana-5924 174 5 |Ο‰| |Ο‰| PROPN cana-5924 174 6 is be AUX cana-5924 174 7 even even ADV cana-5924 174 8 . . PUNCT cana-5924 175 1 then then ADV cana-5924 175 2 𝛿 𝛿 PRON cana-5924 175 3 must must AUX cana-5924 175 4 be be AUX cana-5924 175 5 even even ADV cana-5924 175 6 . . PUNCT cana-5924 176 1 we we PRON cana-5924 176 2 know know VERB cana-5924 176 3 that that SCONJ cana-5924 176 4 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 176 5 ) ) PUNCT cana-5924 176 6 has have VERB cana-5924 176 7 a a DET cana-5924 176 8 partition partition NOUN cana-5924 176 9 such such ADJ cana-5924 176 10 that that SCONJ cana-5924 176 11 { { PUNCT cana-5924 176 12 n1 n1 NOUN cana-5924 176 13 , , PUNCT cana-5924 176 14 n2 n2 NOUN cana-5924 176 15 , , PUNCT cana-5924 176 16 … … PUNCT cana-5924 176 17 , , PUNCT cana-5924 176 18 nΞ· nΞ· NOUN cana-5924 176 19 , , PUNCT cana-5924 176 20 𝑆1 𝑆1 PROPN cana-5924 176 21 , , PUNCT cana-5924 176 22 𝑆2 𝑆2 NOUN cana-5924 176 23 , , PUNCT cana-5924 176 24 … … PUNCT cana-5924 176 25 , , PUNCT cana-5924 176 26 𝑆𝛿 𝑆𝛿 PROPN cana-5924 176 27 } } PUNCT cana-5924 176 28 . . PUNCT cana-5924 177 1 with with ADP cana-5924 177 2 no no DET cana-5924 177 3 loss loss NOUN cana-5924 177 4 of of ADP cana-5924 177 5 generality generality NOUN cana-5924 177 6 , , PUNCT cana-5924 177 7 we we PRON cana-5924 177 8 assume assume VERB cana-5924 177 9 that that SCONJ cana-5924 177 10 πœƒπœ‚ πœƒπœ‚ PROPN cana-5924 177 11 ∈ ∈ PROPN cana-5924 177 12 Ο‰ Ο‰ PROPN cana-5924 177 13 is be AUX cana-5924 177 14 an an DET cana-5924 177 15 element element NOUN cana-5924 177 16 of of ADP cana-5924 177 17 order order NOUN cana-5924 177 18 4β„“ 4β„“ NOUN cana-5924 177 19 , , PUNCT cana-5924 177 20 β„“ β„“ PROPN cana-5924 177 21 β‰  β‰  PROPN cana-5924 177 22 1 1 NUM cana-5924 177 23 . . PUNCT cana-5924 178 1 let let VERB cana-5924 178 2 πœ‘ πœ‘ PRON cana-5924 178 3 be be AUX cana-5924 178 4 the the DET cana-5924 178 5 number number NOUN cana-5924 178 6 of of ADP cana-5924 178 7 cycles cycle NOUN cana-5924 178 8 produced produce VERB cana-5924 178 9 by by ADP cana-5924 178 10 nΞ· nΞ· PROPN cana-5924 178 11 , , PUNCT cana-5924 178 12 the the DET cana-5924 178 13 set set NOUN cana-5924 178 14 of of ADP cana-5924 178 15 all all DET cana-5924 178 16 edges edge NOUN cana-5924 178 17 of of ADP cana-5924 178 18 those those DET cana-5924 178 19 cycles cycle NOUN cana-5924 178 20 are be AUX cana-5924 178 21 generated generate VERB cana-5924 178 22 by by ADP cana-5924 178 23 Ξ· Ξ· PROPN cana-5924 178 24 where where SCONJ cana-5924 178 25 nΞ· nΞ· PROPN cana-5924 178 26 = = SYM cana-5924 178 27 nΞΈ1βˆͺnΞΈ2βˆͺ nΞΈ1βˆͺnΞΈ2βˆͺ PROPN cana-5924 178 28 Β· Β· PUNCT cana-5924 178 29 Β· Β· PUNCT cana-5924 178 30 Β· Β· PUNCT cana-5924 178 31 βˆͺnΞΈΟ† βˆͺnΞΈΟ† NOUN cana-5924 178 32 with with ADP cana-5924 178 33 each each DET cana-5924 178 34 cycle cycle NOUN cana-5924 178 35 of of ADP cana-5924 178 36 length length NOUN cana-5924 178 37 4β„“. 4β„“. NUM cana-5924 178 38 event event NOUN cana-5924 178 39 1 1 X cana-5924 178 40 . . PUNCT cana-5924 179 1 if if SCONJ cana-5924 179 2 πœ‚ πœ‚ PROPN cana-5924 179 3 β‰₯ β‰₯ NUM cana-5924 179 4 1 1 NUM cana-5924 179 5 and and CCONJ cana-5924 179 6 𝛿 𝛿 ADJ cana-5924 179 7 = = ADJ cana-5924 179 8 0 0 NUM cana-5924 179 9 . . NOUN cana-5924 179 10 instance instance NOUN cana-5924 179 11 1.1 1.1 NUM cana-5924 179 12 . . PUNCT cana-5924 180 1 for for ADP cana-5924 180 2 a a DET cana-5924 180 3 odd odd ADJ cana-5924 180 4 πœ‚ πœ‚ NOUN cana-5924 180 5 , , PUNCT cana-5924 180 6 executing execute VERB cana-5924 180 7 a a DET cana-5924 180 8 cordial cordial ADJ cana-5924 180 9 function function NOUN cana-5924 180 10 f f PROPN cana-5924 180 11 from from ADP cana-5924 180 12 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 180 13 ) ) PUNCT cana-5924 180 14 to to ADP cana-5924 180 15 the the DET cana-5924 180 16 set set NOUN cana-5924 180 17 { { PUNCT cana-5924 180 18 0 0 NUM cana-5924 180 19 , , PUNCT cana-5924 180 20 1 1 NUM cana-5924 180 21 } } PUNCT cana-5924 180 22 such such ADJ cana-5924 180 23 that that DET cana-5924 180 24 instance instance NOUN cana-5924 180 25 1.2 1.2 NUM cana-5924 180 26 . . PUNCT cana-5924 181 1 for for ADP cana-5924 181 2 a a DET cana-5924 181 3 even even ADV cana-5924 181 4 πœ‚ πœ‚ NOUN cana-5924 181 5 , , PUNCT cana-5924 181 6 executing execute VERB cana-5924 181 7 a a DET cana-5924 181 8 cordial cordial ADJ cana-5924 181 9 function function NOUN cana-5924 181 10 β„± β„± PROPN cana-5924 181 11 from from ADP cana-5924 181 12 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 181 13 ) ) PUNCT cana-5924 181 14 to to ADP cana-5924 181 15 the the DET cana-5924 181 16 set set NOUN cana-5924 181 17 { { PUNCT cana-5924 181 18 0 0 NUM cana-5924 181 19 , , PUNCT cana-5924 181 20 1 1 NUM cana-5924 181 21 } } PUNCT cana-5924 181 22 such such ADJ cana-5924 181 23 that that SCONJ cana-5924 181 24 communications communication NOUN cana-5924 181 25 on on ADP cana-5924 181 26 applied apply VERB cana-5924 181 27 nonlinear nonlinear ADJ cana-5924 181 28 analysis analysis NOUN cana-5924 181 29 issn issn NOUN cana-5924 181 30 : : PUNCT cana-5924 181 31 1074 1074 NUM cana-5924 181 32 - - PUNCT cana-5924 181 33 133x 133x NUM cana-5924 181 34 vol vol NOUN cana-5924 181 35 31 31 NUM cana-5924 181 36 no no NOUN cana-5924 181 37 . . NOUN cana-5924 181 38 2 2 NUM cana-5924 181 39 ( ( PUNCT cana-5924 181 40 2024 2024 NUM cana-5924 181 41 ) ) PUNCT cana-5924 181 42 485 485 NUM cana-5924 181 43 https://internationalpubls.com https://internationalpubls.com X cana-5924 181 44 in in ADP cana-5924 181 45 the the DET cana-5924 181 46 above above ADJ cana-5924 181 47 two two NUM cana-5924 181 48 instance instance NOUN cana-5924 181 49 of of ADP cana-5924 181 50 this this DET cana-5924 181 51 event event NOUN cana-5924 181 52 , , PUNCT cana-5924 181 53 sum sum NOUN cana-5924 181 54 of of ADP cana-5924 181 55 the the DET cana-5924 181 56 labels label NOUN cana-5924 181 57 of of ADP cana-5924 181 58 incident incident NOUN cana-5924 181 59 edges edge NOUN cana-5924 181 60 in in ADP cana-5924 181 61 every every DET cana-5924 181 62 vertices vertex NOUN cana-5924 181 63 as as SCONJ cana-5924 181 64 said say VERB cana-5924 181 65 to to PART cana-5924 181 66 be be AUX cana-5924 181 67 a a DET cana-5924 181 68 induced induced ADJ cana-5924 181 69 vertex vertex NOUN cana-5924 181 70 sum sum NOUN cana-5924 181 71 of of ADP cana-5924 181 72 𝑉(𝒒 𝑉(𝒒 NOUN cana-5924 181 73 ) ) PUNCT cana-5924 181 74 for for ADP cana-5924 181 75 all all PRON cana-5924 181 76 1 1 NUM cana-5924 181 77 ≀ ≀ NUM cana-5924 181 78 𝑖 𝑖 SYM cana-5924 181 79 ≀ ≀ NUM cana-5924 181 80 𝜌 𝜌 ADP cana-5924 181 81 , , PUNCT cana-5924 181 82 event event NOUN cana-5924 181 83 2 2 NUM cana-5924 181 84 . . PUNCT cana-5924 182 1 if if SCONJ cana-5924 182 2 πœ‚ πœ‚ PROPN cana-5924 182 3 β‰₯ β‰₯ NUM cana-5924 182 4 1 1 NUM cana-5924 182 5 and and CCONJ cana-5924 182 6 𝛿 𝛿 PRON cana-5924 182 7 β‰  β‰  PROPN cana-5924 182 8 0 0 NUM cana-5924 182 9 . . PUNCT cana-5924 182 10 instance instance NOUN cana-5924 182 11 2.1 2.1 NUM cana-5924 182 12 . . PUNCT cana-5924 183 1 for for ADP cana-5924 183 2 a a DET cana-5924 183 3 odd odd ADJ cana-5924 183 4 πœ‚ πœ‚ NOUN cana-5924 183 5 , , PUNCT cana-5924 183 6 executing execute VERB cana-5924 183 7 a a DET cana-5924 183 8 cordial cordial ADJ cana-5924 183 9 function function NOUN cana-5924 183 10 β„± β„± PROPN cana-5924 183 11 from from ADP cana-5924 183 12 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 183 13 ) ) PUNCT cana-5924 183 14 to to ADP cana-5924 183 15 the the DET cana-5924 183 16 set set NOUN cana-5924 183 17 { { PUNCT cana-5924 183 18 0 0 NUM cana-5924 183 19 , , PUNCT cana-5924 183 20 1 1 NUM cana-5924 183 21 } } PUNCT cana-5924 183 22 such such ADJ cana-5924 183 23 that that DET cana-5924 183 24 instance instance NOUN cana-5924 183 25 2.2 2.2 NUM cana-5924 183 26 . . PUNCT cana-5924 184 1 for for ADP cana-5924 184 2 a a DET cana-5924 184 3 even even ADV cana-5924 184 4 πœ‚ πœ‚ NOUN cana-5924 184 5 , , PUNCT cana-5924 184 6 executing execute VERB cana-5924 184 7 a a DET cana-5924 184 8 cordial cordial ADJ cana-5924 184 9 function function NOUN cana-5924 184 10 β„± β„± PROPN cana-5924 184 11 from from ADP cana-5924 184 12 l(𝒒 l(𝒒 PROPN cana-5924 184 13 ) ) PUNCT cana-5924 184 14 to to ADP cana-5924 184 15 the the DET cana-5924 184 16 set set NOUN cana-5924 184 17 { { PUNCT cana-5924 184 18 0 0 NUM cana-5924 184 19 , , PUNCT cana-5924 184 20 1 1 NUM cana-5924 184 21 } } PUNCT cana-5924 184 22 such such ADJ cana-5924 184 23 that that SCONJ cana-5924 184 24 , , PUNCT cana-5924 184 25 communications communication NOUN cana-5924 184 26 on on ADP cana-5924 184 27 applied apply VERB cana-5924 184 28 nonlinear nonlinear ADJ cana-5924 184 29 analysis analysis NOUN cana-5924 184 30 issn issn NOUN cana-5924 184 31 : : PUNCT cana-5924 184 32 1074 1074 NUM cana-5924 184 33 - - PUNCT cana-5924 184 34 133x 133x NUM cana-5924 184 35 vol vol NOUN cana-5924 184 36 31 31 NUM cana-5924 184 37 no no NOUN cana-5924 184 38 . . NOUN cana-5924 184 39 2 2 NUM cana-5924 184 40 ( ( PUNCT cana-5924 184 41 2024 2024 NUM cana-5924 184 42 ) ) PUNCT cana-5924 184 43 486 486 NUM cana-5924 184 44 https://internationalpubls.com https://internationalpubls.com X cana-5924 184 45 at at ADP cana-5924 184 46 this this DET cana-5924 184 47 event event NOUN cana-5924 184 48 , , PUNCT cana-5924 184 49 by by ADP cana-5924 184 50 the the DET cana-5924 184 51 above above ADJ cana-5924 184 52 labelled label VERB cana-5924 184 53 assignments assignment NOUN cana-5924 184 54 of of ADP cana-5924 184 55 both both DET cana-5924 184 56 instance instance NOUN cana-5924 184 57 we we PRON cana-5924 184 58 will will AUX cana-5924 184 59 obtained obtain VERB cana-5924 184 60 the the DET cana-5924 184 61 result result NOUN cana-5924 184 62 , , PUNCT cana-5924 184 63 which which PRON cana-5924 184 64 will will AUX cana-5924 184 65 meet meet VERB cana-5924 184 66 the the DET cana-5924 184 67 edge edge NOUN cana-5924 184 68 cordiality cordiality NOUN cana-5924 184 69 such such ADJ cana-5924 184 70 that that SCONJ cana-5924 184 71 , , PUNCT cana-5924 184 72 𝑣ℱ(0 𝑣ℱ(0 NUM cana-5924 184 73 ) ) PUNCT cana-5924 184 74 = = SYM cana-5924 184 75 𝑣ℱ(1 𝑣ℱ(1 NOUN cana-5924 184 76 ) ) PUNCT cana-5924 184 77 = = PUNCT cana-5924 184 78 𝜌 𝜌 ADP cana-5924 184 79 2 2 NUM cana-5924 184 80 and and CCONJ cana-5924 184 81 𝑒ℱ(0 𝑒ℱ(0 NUM cana-5924 184 82 ) ) PUNCT cana-5924 184 83 = = PUNCT cana-5924 184 84 𝑒ℱ(1 𝑒ℱ(1 NOUN cana-5924 184 85 ) ) PUNCT cana-5924 184 86 = = PUNCT cana-5924 184 87 𝜌|Ο‰| 𝜌|Ο‰| VERB cana-5924 184 88 4 4 NUM cana-5924 184 89 . . PUNCT cana-5924 185 1 event event NOUN cana-5924 185 2 3 3 NUM cana-5924 185 3 . . PUNCT cana-5924 185 4 suppose suppose VERB cana-5924 185 5 πœ‚ πœ‚ NOUN cana-5924 185 6 = = SYM cana-5924 185 7 0 0 NUM cana-5924 185 8 and and CCONJ cana-5924 185 9 even even ADV cana-5924 185 10 𝛿. 𝛿. ADJ cana-5924 185 11 at at ADP cana-5924 185 12 this this DET cana-5924 185 13 instance instance NOUN cana-5924 185 14 the the DET cana-5924 185 15 generating generating NOUN cana-5924 185 16 subset subset NOUN cana-5924 185 17 Ο‰ Ο‰ PROPN cana-5924 185 18 contains contain VERB cana-5924 185 19 only only ADV cana-5924 185 20 the the DET cana-5924 185 21 self self NOUN cana-5924 185 22 inverse(order inverse(order PROPN cana-5924 185 23 two two NUM cana-5924 185 24 ) ) PUNCT cana-5924 185 25 elements element NOUN cana-5924 185 26 of of ADP cana-5924 185 27 𝛀. 𝛀. PROPN cana-5924 185 28 in in ADP cana-5924 185 29 this this DET cana-5924 185 30 case case NOUN cana-5924 185 31 by by ADP cana-5924 185 32 proposition proposition NOUN cana-5924 185 33 2.1 2.1 NUM cana-5924 185 34 , , PUNCT cana-5924 185 35 the the DET cana-5924 185 36 proof proof NOUN cana-5924 185 37 is be AUX cana-5924 185 38 immediate immediate ADJ cana-5924 185 39 . . PUNCT cana-5924 186 1 hence hence ADV cana-5924 186 2 by by ADP cana-5924 186 3 the the DET cana-5924 186 4 above above ADJ cana-5924 186 5 three three NUM cana-5924 186 6 labeling labeling NOUN cana-5924 186 7 events event NOUN cana-5924 186 8 , , PUNCT cana-5924 186 9 it it PRON cana-5924 186 10 is be AUX cana-5924 186 11 clear clear ADJ cana-5924 186 12 that that SCONJ cana-5924 186 13 the the DET cana-5924 186 14 cayley cayley NOUN cana-5924 186 15 graph(𝒒 graph(𝒒 PROPN cana-5924 186 16 ) ) PUNCT cana-5924 186 17 is be AUX cana-5924 186 18 binary binary NOUN cana-5924 186 19 coded code VERB cana-5924 186 20 . . PUNCT cana-5924 186 21 example example NOUN cana-5924 186 22 2.5 2.5 NUM cana-5924 186 23 . . PUNCT cana-5924 187 1 consider consider VERB cana-5924 187 2 the the DET cana-5924 187 3 cayley cayley ADJ cana-5924 187 4 graph graph NOUN cana-5924 187 5 corresponding correspond VERB cana-5924 187 6 to to ADP cana-5924 187 7 the the DET cana-5924 187 8 dihedral dihedral ADJ cana-5924 187 9 group group NOUN cana-5924 187 10 𝐷16 𝐷16 PROPN cana-5924 187 11 which which PRON cana-5924 187 12 has have VERB cana-5924 187 13 more more ADJ cana-5924 187 14 than than ADP cana-5924 187 15 one one NUM cana-5924 187 16 self self NOUN cana-5924 187 17 inverse inverse NOUN cana-5924 187 18 element element NOUN cana-5924 187 19 as as ADV cana-5924 187 20 well well ADV cana-5924 187 21 as as ADP cana-5924 187 22 the the DET cana-5924 187 23 non non ADJ cana-5924 187 24 self self NOUN cana-5924 187 25 inverse inverse NOUN cana-5924 187 26 element element NOUN cana-5924 187 27 . . PUNCT cana-5924 188 1 by by ADP cana-5924 188 2 applying apply VERB cana-5924 188 3 the the DET cana-5924 188 4 binary binary NOUN cana-5924 188 5 encoding encoding NOUN cana-5924 188 6 labeling labeling NOUN cana-5924 188 7 function function NOUN cana-5924 188 8 on on ADP cana-5924 188 9 it it PRON cana-5924 188 10 through through ADP cana-5924 188 11 the the DET cana-5924 188 12 theorem theorem ADJ cana-5924 188 13 2.4 2.4 NUM cana-5924 188 14 . . PUNCT cana-5924 189 1 the the DET cana-5924 189 2 dotted dotted ADJ cana-5924 189 3 edges edge NOUN cana-5924 189 4 and and CCONJ cana-5924 189 5 non non ADJ cana-5924 189 6 - - ADJ cana-5924 189 7 darkened darken VERB cana-5924 189 8 vertices vertex NOUN cana-5924 189 9 indicates indicate VERB cana-5924 189 10 the the DET cana-5924 189 11 holding hold VERB cana-5924 189 12 weight weight NOUN cana-5924 189 13 is be AUX cana-5924 189 14 zero zero NUM cana-5924 189 15 and and CCONJ cana-5924 189 16 the the DET cana-5924 189 17 non non ADJ cana-5924 189 18 - - ADJ cana-5924 189 19 dotted dotted ADJ cana-5924 189 20 lines line NOUN cana-5924 189 21 and and CCONJ cana-5924 189 22 darkened darken VERB cana-5924 189 23 nodes node NOUN cana-5924 189 24 indicates indicate VERB cana-5924 189 25 the the DET cana-5924 189 26 holding hold VERB cana-5924 189 27 weight weight NOUN cana-5924 189 28 is be AUX cana-5924 189 29 one one NUM cana-5924 189 30 . . PUNCT cana-5924 190 1 communications communication NOUN cana-5924 190 2 on on ADP cana-5924 190 3 applied apply VERB cana-5924 190 4 nonlinear nonlinear ADJ cana-5924 190 5 analysis analysis NOUN cana-5924 190 6 issn issn NOUN cana-5924 190 7 : : PUNCT cana-5924 190 8 1074 1074 NUM cana-5924 190 9 - - PUNCT cana-5924 190 10 133x 133x NUM cana-5924 190 11 vol vol NOUN cana-5924 190 12 31 31 NUM cana-5924 190 13 no no NOUN cana-5924 190 14 . . NOUN cana-5924 190 15 2 2 NUM cana-5924 190 16 ( ( PUNCT cana-5924 190 17 2024 2024 NUM cana-5924 190 18 ) ) PUNCT cana-5924 190 19 487 487 NUM cana-5924 190 20 https://internationalpubls.com https://internationalpubls.com X cana-5924 190 21 theorem theorem VERB cana-5924 190 22 2.6 2.6 NUM cana-5924 190 23 . . PUNCT cana-5924 191 1 let let VERB cana-5924 191 2 Ο‰ Ο‰ PRON cana-5924 191 3 be be AUX cana-5924 191 4 a a DET cana-5924 191 5 generating generating NOUN cana-5924 191 6 subset subset NOUN cana-5924 191 7 of of ADP cana-5924 191 8 a a DET cana-5924 191 9 group group NOUN cana-5924 191 10 𝛀 𝛀 PROPN cana-5924 191 11 of of ADP cana-5924 191 12 order order NOUN cana-5924 191 13 𝜌. 𝜌. NOUN cana-5924 191 14 then then ADV cana-5924 191 15 the the DET cana-5924 191 16 cayley cayley ADJ cana-5924 191 17 network network NOUN cana-5924 191 18 𝒒 𝒒 PROPN cana-5924 191 19 is be AUX cana-5924 191 20 binary binary ADJ cana-5924 191 21 codable codable NOUN cana-5924 191 22 , , PUNCT cana-5924 191 23 if if SCONJ cana-5924 191 24 𝜌 𝜌 ADP cana-5924 191 25 ∈ ∈ PROPN cana-5924 191 26 [ [ X cana-5924 191 27 1]4 1]4 NOUN cana-5924 191 28 and and CCONJ cana-5924 191 29 𝜌 𝜌 X cana-5924 191 30 ∈ ∈ PROPN cana-5924 192 1 [ [ X cana-5924 192 2 3]4 3]4 NOUN cana-5924 192 3 . . PUNCT cana-5924 193 1 proof proof NOUN cana-5924 193 2 . . PUNCT cana-5924 194 1 since since SCONJ cana-5924 194 2 |Ο‰| |Ο‰| NUM cana-5924 194 3 is be AUX cana-5924 194 4 even even ADV cana-5924 194 5 and and CCONJ cana-5924 194 6 Ο‰ Ο‰ NUM cana-5924 194 7 = = SYM cana-5924 194 8 { { PUNCT cana-5924 194 9 πœƒ1 πœƒ1 NOUN cana-5924 194 10 , , PUNCT cana-5924 194 11 πœƒ2 πœƒ2 PROPN cana-5924 194 12 , , PUNCT cana-5924 194 13 . . PUNCT cana-5924 194 14 . . PUNCT cana-5924 195 1 . . PUNCT cana-5924 196 1 , , PUNCT cana-5924 196 2 πœƒπœ‚ πœƒπœ‚ INTJ cana-5924 196 3 , , PUNCT cana-5924 196 4 πœƒπœ‚+1 πœƒπœ‚+1 NUM cana-5924 196 5 , , PUNCT cana-5924 196 6 πœƒπœ‚+2 πœƒπœ‚+2 NUM cana-5924 196 7 , , PUNCT cana-5924 196 8 . . PUNCT cana-5924 196 9 . . PUNCT cana-5924 197 1 . . PUNCT cana-5924 198 1 , , PUNCT cana-5924 198 2 πœƒ2πœ‚ πœƒ2πœ‚ ADP cana-5924 198 3 } } PUNCT cana-5924 198 4 , , PUNCT cana-5924 198 5 πœƒπ‘–βˆ’1 πœƒπ‘–βˆ’1 ADJ cana-5924 198 6 = = SYM cana-5924 198 7 πœƒπ‘–+πœ‚ πœƒπ‘–+πœ‚ NOUN cana-5924 198 8 , , PUNCT cana-5924 198 9 1 1 NUM cana-5924 198 10 ≀ ≀ NUM cana-5924 198 11 𝑖 𝑖 SYM cana-5924 198 12 ≀ ≀ NOUN cana-5924 198 13 πœ‚. πœ‚. NOUN cana-5924 198 14 event event NOUN cana-5924 198 15 1 1 NUM cana-5924 198 16 . . PUNCT cana-5924 199 1 if if SCONJ cana-5924 199 2 πœ‚ πœ‚ NOUN cana-5924 199 3 is be AUX cana-5924 199 4 odd odd ADJ cana-5924 199 5 . . PUNCT cana-5924 200 1 executing execute VERB cana-5924 200 2 a a DET cana-5924 200 3 cordial cordial ADJ cana-5924 200 4 function function NOUN cana-5924 200 5 β„± β„± PROPN cana-5924 200 6 from from ADP cana-5924 200 7 𝐿(𝒒 𝐿(𝒒 NOUN cana-5924 200 8 ) ) PUNCT cana-5924 200 9 to to ADP cana-5924 200 10 the the DET cana-5924 200 11 set set NOUN cana-5924 200 12 { { PUNCT cana-5924 200 13 0 0 NUM cana-5924 200 14 , , PUNCT cana-5924 200 15 1 1 NUM cana-5924 200 16 } } PUNCT cana-5924 200 17 such such ADJ cana-5924 200 18 that that DET cana-5924 200 19 event event NOUN cana-5924 200 20 2 2 X cana-5924 200 21 . . PUNCT cana-5924 201 1 if if SCONJ cana-5924 201 2 Ξ· Ξ· PROPN cana-5924 201 3 is be AUX cana-5924 201 4 even even ADV cana-5924 201 5 . . PUNCT cana-5924 202 1 executing execute VERB cana-5924 202 2 a a DET cana-5924 202 3 cordial cordial ADJ cana-5924 202 4 function function NOUN cana-5924 202 5 β„± β„± PROPN cana-5924 202 6 , , PUNCT cana-5924 202 7 from from ADP cana-5924 202 8 l(𝒒 l(𝒒 PROPN cana-5924 202 9 ) ) PUNCT cana-5924 202 10 to to ADP cana-5924 202 11 the the DET cana-5924 202 12 set set NOUN cana-5924 202 13 { { PUNCT cana-5924 202 14 0 0 NUM cana-5924 202 15 , , PUNCT cana-5924 202 16 1 1 NUM cana-5924 202 17 } } PUNCT cana-5924 202 18 such such ADJ cana-5924 202 19 as as ADP cana-5924 202 20 by by ADP cana-5924 202 21 the the DET cana-5924 202 22 above above ADJ cana-5924 202 23 labeled label VERB cana-5924 202 24 events event NOUN cana-5924 202 25 shows show VERB cana-5924 202 26 that that SCONJ cana-5924 202 27 for for ADP cana-5924 202 28 all all DET cana-5924 202 29 1 1 NUM cana-5924 202 30 ≀ ≀ NUM cana-5924 202 31 𝑖 𝑖 SYM cana-5924 202 32 ≀ ≀ NUM cana-5924 202 33 𝜌 𝜌 ADP cana-5924 202 34 , , PUNCT cana-5924 202 35 3 3 X cana-5924 202 36 . . X cana-5924 202 37 conclusion conclusion NOUN cana-5924 202 38 basically basically ADV cana-5924 202 39 , , PUNCT cana-5924 202 40 cayley cayley ADJ cana-5924 202 41 graphs graph NOUN cana-5924 202 42 are be AUX cana-5924 202 43 the the DET cana-5924 202 44 good good ADJ cana-5924 202 45 network network NOUN cana-5924 202 46 model model NOUN cana-5924 202 47 , , PUNCT cana-5924 202 48 by by ADP cana-5924 202 49 implementing implement VERB cana-5924 202 50 this this DET cana-5924 202 51 edge edge NOUN cana-5924 202 52 binary binary NOUN cana-5924 202 53 encoding encoding NOUN cana-5924 202 54 analogue analogue NOUN cana-5924 202 55 on on ADP cana-5924 202 56 any any DET cana-5924 202 57 practical practical ADJ cana-5924 202 58 problem problem NOUN cana-5924 202 59 , , PUNCT cana-5924 202 60 whose whose DET cana-5924 202 61 configuration configuration NOUN cana-5924 202 62 resembles resemble VERB cana-5924 202 63 the the DET cana-5924 202 64 cayley cayley ADJ cana-5924 202 65 graph graph NOUN cana-5924 202 66 at at ADP cana-5924 202 67 that that DET cana-5924 202 68 instance instance NOUN cana-5924 202 69 , , PUNCT cana-5924 202 70 multiple multiple ADJ cana-5924 202 71 non non ADJ cana-5924 202 72 - - ADJ cana-5924 202 73 binary binary ADJ cana-5924 202 74 output output NOUN cana-5924 202 75 can can AUX cana-5924 202 76 be be AUX cana-5924 202 77 gained gain VERB cana-5924 202 78 by by ADP cana-5924 202 79 the the DET cana-5924 202 80 polynomial polynomial ADJ cana-5924 202 81 time time NOUN cana-5924 202 82 algorithm algorithm NOUN cana-5924 202 83 as as ADP cana-5924 202 84 a a DET cana-5924 202 85 decoder decoder NOUN cana-5924 202 86 . . PUNCT cana-5924 203 1 further far ADV cana-5924 203 2 , , PUNCT cana-5924 203 3 this this DET cana-5924 203 4 research research NOUN cana-5924 203 5 can can AUX cana-5924 203 6 be be AUX cana-5924 203 7 extended extend VERB cana-5924 203 8 to to PART cana-5924 203 9 encode encode VERB cana-5924 203 10 the the DET cana-5924 203 11 various various ADJ cana-5924 203 12 families family NOUN cana-5924 203 13 of of ADP cana-5924 203 14 cayley cayley ADJ cana-5924 203 15 graphs graph NOUN cana-5924 203 16 such such ADJ cana-5924 203 17 as as ADP cana-5924 203 18 unitary unitary ADJ cana-5924 203 19 cayley cayley NOUN cana-5924 203 20 , , PUNCT cana-5924 203 21 unitary unitary ADJ cana-5924 203 22 addition addition NOUN cana-5924 203 23 cayley cayley NOUN cana-5924 203 24 and and CCONJ cana-5924 203 25 euler euler PROPN cana-5924 203 26 totient totient PROPN cana-5924 203 27 cayley cayley PROPN cana-5924 203 28 graphs graph NOUN cana-5924 203 29 , , PUNCT cana-5924 203 30 etc etc X cana-5924 203 31 . . X cana-5924 203 32 , , PUNCT cana-5924 203 33 with with ADP cana-5924 203 34 the the DET cana-5924 203 35 different different ADJ cana-5924 203 36 algebraic algebraic ADJ cana-5924 203 37 structure structure NOUN cana-5924 203 38 as as ADV cana-5924 203 39 well well ADV cana-5924 203 40 as as ADP cana-5924 203 41 the the DET cana-5924 203 42 different different ADJ cana-5924 203 43 adjacency adjacency NOUN cana-5924 203 44 constraints constraint NOUN cana-5924 203 45 . . PUNCT cana-5924 204 1 communications communication NOUN cana-5924 204 2 on on ADP cana-5924 204 3 applied apply VERB cana-5924 204 4 nonlinear nonlinear ADJ cana-5924 204 5 analysis analysis NOUN cana-5924 204 6 issn issn NOUN cana-5924 204 7 : : PUNCT cana-5924 204 8 1074 1074 NUM cana-5924 204 9 - - PUNCT cana-5924 204 10 133x 133x NUM cana-5924 204 11 vol vol NOUN cana-5924 204 12 31 31 NUM cana-5924 204 13 no no NOUN cana-5924 204 14 . . NOUN cana-5924 204 15 2 2 NUM cana-5924 204 16 ( ( PUNCT cana-5924 204 17 2024 2024 NUM cana-5924 204 18 ) ) PUNCT cana-5924 204 19 488 488 NUM cana-5924 204 20 https://internationalpubls.com https://internationalpubls.com NUM cana-5924 204 21 references reference NOUN cana-5924 204 22 [ [ X cana-5924 204 23 1 1 NUM cana-5924 204 24 ] ] PUNCT cana-5924 204 25 i. i. PROPN cana-5924 204 26 cahit cahit PROPN cana-5924 204 27 , , PUNCT cana-5924 204 28 cordial cordial ADJ cana-5924 204 29 graphs graph NOUN cana-5924 204 30 : : PUNCT cana-5924 204 31 a a DET cana-5924 204 32 weaker weak ADJ cana-5924 204 33 version version NOUN cana-5924 204 34 of of ADP cana-5924 204 35 graceful graceful ADJ cana-5924 204 36 and and CCONJ cana-5924 204 37 harmonious harmonious ADJ cana-5924 204 38 graphs graph NOUN cana-5924 204 39 , , PUNCT cana-5924 204 40 ars ar VERB cana-5924 204 41 combin combin NOUN cana-5924 204 42 . . PUNCT cana-5924 204 43 , , PUNCT cana-5924 204 44 23 23 NUM cana-5924 204 45 ( ( PUNCT cana-5924 204 46 1987 1987 NUM cana-5924 204 47 ) ) PUNCT cana-5924 204 48 , , PUNCT cana-5924 204 49 201–207 201–207 NUM cana-5924 204 50 . . PUNCT cana-5924 205 1 [ [ X cana-5924 205 2 2 2 X cana-5924 205 3 ] ] PUNCT cana-5924 205 4 j. j. PROPN cana-5924 205 5 a. a. PROPN cana-5924 205 6 gallian gallian PROPN cana-5924 205 7 , , PUNCT cana-5924 205 8 a a DET cana-5924 205 9 dynamic dynamic ADJ cana-5924 205 10 survey survey NOUN cana-5924 205 11 of of ADP cana-5924 205 12 graph graph NOUN cana-5924 205 13 labeling labeling NOUN cana-5924 205 14 , , PUNCT cana-5924 205 15 electron electron NOUN cana-5924 205 16 . . PUNCT cana-5924 206 1 j. j. PROPN cana-5924 206 2 combin combin PROPN cana-5924 206 3 . . PROPN cana-5924 206 4 , , PUNCT cana-5924 206 5 18 18 NUM cana-5924 206 6 ( ( PUNCT cana-5924 206 7 2011 2011 NUM cana-5924 206 8 ) ) PUNCT cana-5924 206 9 , , PUNCT cana-5924 206 10 # # NOUN cana-5924 206 11 ds6 ds6 NOUN cana-5924 206 12 . . PUNCT cana-5924 207 1 [ [ X cana-5924 207 2 3 3 NUM cana-5924 207 3 ] ] X cana-5924 207 4 m. m. NOUN cana-5924 207 5 heydemann heydemann PROPN cana-5924 207 6 , , PUNCT cana-5924 207 7 cayley cayley ADJ cana-5924 207 8 graphs graph NOUN cana-5924 207 9 and and CCONJ cana-5924 207 10 interconnection interconnection NOUN cana-5924 207 11 networks network NOUN cana-5924 207 12 . . PUNCT cana-5924 208 1 in in ADP cana-5924 208 2 : : PUNCT cana-5924 208 3 g. g. PROPN cana-5924 208 4 hahn hahn PROPN cana-5924 208 5 and and CCONJ cana-5924 208 6 g. g. PROPN cana-5924 208 7 sabidussi sabidussi PROPN cana-5924 208 8 ( ( PUNCT cana-5924 208 9 eds ed NOUN cana-5924 208 10 . . PUNCT cana-5924 208 11 ) ) PUNCT cana-5924 208 12 , , PUNCT cana-5924 208 13 graph graph NOUN cana-5924 208 14 symmetry symmetry NOUN cana-5924 208 15 : : PUNCT cana-5924 208 16 algebraic algebraic ADJ cana-5924 208 17 methods method NOUN cana-5924 208 18 and and CCONJ cana-5924 208 19 applications application NOUN cana-5924 208 20 , , PUNCT cana-5924 208 21 ( ( PUNCT cana-5924 208 22 1997 1997 NUM cana-5924 208 23 ) ) PUNCT cana-5924 208 24 , , PUNCT cana-5924 208 25 167–224 167–224 NUM cana-5924 208 26 . . PUNCT cana-5924 209 1 [ [ X cana-5924 209 2 4 4 NUM cana-5924 209 3 ] ] PUNCT cana-5924 209 4 a. a. NOUN cana-5924 209 5 rosa rosa PROPN cana-5924 209 6 , , PUNCT cana-5924 209 7 on on ADP cana-5924 209 8 certain certain ADJ cana-5924 209 9 valuation valuation NOUN cana-5924 209 10 of of ADP cana-5924 209 11 the the DET cana-5924 209 12 vertices vertex NOUN cana-5924 209 13 of of ADP cana-5924 209 14 a a DET cana-5924 209 15 graph graph NOUN cana-5924 209 16 , , PUNCT cana-5924 209 17 theory theory NOUN cana-5924 209 18 of of ADP cana-5924 209 19 graphs(international graphs(international PROPN cana-5924 209 20 . . PROPN cana-5924 209 21 symposium symposium PROPN cana-5924 209 22 , , PUNCT cana-5924 209 23 rome rome PROPN cana-5924 209 24 , , PUNCT cana-5924 209 25 july july PROPN cana-5924 209 26 1966 1966 NUM cana-5924 209 27 ) ) PUNCT cana-5924 209 28 , , PUNCT cana-5924 209 29 gordon gordon PROPN cana-5924 209 30 and and CCONJ cana-5924 209 31 breach breach NOUN cana-5924 209 32 , , PUNCT cana-5924 209 33 n. n. PROPN cana-5924 209 34 y. y. PROPN cana-5924 209 35 and and CCONJ cana-5924 209 36 dunodparis dunodparis PROPN cana-5924 209 37 , , PUNCT cana-5924 209 38 the the DET cana-5924 209 39 electronic electronic ADJ cana-5924 209 40 journal journal NOUN cana-5924 209 41 of of ADP cana-5924 209 42 combinatorics combinatoric NOUN cana-5924 209 43 , , PUNCT cana-5924 209 44 16 16 NUM cana-5924 209 45 ( ( PUNCT cana-5924 209 46 1967 1967 NUM cana-5924 209 47 ) ) PUNCT cana-5924 209 48 , , PUNCT cana-5924 209 49 349–355 349–355 NUM cana-5924 209 50 , , PUNCT cana-5924 209 51 # # SYM cana-5924 209 52 ds6 ds6 NOUN cana-5924 209 53 . . PUNCT cana-5924 210 1 [ [ X cana-5924 210 2 5 5 X cana-5924 210 3 ] ] PUNCT cana-5924 210 4 t. t. PROPN cana-5924 210 5 tamizh tamizh PROPN cana-5924 210 6 chelvam chelvam PROPN cana-5924 210 7 , , PUNCT cana-5924 210 8 n. n. PROPN cana-5924 210 9 mohamed mohamed PROPN cana-5924 210 10 rilwan rilwan PROPN cana-5924 210 11 and and CCONJ cana-5924 210 12 k. k. PROPN cana-5924 210 13 kalaimurugan kalaimurugan PROPN cana-5924 210 14 , , PUNCT cana-5924 210 15 antimagic antimagic ADJ cana-5924 210 16 and and CCONJ cana-5924 210 17 magic magic ADJ cana-5924 210 18 labelings labeling NOUN cana-5924 210 19 in in ADP cana-5924 210 20 cayley cayley ADJ cana-5924 210 21 digraphs digraph NOUN cana-5924 210 22 , , PUNCT cana-5924 210 23 australian australian ADJ cana-5924 210 24 journal journal NOUN cana-5924 210 25 of of ADP cana-5924 210 26 combinatorics combinatoric NOUN cana-5924 210 27 , , PUNCT cana-5924 210 28 55 55 NUM cana-5924 210 29 ( ( PUNCT cana-5924 210 30 2013 2013 NUM cana-5924 210 31 ) ) PUNCT cana-5924 210 32 , , PUNCT cana-5924 210 33 65–71 65–71 NOUN cana-5924 210 34 . . PUNCT cana-5924 211 1 [ [ X cana-5924 211 2 6 6 NUM cana-5924 211 3 ] ] PUNCT cana-5924 211 4 k. k. PROPN cana-5924 211 5 thirusangu thirusangu PROPN cana-5924 211 6 , , PUNCT cana-5924 211 7 a. a. PROPN cana-5924 211 8 k. k. PROPN cana-5924 211 9 nagar nagar PROPN cana-5924 211 10 and and CCONJ cana-5924 211 11 r. r. PROPN cana-5924 211 12 rajeswari rajeswari NOUN cana-5924 211 13 , , PUNCT cana-5924 211 14 labelings labeling NOUN cana-5924 211 15 in in ADP cana-5924 211 16 cayley cayley ADJ cana-5924 211 17 digraphs digraph NOUN cana-5924 211 18 , , PUNCT cana-5924 211 19 european european PROPN cana-5924 211 20 j. j. PROPN cana-5924 211 21 combin combin PROPN cana-5924 211 22 . . PROPN cana-5924 211 23 , , PUNCT cana-5924 211 24 32(1 32(1 NUM cana-5924 211 25 ) ) PUNCT cana-5924 211 26 ( ( PUNCT cana-5924 211 27 2011 2011 NUM cana-5924 211 28 ) ) PUNCT cana-5924 211 29 , , PUNCT cana-5924 211 30 133–139 133–139 NUM cana-5924 211 31 . . PUNCT cana-5924 212 1 [ [ X cana-5924 212 2 7 7 X cana-5924 212 3 ] ] X cana-5924 212 4 r. r. PROPN cana-5924 212 5 yilmaz yilmaz PROPN cana-5924 212 6 and and CCONJ cana-5924 212 7 i. i. PROPN cana-5924 212 8 cahit cahit PROPN cana-5924 212 9 , , PUNCT cana-5924 212 10 e e ADJ cana-5924 212 11 - - ADJ cana-5924 212 12 cordial cordial ADJ cana-5924 212 13 graphs graph NOUN cana-5924 212 14 , , PUNCT cana-5924 212 15 ars ar VERB cana-5924 212 16 combin combin NOUN cana-5924 212 17 . . PROPN cana-5924 212 18 , , PUNCT cana-5924 212 19 46(1997 46(1997 NUM cana-5924 212 20 ) ) PUNCT cana-5924 212 21 , , PUNCT cana-5924 212 22 251–266 251–266 NUM cana-5924 212 23 . . PUNCT