id sid tid token lemma pos cana-4368 1 1 communications communication NOUN cana-4368 1 2 on on ADP cana-4368 1 3 applied apply VERB cana-4368 1 4 nonlinear nonlinear ADJ cana-4368 1 5 analysis analysis NOUN cana-4368 1 6 issn issn NOUN cana-4368 1 7 : : PUNCT cana-4368 1 8 1074 1074 NUM cana-4368 1 9 - - PUNCT cana-4368 1 10 133x 133x NUM cana-4368 1 11 vol vol NOUN cana-4368 1 12 32 32 NUM cana-4368 1 13 no no NOUN cana-4368 1 14 . . PUNCT cana-4368 2 1 9s 9s NUM cana-4368 2 2 ( ( PUNCT cana-4368 2 3 2025 2025 NUM cana-4368 2 4 ) ) PUNCT cana-4368 2 5 1928 1928 NUM cana-4368 2 6 https://internationalpubls.com https://internationalpubls.com X cana-4368 2 7 odd odd ADJ cana-4368 2 8 hamming hamming NOUN cana-4368 2 9 distance distance NOUN cana-4368 2 10 labeling labeling NOUN cana-4368 2 11 of of ADP cana-4368 2 12 some some DET cana-4368 2 13 path path NOUN cana-4368 2 14 related relate VERB cana-4368 2 15 graphs graph NOUN cana-4368 2 16 e.esakkiammal e.esakkiammal ADJ cana-4368 2 17 1 1 NUM cana-4368 2 18 , , PUNCT cana-4368 2 19 k.thirusangu2 k.thirusangu2 NOUN cana-4368 2 20 and and CCONJ cana-4368 2 21 s.seethalakshmi s.seethalakshmi VERB cana-4368 2 22 3 3 NUM cana-4368 2 23 1,2 1,2 NUM cana-4368 2 24 department department NOUN cana-4368 2 25 of of ADP cana-4368 2 26 mathematics mathematic NOUN cana-4368 2 27 , , PUNCT cana-4368 2 28 s.i.v.e.t s.i.v.e.t NOUN cana-4368 2 29 . . PUNCT cana-4368 3 1 college college NOUN cana-4368 3 2 , , PUNCT cana-4368 3 3 gowrivakkam gowrivakkam NOUN cana-4368 3 4 , , PUNCT cana-4368 3 5 chennai chennai PROPN cana-4368 3 6 , , PUNCT cana-4368 3 7 india india PROPN cana-4368 3 8 . . PUNCT cana-4368 4 1 3department 3department NUM cana-4368 4 2 of of ADP cana-4368 4 3 mathematics mathematic NOUN cana-4368 4 4 , , PUNCT cana-4368 4 5 r.v r.v PROPN cana-4368 4 6 . . PROPN cana-4368 4 7 govt govt PROPN cana-4368 4 8 . . PUNCT cana-4368 5 1 arts art NOUN cana-4368 5 2 , , PUNCT cana-4368 5 3 chengalpattu chengalpattu ADV cana-4368 5 4 , , PUNCT cana-4368 5 5 chennai chennai PROPN cana-4368 5 6 , , PUNCT cana-4368 5 7 india india PROPN cana-4368 5 8 . . PUNCT cana-4368 6 1 1esakkiammal2682@gmail.com,2kthirusangu@gmail.com,3seetha0687@gmail.com 1esakkiammal2682@gmail.com,2kthirusangu@gmail.com,3seetha0687@gmail.com NUM cana-4368 6 2 article article NOUN cana-4368 6 3 history history NOUN cana-4368 6 4 : : PUNCT cana-4368 6 5 received receive VERB cana-4368 6 6 : : PUNCT cana-4368 6 7 12 12 NUM cana-4368 6 8 - - SYM cana-4368 6 9 01 01 NUM cana-4368 6 10 - - PUNCT cana-4368 6 11 2025 2025 NUM cana-4368 6 12 revised revise VERB cana-4368 6 13 : : PUNCT cana-4368 6 14 15 15 NUM cana-4368 6 15 - - NUM cana-4368 6 16 02 02 NUM cana-4368 6 17 - - PUNCT cana-4368 6 18 2025 2025 NUM cana-4368 6 19 accepted accept VERB cana-4368 6 20 : : PUNCT cana-4368 6 21 01 01 NUM cana-4368 6 22 - - SYM cana-4368 6 23 03 03 NUM cana-4368 6 24 - - PUNCT cana-4368 6 25 2025 2025 NUM cana-4368 6 26 abstract abstract ADJ cana-4368 6 27 binary binary ADJ cana-4368 6 28 data datum NOUN cana-4368 6 29 strings string NOUN cana-4368 6 30 of of ADP cana-4368 6 31 equal equal ADJ cana-4368 6 32 length length NOUN cana-4368 6 33 are be AUX cana-4368 6 34 compared compare VERB cana-4368 6 35 using use VERB cana-4368 6 36 the the DET cana-4368 6 37 metric metric NOUN cana-4368 6 38 called call VERB cana-4368 6 39 hamming hamming NOUN cana-4368 6 40 distance distance NOUN cana-4368 6 41 . . PUNCT cana-4368 7 1 it it PRON cana-4368 7 2 is be AUX cana-4368 7 3 the the DET cana-4368 7 4 number number NOUN cana-4368 7 5 of of ADP cana-4368 7 6 bit bit NOUN cana-4368 7 7 positions position NOUN cana-4368 7 8 in in ADP cana-4368 7 9 which which PRON cana-4368 7 10 the the DET cana-4368 7 11 two two NUM cana-4368 7 12 binary binary ADJ cana-4368 7 13 strings string NOUN cana-4368 7 14 differ differ VERB cana-4368 7 15 . . PUNCT cana-4368 8 1 the the DET cana-4368 8 2 hamming hamming NOUN cana-4368 8 3 distance distance NOUN cana-4368 8 4 between between ADP cana-4368 8 5 two two NUM cana-4368 8 6 binary binary ADJ cana-4368 8 7 strings string NOUN cana-4368 8 8 m m PROPN cana-4368 8 9 and and CCONJ cana-4368 8 10 n n PROPN cana-4368 8 11 of of ADP cana-4368 8 12 equal equal ADJ cana-4368 8 13 length length NOUN cana-4368 8 14 is be AUX cana-4368 8 15 denoted denote VERB cana-4368 8 16 by by ADP cana-4368 8 17 hd(m hd(m NOUN cana-4368 8 18 , , PUNCT cana-4368 8 19 n n CCONJ cana-4368 8 20 ) ) PUNCT cana-4368 8 21 . . PUNCT cana-4368 9 1 we we PRON cana-4368 9 2 introduced introduce VERB cana-4368 9 3 the the DET cana-4368 9 4 concept concept NOUN cana-4368 9 5 of of ADP cana-4368 9 6 hamming hamming NOUN cana-4368 9 7 distance distance NOUN cana-4368 9 8 labeling labeling NOUN cana-4368 9 9 and and CCONJ cana-4368 9 10 odd odd ADJ cana-4368 9 11 hamming hamming NOUN cana-4368 9 12 distance distance NOUN cana-4368 9 13 labeling labeling NOUN cana-4368 9 14 . . PUNCT cana-4368 10 1 in in ADP cana-4368 10 2 this this DET cana-4368 10 3 paper paper NOUN cana-4368 10 4 , , PUNCT cana-4368 10 5 it it PRON cana-4368 10 6 is be AUX cana-4368 10 7 shown show VERB cana-4368 10 8 that that SCONJ cana-4368 10 9 path path NOUN cana-4368 10 10 graph graph NOUN cana-4368 10 11 , , PUNCT cana-4368 10 12 star star NOUN cana-4368 10 13 graph graph NOUN cana-4368 10 14 , , PUNCT cana-4368 10 15 one one NUM cana-4368 10 16 point point NOUN cana-4368 10 17 union union NOUN cana-4368 10 18 of of ADP cana-4368 10 19 path path NOUN cana-4368 10 20 graphs graph NOUN cana-4368 10 21 , , PUNCT cana-4368 10 22 coconut coconut NOUN cana-4368 10 23 tree tree NOUN cana-4368 10 24 , , PUNCT cana-4368 10 25 are be AUX cana-4368 10 26 odd odd ADJ cana-4368 10 27 hamming hamming NOUN cana-4368 10 28 distance distance NOUN cana-4368 10 29 labeled label VERB cana-4368 10 30 graphs graph NOUN cana-4368 10 31 and and CCONJ cana-4368 10 32 obtained obtain VERB cana-4368 10 33 their their PRON cana-4368 10 34 odd odd ADJ cana-4368 10 35 hamming hamming NOUN cana-4368 10 36 distance distance NOUN cana-4368 10 37 number number NOUN cana-4368 10 38 . . PUNCT cana-4368 11 1 both both PRON cana-4368 11 2 hamming ham VERB cana-4368 11 3 and and CCONJ cana-4368 11 4 odd odd ADJ cana-4368 11 5 hamming hamming NOUN cana-4368 11 6 distance distance NOUN cana-4368 11 7 labeling labeling NOUN cana-4368 11 8 are be AUX cana-4368 11 9 used use VERB cana-4368 11 10 to to PART cana-4368 11 11 send send VERB cana-4368 11 12 secret secret ADJ cana-4368 11 13 messages message NOUN cana-4368 11 14 in in ADP cana-4368 11 15 cryptography cryptography NOUN cana-4368 11 16 . . PUNCT cana-4368 12 1 keywords keyword NOUN cana-4368 12 2 : : PUNCT cana-4368 12 3 hamming ham VERB cana-4368 12 4 distance distance NOUN cana-4368 12 5 , , PUNCT cana-4368 12 6 odd odd ADJ cana-4368 12 7 hamming hamming NOUN cana-4368 12 8 distance distance NOUN cana-4368 12 9 labeling labeling NOUN cana-4368 12 10 , , PUNCT cana-4368 12 11 path path NOUN cana-4368 12 12 graph graph NOUN cana-4368 12 13 , , PUNCT cana-4368 12 14 star star NOUN cana-4368 12 15 graph graph NOUN cana-4368 12 16 , , PUNCT cana-4368 12 17 one one NUM cana-4368 12 18 - - PUNCT cana-4368 12 19 point point NOUN cana-4368 12 20 union union NOUN cana-4368 12 21 for for ADP cana-4368 12 22 path path NOUN cana-4368 12 23 of of ADP cana-4368 12 24 graphs graph NOUN cana-4368 12 25 , , PUNCT cana-4368 12 26 coconut coconut NOUN cana-4368 12 27 tree tree NOUN cana-4368 12 28 graph graph NOUN cana-4368 12 29 . . PUNCT cana-4368 13 1 1 1 X cana-4368 13 2 . . X cana-4368 13 3 introduction introduction NOUN cana-4368 13 4 let let VERB cana-4368 13 5 g g PROPN cana-4368 13 6 = = SYM cana-4368 13 7 ( ( PUNCT cana-4368 13 8 v v NOUN cana-4368 13 9 , , PUNCT cana-4368 13 10 e e NOUN cana-4368 13 11 ) ) PUNCT cana-4368 13 12 be be AUX cana-4368 13 13 a a DET cana-4368 13 14 graph graph NOUN cana-4368 13 15 with with ADP cana-4368 13 16 vertex vertex NOUN cana-4368 13 17 set set VERB cana-4368 13 18 v v NOUN cana-4368 13 19 and and CCONJ cana-4368 13 20 edge edge NOUN cana-4368 13 21 set set VERB cana-4368 13 22 e. e. PROPN cana-4368 14 1 a a DET cana-4368 14 2 path path NOUN cana-4368 14 3 graph graph NOUN cana-4368 14 4 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 14 5 , , PUNCT cana-4368 14 6 π‘š π‘š X cana-4368 14 7 β‰₯ β‰₯ NUM cana-4368 14 8 1 1 NUM cana-4368 14 9 is be AUX cana-4368 14 10 an an DET cana-4368 14 11 alternating alternate VERB cana-4368 14 12 sequence sequence NOUN cana-4368 14 13 of of ADP cana-4368 14 14 vertices vertex NOUN cana-4368 14 15 and and CCONJ cana-4368 14 16 edges edge NOUN cana-4368 14 17 , , PUNCT cana-4368 14 18 beginning begin VERB cana-4368 14 19 and and CCONJ cana-4368 14 20 ending end VERB cana-4368 14 21 with with ADP cana-4368 14 22 vertices vertex NOUN cana-4368 14 23 in in ADP cana-4368 14 24 which which PRON cana-4368 14 25 each each DET cana-4368 14 26 edge edge NOUN cana-4368 14 27 is be AUX cana-4368 14 28 incident incident NOUN cana-4368 14 29 with with ADP cana-4368 14 30 two two NUM cana-4368 14 31 vertices vertex NOUN cana-4368 14 32 immediately immediately ADV cana-4368 14 33 preceding precede VERB cana-4368 14 34 and and CCONJ cana-4368 14 35 following follow VERB cana-4368 14 36 it it PRON cana-4368 14 37 . . PUNCT cana-4368 15 1 edges edge NOUN cana-4368 15 2 and and CCONJ cana-4368 15 3 vertices vertex NOUN cana-4368 15 4 appear appear VERB cana-4368 15 5 only only ADV cana-4368 15 6 once once ADV cana-4368 15 7 in in ADP cana-4368 15 8 a a DET cana-4368 15 9 path path NOUN cana-4368 15 10 . . PUNCT cana-4368 16 1 a a DET cana-4368 16 2 path path NOUN cana-4368 16 3 graph graph NOUN cana-4368 16 4 of of ADP cana-4368 16 5 length length NOUN cana-4368 16 6 m m VERB cana-4368 16 7 has have VERB cana-4368 16 8 m+1 m+1 NUM cana-4368 16 9 vertices vertex NOUN cana-4368 16 10 and and CCONJ cana-4368 16 11 m m VERB cana-4368 16 12 edges[2].the edges[2].the PRON cana-4368 16 13 complete complete ADJ cana-4368 16 14 bipartite bipartite NOUN cana-4368 16 15 graph graph NOUN cana-4368 16 16 of of ADP cana-4368 16 17 the the DET cana-4368 16 18 form form NOUN cana-4368 16 19 𝐾1,𝑛 𝐾1,𝑛 PROPN cana-4368 16 20 is be AUX cana-4368 16 21 a a DET cana-4368 16 22 star star NOUN cana-4368 16 23 graph graph NOUN cana-4368 16 24 with with ADP cana-4368 16 25 n+1 n+1 PROPN cana-4368 16 26 vertices vertex NOUN cana-4368 16 27 and and CCONJ cana-4368 16 28 it it PRON cana-4368 16 29 is be AUX cana-4368 16 30 denoted denote VERB cana-4368 16 31 by by ADP cana-4368 16 32 𝑆𝑛 𝑆𝑛 PROPN cana-4368 16 33 , , PUNCT cana-4368 16 34 𝑛 𝑛 PRON cana-4368 16 35 β‰₯ β‰₯ NOUN cana-4368 16 36 1[3].the 1[3].the DET cana-4368 16 37 one one NUM cana-4368 16 38 point point NOUN cana-4368 16 39 union union NOUN cana-4368 16 40 of of ADP cana-4368 16 41 path path NOUN cana-4368 16 42 graph graph NOUN cana-4368 16 43 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 16 44 𝑛 𝑛 PRON cana-4368 16 45 𝑛,π‘š 𝑛,π‘š NOUN cana-4368 16 46 β‰₯ β‰₯ NOUN cana-4368 16 47 2 2 NUM cana-4368 16 48 , , PUNCT cana-4368 16 49 is be AUX cana-4368 16 50 obtain obtain ADJ cana-4368 16 51 by by ADP cana-4368 16 52 replacing replace VERB cana-4368 16 53 each each DET cana-4368 16 54 edge edge NOUN cana-4368 16 55 of of ADP cana-4368 16 56 a a DET cana-4368 16 57 star star NOUN cana-4368 16 58 graph graph NOUN cana-4368 16 59 𝐾1,𝑛 𝐾1,𝑛 VERB cana-4368 16 60 by by ADP cana-4368 16 61 path path NOUN cana-4368 16 62 graph graph NOUN cana-4368 16 63 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 16 64 , , PUNCT cana-4368 16 65 where where SCONJ cana-4368 16 66 n n X cana-4368 16 67 is be AUX cana-4368 16 68 the the DET cana-4368 16 69 number number NOUN cana-4368 16 70 of of ADP cana-4368 16 71 pendant pendant ADJ cana-4368 16 72 edges edge NOUN cana-4368 16 73 in in ADP cana-4368 16 74 star star NOUN cana-4368 16 75 graph graph NOUN cana-4368 16 76 and and CCONJ cana-4368 16 77 m m NOUN cana-4368 16 78 is be AUX cana-4368 16 79 the the DET cana-4368 16 80 length length NOUN cana-4368 16 81 of of ADP cana-4368 16 82 the the DET cana-4368 16 83 path path NOUN cana-4368 16 84 graph[6].a graph[6].a PROPN cana-4368 16 85 coconut coconut NOUN cana-4368 16 86 tree tree NOUN cana-4368 16 87 ct(n ct(n NOUN cana-4368 16 88 , , PUNCT cana-4368 16 89 m m NOUN cana-4368 16 90 ) ) PUNCT cana-4368 16 91 , , PUNCT cana-4368 16 92 𝑛 𝑛 PRON cana-4368 16 93 β‰₯ β‰₯ NOUN cana-4368 16 94 2,π‘š 2,π‘š PROPN cana-4368 16 95 β‰₯ β‰₯ NUM cana-4368 16 96 1 1 NUM cana-4368 16 97 is be AUX cana-4368 16 98 the the DET cana-4368 16 99 graph graph NOUN cana-4368 16 100 obtained obtain VERB cana-4368 16 101 from from ADP cana-4368 16 102 the the DET cana-4368 16 103 path path NOUN cana-4368 16 104 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 16 105 by by ADP cana-4368 16 106 appending append VERB cana-4368 16 107 n n DET cana-4368 16 108 new new ADJ cana-4368 16 109 pendant pendant ADJ cana-4368 16 110 edges edge NOUN cana-4368 16 111 at at ADP cana-4368 16 112 an an DET cana-4368 16 113 end end NOUN cana-4368 16 114 vertex vertex NOUN cana-4368 16 115 of of ADP cana-4368 16 116 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 16 117 [ [ X cana-4368 16 118 7 7 NUM cana-4368 16 119 ] ] PUNCT cana-4368 16 120 . . PUNCT cana-4368 17 1 graph graph NOUN cana-4368 17 2 labeling labeling NOUN cana-4368 17 3 is be AUX cana-4368 17 4 a a DET cana-4368 17 5 function function NOUN cana-4368 17 6 defined define VERB cana-4368 17 7 on on ADP cana-4368 17 8 the the DET cana-4368 17 9 vertex vertex NOUN cana-4368 17 10 set set NOUN cana-4368 17 11 or or CCONJ cana-4368 17 12 edge edge NOUN cana-4368 17 13 set set VERB cana-4368 17 14 subject subject ADJ cana-4368 17 15 to to ADP cana-4368 17 16 certain certain ADJ cana-4368 17 17 conditions condition NOUN cana-4368 17 18 enforced enforce VERB cana-4368 17 19 on on ADP cana-4368 17 20 the the DET cana-4368 17 21 number number NOUN cana-4368 17 22 of of ADP cana-4368 17 23 vertices vertex NOUN cana-4368 17 24 p p NOUN cana-4368 17 25 or or CCONJ cana-4368 17 26 on on ADP cana-4368 17 27 the the DET cana-4368 17 28 number number NOUN cana-4368 17 29 of of ADP cana-4368 17 30 edges edge NOUN cana-4368 17 31 q q PROPN cana-4368 17 32 or or CCONJ cana-4368 17 33 on on ADP cana-4368 17 34 both both CCONJ cana-4368 17 35 p p NOUN cana-4368 17 36 and and CCONJ cana-4368 17 37 q[1 q[1 PROPN cana-4368 17 38 ] ] PUNCT cana-4368 17 39 . . PUNCT cana-4368 18 1 the the DET cana-4368 18 2 concept concept NOUN cana-4368 18 3 of of ADP cana-4368 18 4 graph graph NOUN cana-4368 18 5 labeling labeling NOUN cana-4368 18 6 was be AUX cana-4368 18 7 introduced introduce VERB cana-4368 18 8 in in ADP cana-4368 18 9 the the DET cana-4368 18 10 year year NOUN cana-4368 18 11 1967 1967 NUM cana-4368 18 12 by by ADP cana-4368 18 13 rosa rosa PROPN cana-4368 18 14 and and CCONJ cana-4368 18 15 it it PRON cana-4368 18 16 was be AUX cana-4368 18 17 further far ADV cana-4368 18 18 developed develop VERB cana-4368 18 19 by by ADP cana-4368 18 20 graham graham PROPN cana-4368 18 21 and and CCONJ cana-4368 18 22 sloane sloane NOUN cana-4368 18 23 in in ADP cana-4368 18 24 1980[4 1980[4 NUM cana-4368 18 25 ] ] PUNCT cana-4368 18 26 . . PUNCT cana-4368 19 1 we we PRON cana-4368 19 2 introduced introduce VERB cana-4368 19 3 the the DET cana-4368 19 4 concept concept NOUN cana-4368 19 5 of of ADP cana-4368 19 6 hamming hamming NOUN cana-4368 19 7 distance distance NOUN cana-4368 19 8 labeling labeling NOUN cana-4368 19 9 and and CCONJ cana-4368 19 10 proved prove VERB cana-4368 19 11 that that SCONJ cana-4368 19 12 some some DET cana-4368 19 13 path path NOUN cana-4368 19 14 related relate VERB cana-4368 19 15 graphs graph NOUN cana-4368 19 16 are be AUX cana-4368 19 17 hamming ham VERB cana-4368 19 18 distance distance NOUN cana-4368 19 19 graphs graph NOUN cana-4368 19 20 [ [ X cana-4368 19 21 8 8 NUM cana-4368 19 22 ] ] PUNCT cana-4368 19 23 . . PUNCT cana-4368 20 1 in in ADP cana-4368 20 2 this this DET cana-4368 20 3 paper paper NOUN cana-4368 20 4 , , PUNCT cana-4368 20 5 prove prove VERB cana-4368 20 6 the the DET cana-4368 20 7 existence existence NOUN cana-4368 20 8 of of ADP cana-4368 20 9 odd odd ADJ cana-4368 20 10 hamming hamming NOUN cana-4368 20 11 distance distance NOUN cana-4368 20 12 labeling labeling NOUN cana-4368 20 13 of of ADP cana-4368 20 14 some some DET cana-4368 20 15 path path NOUN cana-4368 20 16 related relate VERB cana-4368 20 17 graphs graph NOUN cana-4368 20 18 . . PUNCT cana-4368 21 1 here here ADV cana-4368 21 2 the the DET cana-4368 21 3 notation notation NOUN cana-4368 21 4 [ [ X cana-4368 21 5 π‘₯]2 π‘₯]2 PROPN cana-4368 21 6 denotes denote VERB cana-4368 21 7 the the DET cana-4368 21 8 binary binary PROPN cana-4368 21 9 conversion conversion NOUN cana-4368 21 10 of of ADP cana-4368 21 11 the the DET cana-4368 21 12 number number NOUN cana-4368 22 1 π‘₯. π‘₯. NOUN cana-4368 22 2 mailto:esakkiammal2682@gmail.com mailto:esakkiammal2682@gmail.com X cana-4368 23 1 mailto:kthirusangu@gmail.com mailto:kthirusangu@gmail.com NOUN cana-4368 23 2 communications communication NOUN cana-4368 23 3 on on ADP cana-4368 23 4 applied apply VERB cana-4368 23 5 nonlinear nonlinear ADJ cana-4368 23 6 analysis analysis NOUN cana-4368 23 7 issn issn NOUN cana-4368 23 8 : : PUNCT cana-4368 23 9 1074 1074 NUM cana-4368 23 10 - - PUNCT cana-4368 23 11 133x 133x NUM cana-4368 23 12 vol vol NOUN cana-4368 23 13 32 32 NUM cana-4368 23 14 no no NOUN cana-4368 23 15 . . PUNCT cana-4368 24 1 9s 9s NUM cana-4368 24 2 ( ( PUNCT cana-4368 24 3 2025 2025 NUM cana-4368 24 4 ) ) PUNCT cana-4368 24 5 1929 1929 NUM cana-4368 25 1 https://internationalpubls.com https://internationalpubls.com X cana-4368 25 2 2 2 NUM cana-4368 25 3 . . NOUN cana-4368 25 4 odd odd ADJ cana-4368 25 5 hamming hamming NOUN cana-4368 25 6 distance distance NOUN cana-4368 25 7 labelling labelling NOUN cana-4368 25 8 of of ADP cana-4368 25 9 some some DET cana-4368 25 10 graphs graph NOUN cana-4368 25 11 2.1 2.1 NUM cana-4368 25 12 . . PUNCT cana-4368 26 1 definition definition NOUN cana-4368 26 2 let let VERB cana-4368 26 3 g g PROPN cana-4368 26 4 = = SYM cana-4368 26 5 ( ( PUNCT cana-4368 26 6 v v NOUN cana-4368 26 7 , , PUNCT cana-4368 26 8 e e NOUN cana-4368 26 9 ) ) PUNCT cana-4368 26 10 be be AUX cana-4368 26 11 a a DET cana-4368 26 12 graph graph NOUN cana-4368 26 13 . . PUNCT cana-4368 27 1 a a DET cana-4368 27 2 function function NOUN cana-4368 27 3 𝑓 𝑓 NOUN cana-4368 27 4 : : PUNCT cana-4368 27 5 𝑉 𝑉 PROPN cana-4368 27 6 β†’ β†’ PUNCT cana-4368 27 7 𝑁 𝑁 PROPN cana-4368 27 8 βˆͺ βˆͺ ADJ cana-4368 27 9 { { PUNCT cana-4368 27 10 0 0 NUM cana-4368 27 11 } } PUNCT cana-4368 27 12 is be AUX cana-4368 27 13 said say VERB cana-4368 27 14 to to PART cana-4368 27 15 be be AUX cana-4368 27 16 an an DET cana-4368 27 17 odd odd ADJ cana-4368 27 18 hamming hamming NOUN cana-4368 27 19 distance distance NOUN cana-4368 27 20 labeling labeling NOUN cana-4368 27 21 if if SCONJ cana-4368 27 22 there there PRON cana-4368 27 23 exist exist VERB cana-4368 27 24 an an DET cana-4368 27 25 induced induced ADJ cana-4368 27 26 function function NOUN cana-4368 27 27 π‘“βˆ— π‘“βˆ— NOUN cana-4368 27 28 ∢ ∢ PROPN cana-4368 27 29 𝐸 𝐸 PROPN cana-4368 27 30 β†’ β†’ SYM cana-4368 27 31 { { PUNCT cana-4368 27 32 1,3,5 1,3,5 NUM cana-4368 27 33 , , PUNCT cana-4368 27 34 … … PUNCT cana-4368 27 35 , , PUNCT cana-4368 27 36 n n CCONJ cana-4368 27 37 } } PUNCT cana-4368 27 38 such such ADJ cana-4368 27 39 that that PRON cana-4368 27 40 for for ADP cana-4368 27 41 every every DET cana-4368 27 42 𝑒𝑣 𝑒𝑣 PROPN cana-4368 27 43 ∈ ∈ PROPN cana-4368 27 44 𝐸 𝐸 PROPN cana-4368 27 45 , , PUNCT cana-4368 27 46 π‘“βˆ—(𝑒𝑣 π‘“βˆ—(𝑒𝑣 PROPN cana-4368 27 47 ) ) PUNCT cana-4368 27 48 = = SYM cana-4368 27 49 β„Žπ‘‘([𝑓(𝑒)]2 β„Žπ‘‘([𝑓(𝑒)]2 ADJ cana-4368 27 50 , , PUNCT cana-4368 27 51 [ [ X cana-4368 27 52 𝑓(𝑣)]2 𝑓(𝑣)]2 X cana-4368 27 53 ) ) PUNCT cana-4368 27 54 satisfying satisfy VERB cana-4368 27 55 the the DET cana-4368 27 56 following follow VERB cana-4368 27 57 conditions condition NOUN cana-4368 27 58 : : PUNCT cana-4368 27 59 ( ( PUNCT cana-4368 27 60 i i NOUN cana-4368 27 61 ) ) PUNCT cana-4368 27 62 for for ADP cana-4368 27 63 every every DET cana-4368 27 64 vertex vertex NOUN cana-4368 27 65 𝑣 𝑣 ADP cana-4368 27 66 πœ– πœ– PROPN cana-4368 27 67 𝑉 𝑉 PROPN cana-4368 27 68 , , PUNCT cana-4368 27 69 the the DET cana-4368 27 70 set set NOUN cana-4368 27 71 of of ADP cana-4368 27 72 all all DET cana-4368 27 73 edges edge NOUN cana-4368 27 74 incident incident NOUN cana-4368 27 75 with with ADP cana-4368 27 76 𝑣 𝑣 PART cana-4368 27 77 receive receive VERB cana-4368 27 78 distinct distinct ADJ cana-4368 27 79 odd odd ADJ cana-4368 27 80 labels label NOUN cana-4368 27 81 . . PUNCT cana-4368 28 1 ( ( PUNCT cana-4368 28 2 ii ii NOUN cana-4368 28 3 ) ) PUNCT cana-4368 28 4 for for ADP cana-4368 28 5 every every DET cana-4368 28 6 edge edge NOUN cana-4368 28 7 𝑒 𝑒 ADP cana-4368 28 8 = = SYM cana-4368 28 9 𝑒𝑣 𝑒𝑣 PROPN cana-4368 28 10 , , PUNCT cana-4368 28 11 the the DET cana-4368 28 12 adjacent adjacent ADJ cana-4368 28 13 vertices vertice VERB cana-4368 28 14 𝑒 𝑒 PRON cana-4368 28 15 and and CCONJ cana-4368 28 16 𝑣 𝑣 PART cana-4368 28 17 receive receive VERB cana-4368 28 18 distinct distinct ADJ cana-4368 28 19 labels label NOUN cana-4368 28 20 . . PUNCT cana-4368 29 1 a a DET cana-4368 29 2 graph graph NOUN cana-4368 29 3 which which PRON cana-4368 29 4 admits admit VERB cana-4368 29 5 odd odd ADJ cana-4368 29 6 hamming hamming NOUN cana-4368 29 7 distance distance NOUN cana-4368 29 8 labeling labeling NOUN cana-4368 29 9 is be AUX cana-4368 29 10 called call VERB cana-4368 29 11 odd odd ADJ cana-4368 29 12 hamming hamming NOUN cana-4368 29 13 distance distance NOUN cana-4368 29 14 graph.the graph.the DET cana-4368 29 15 odd odd ADJ cana-4368 29 16 hamming hamming NOUN cana-4368 29 17 distance distance NOUN cana-4368 29 18 number number NOUN cana-4368 29 19 of of ADP cana-4368 29 20 a a DET cana-4368 29 21 graph graph NOUN cana-4368 29 22 g g NOUN cana-4368 29 23 is be AUX cana-4368 29 24 the the DET cana-4368 29 25 least least ADV cana-4368 29 26 positive positive ADJ cana-4368 29 27 integer integer NOUN cana-4368 29 28 n n CCONJ cana-4368 29 29 such such ADJ cana-4368 29 30 that that SCONJ cana-4368 29 31 2𝑛 2𝑛 PROPN cana-4368 30 1 βˆ’ βˆ’ PROPN cana-4368 30 2 1 1 NUM cana-4368 30 3 β‰₯ β‰₯ NOUN cana-4368 30 4 π‘˜ π‘˜ NOUN cana-4368 30 5 , , PUNCT cana-4368 30 6 where where SCONJ cana-4368 30 7 π‘˜ π‘˜ PROPN cana-4368 30 8 = = X cana-4368 30 9 max max PROPN cana-4368 30 10 { { PUNCT cana-4368 30 11 𝑓(𝑣)/𝑣 𝑓(𝑣)/𝑣 PROPN cana-4368 30 12 ∈ ∈ PROPN cana-4368 30 13 𝑉 𝑉 PROPN cana-4368 30 14 } } PUNCT cana-4368 30 15 and and CCONJ cana-4368 30 16 it it PRON cana-4368 30 17 is be AUX cana-4368 30 18 denoted denote VERB cana-4368 30 19 by by ADP cana-4368 30 20 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 30 21 β€² β€² NUM cana-4368 30 22 ( ( PUNCT cana-4368 30 23 g g NOUN cana-4368 30 24 ) ) PUNCT cana-4368 30 25 . . PUNCT cana-4368 31 1 2.2.1 2.2.1 NUM cana-4368 31 2 . . PUNCT cana-4368 31 3 algorithm algorithm NOUN cana-4368 31 4 : : PUNCT cana-4368 31 5 odd odd ADJ cana-4368 31 6 hamming hamming NOUN cana-4368 31 7 distance distance NOUN cana-4368 31 8 labeling labeling NOUN cana-4368 31 9 of of ADP cana-4368 31 10 𝐏𝐦 𝐏𝐦 PROPN cana-4368 31 11 graph graph NOUN cana-4368 31 12 procedure procedure NOUN cana-4368 31 13 : : PUNCT cana-4368 31 14 vertex vertex NOUN cana-4368 31 15 labeling labeling NOUN cana-4368 31 16 of of ADP cana-4368 31 17 pm pm NOUN cana-4368 31 18 graph graph NOUN cana-4368 31 19 , , PUNCT cana-4368 31 20 m m VERB cana-4368 31 21 β‰₯ β‰₯ NOUN cana-4368 31 22 1 1 NUM cana-4368 31 23 input input NOUN cana-4368 31 24 : : PUNCT cana-4368 31 25 path path NOUN cana-4368 31 26 graph graph NOUN cana-4368 31 27 pm pm PROPN cana-4368 31 28 v v X cana-4368 31 29 ← ← PROPN cana-4368 31 30 { { PUNCT cana-4368 31 31 vi vi NOUN cana-4368 31 32 /0 /0 NOUN cana-4368 31 33 ≀ ≀ NUM cana-4368 32 1 i i PRON cana-4368 32 2 ≀ ≀ NUM cana-4368 32 3 m m VERB cana-4368 32 4 } } PUNCT cana-4368 32 5 v0 v0 NOUN cana-4368 32 6 ← ← PROPN cana-4368 32 7 0 0 NUM cana-4368 32 8 ; ; PUNCT cana-4368 32 9 for for ADP cana-4368 32 10 i i PRON cana-4368 32 11 = = SYM cana-4368 32 12 1 1 NUM cana-4368 32 13 to to PART cana-4368 32 14 m m PROPN cana-4368 32 15 do do AUX cana-4368 32 16 vi vi PROPN cana-4368 32 17 ← ← PROPN cana-4368 32 18 { { PUNCT cana-4368 32 19 1 1 NUM cana-4368 32 20 if if SCONJ cana-4368 32 21 i i PRON cana-4368 32 22 ≑ ≑ PROPN cana-4368 32 23 1(mod4 1(mod4 NUM cana-4368 32 24 ) ) PUNCT cana-4368 32 25 6 6 NUM cana-4368 32 26 if if SCONJ cana-4368 32 27 i i PRON cana-4368 32 28 ≑ ≑ PROPN cana-4368 32 29 2(mod4 2(mod4 NUM cana-4368 32 30 ) ) PUNCT cana-4368 32 31 2 2 NUM cana-4368 32 32 if if SCONJ cana-4368 32 33 i i PRON cana-4368 32 34 ≑ ≑ PROPN cana-4368 32 35 3(mod4 3(mod4 NUM cana-4368 32 36 ) ) PUNCT cana-4368 32 37 5 5 NUM cana-4368 32 38 if if SCONJ cana-4368 32 39 i i PRON cana-4368 32 40 ≑ ≑ PROPN cana-4368 32 41 0(mod4 0(mod4 NUM cana-4368 32 42 ) ) PUNCT cana-4368 32 43 ; ; PUNCT cana-4368 32 44 end end VERB cana-4368 32 45 for for ADP cana-4368 32 46 end end NOUN cana-4368 32 47 procedure procedure NOUN cana-4368 32 48 output output NOUN cana-4368 32 49 : : PUNCT cana-4368 32 50 the the DET cana-4368 32 51 labeled label VERB cana-4368 32 52 vertices vertex NOUN cana-4368 32 53 of of ADP cana-4368 32 54 path path NOUN cana-4368 32 55 graph graph NOUN cana-4368 32 56 pm pm NOUN cana-4368 32 57 . . PUNCT cana-4368 33 1 2.3.2.theorem 2.3.2.theorem NUM cana-4368 33 2 the the DET cana-4368 33 3 path path NOUN cana-4368 33 4 graph graph NOUN cana-4368 33 5 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 33 6 , , PUNCT cana-4368 33 7 π‘š π‘š X cana-4368 33 8 β‰₯ β‰₯ NUM cana-4368 33 9 1 1 NUM cana-4368 33 10 is be AUX cana-4368 33 11 an an DET cana-4368 33 12 odd odd ADJ cana-4368 33 13 hamming hamming NOUN cana-4368 33 14 distance distance NOUN cana-4368 33 15 graph graph NOUN cana-4368 33 16 and and CCONJ cana-4368 33 17 the the DET cana-4368 33 18 odd odd ADJ cana-4368 33 19 hamming hamming NOUN cana-4368 33 20 distance distance NOUN cana-4368 33 21 number number NOUN cana-4368 33 22 is be AUX cana-4368 33 23 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 33 24 β€² β€² NUM cana-4368 33 25 ( ( PUNCT cana-4368 33 26 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 33 27 ) ) PUNCT cana-4368 33 28 = = PUNCT cana-4368 33 29 { { PUNCT cana-4368 33 30 1 1 NUM cana-4368 33 31 𝑖𝑓 𝑖𝑓 ADP cana-4368 33 32 π‘š π‘š NOUN cana-4368 33 33 = = SYM cana-4368 33 34 1 1 NUM cana-4368 33 35 3 3 NUM cana-4368 33 36 𝑖𝑓 𝑖𝑓 ADP cana-4368 33 37 π‘š π‘š X cana-4368 33 38 > > X cana-4368 33 39 1 1 NUM cana-4368 33 40 . . PUNCT cana-4368 34 1 proof proof NOUN cana-4368 34 2 : : PUNCT cana-4368 34 3 let let VERB cana-4368 34 4 us we PRON cana-4368 34 5 consider consider VERB cana-4368 34 6 the the DET cana-4368 34 7 path path NOUN cana-4368 34 8 graph graph NOUN cana-4368 34 9 pm pm NOUN cana-4368 34 10 with with ADP cana-4368 34 11 vertex vertex NOUN cana-4368 34 12 set set VERB cana-4368 34 13 v v NOUN cana-4368 34 14 = = SYM cana-4368 34 15 { { PUNCT cana-4368 34 16 vi/0 vi/0 NOUN cana-4368 34 17 ≀ ≀ NUM cana-4368 34 18 i i PRON cana-4368 34 19 ≀ ≀ NOUN cana-4368 34 20 m m VERB cana-4368 34 21 } } PUNCT cana-4368 34 22 and and CCONJ cana-4368 34 23 edge edge VERB cana-4368 34 24 set set VERB cana-4368 34 25 e e NOUN cana-4368 34 26 = = PUNCT cana-4368 34 27 { { PUNCT cana-4368 34 28 vivi+1 vivi+1 PROPN cana-4368 34 29 / / SYM cana-4368 34 30 0 0 NUM cana-4368 34 31 ≀ ≀ NOUN cana-4368 35 1 i i PRON cana-4368 35 2 ≀ ≀ ADJ cana-4368 35 3 mβˆ’ mβˆ’ PROPN cana-4368 35 4 1 1 NUM cana-4368 35 5 } } PUNCT cana-4368 35 6 . . PUNCT cana-4368 36 1 define define VERB cana-4368 36 2 a a DET cana-4368 36 3 function function NOUN cana-4368 36 4 𝑓 𝑓 PRON cana-4368 36 5 : : PUNCT cana-4368 36 6 v v NOUN cana-4368 36 7 β†’ β†’ SYM cana-4368 36 8 n n CCONJ cana-4368 36 9 βˆͺ βˆͺ X cana-4368 36 10 { { PUNCT cana-4368 36 11 0 0 NUM cana-4368 36 12 } } PUNCT cana-4368 36 13 such such ADJ cana-4368 36 14 that that SCONJ cana-4368 36 15 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-4368 36 16 ) ) PUNCT cana-4368 36 17 β‰  β‰  PROPN cana-4368 36 18 𝑓(𝑣𝑖+1 𝑓(𝑣𝑖+1 NUM cana-4368 36 19 ) ) PUNCT cana-4368 36 20 , , PUNCT cana-4368 36 21 0 0 NUM cana-4368 36 22 ≀ ≀ PUNCT cana-4368 37 1 i i PRON cana-4368 37 2 ≀ ≀ NUM cana-4368 37 3 m m VERB cana-4368 37 4 βˆ’ βˆ’ NOUN cana-4368 37 5 1 1 NUM cana-4368 37 6 as as SCONJ cana-4368 37 7 given give VERB cana-4368 37 8 in in ADP cana-4368 37 9 the the DET cana-4368 37 10 above above ADJ cana-4368 37 11 algorithm algorithm NOUN cana-4368 37 12 3.3.1 3.3.1 NUM cana-4368 37 13 . . PUNCT cana-4368 38 1 hence hence ADV cana-4368 38 2 the the DET cana-4368 38 3 adjacent adjacent ADJ cana-4368 38 4 vertices vertex NOUN cana-4368 38 5 receive receive VERB cana-4368 38 6 distinct distinct ADJ cana-4368 38 7 labels.the labels.the DET cana-4368 38 8 edge edge NOUN cana-4368 38 9 labels label NOUN cana-4368 38 10 are be AUX cana-4368 38 11 obtained obtain VERB cana-4368 38 12 as as SCONJ cana-4368 38 13 follows follow VERB cana-4368 38 14 : : PUNCT cana-4368 38 15 f f PROPN cana-4368 38 16 βˆ—(v0v1 βˆ—(v0v1 PROPN cana-4368 38 17 ) ) PUNCT cana-4368 39 1 = = SYM cana-4368 39 2 hd([f(v0)]2 hd([f(v0)]2 NOUN cana-4368 39 3 , , PUNCT cana-4368 39 4 [ [ X cana-4368 39 5 f(v1)]2 f(v1)]2 NOUN cana-4368 39 6 ) ) PUNCT cana-4368 39 7 = = SYM cana-4368 40 1 hd([0]2 hd([0]2 NOUN cana-4368 40 2 , , PUNCT cana-4368 40 3 [ [ X cana-4368 40 4 1]2 1]2 NUM cana-4368 40 5 ) ) PUNCT cana-4368 40 6 = = SYM cana-4368 40 7 hd(00000 hd(00000 PROPN cana-4368 40 8 , , PUNCT cana-4368 40 9 00001 00001 NUM cana-4368 40 10 ) ) PUNCT cana-4368 41 1 = = NOUN cana-4368 41 2 1 1 NUM cana-4368 41 3 for for ADP cana-4368 41 4 1 1 NUM cana-4368 41 5 ≀ ≀ NUM cana-4368 41 6 i i PRON cana-4368 41 7 ≀ ≀ NOUN cana-4368 41 8 m m VERB cana-4368 41 9 βˆ’ βˆ’ NUM cana-4368 41 10 1 1 NUM cana-4368 41 11 case case NOUN cana-4368 41 12 ( ( PUNCT cana-4368 41 13 i i NOUN cana-4368 41 14 ): ): PUNCT cana-4368 41 15 when when SCONJ cana-4368 41 16 i i PRON cana-4368 41 17 ≑ ≑ PROPN cana-4368 41 18 1(mod 1(mod NUM cana-4368 42 1 4);f 4);f NUM cana-4368 42 2 βˆ—(vivi+1 βˆ—(vivi+1 X cana-4368 42 3 ) ) PUNCT cana-4368 42 4 = = SYM cana-4368 43 1 hd([f(vi)]2 hd([f(vi)]2 NUM cana-4368 43 2 , , PUNCT cana-4368 43 3 [ [ X cana-4368 43 4 f(vi+1)]2 f(vi+1)]2 X cana-4368 43 5 ) ) PUNCT cana-4368 43 6 = = PUNCT cana-4368 44 1 hd([1]2 hd([1]2 PROPN cana-4368 44 2 , , PUNCT cana-4368 44 3 [ [ X cana-4368 44 4 6]2 6]2 NOUN cana-4368 44 5 ) ) PUNCT cana-4368 44 6 = = SYM cana-4368 44 7 3 3 NUM cana-4368 44 8 case case NOUN cana-4368 44 9 ( ( PUNCT cana-4368 44 10 ii ii NOUN cana-4368 44 11 ): ): PUNCT cana-4368 44 12 when when SCONJ cana-4368 44 13 i i PRON cana-4368 44 14 ≑ ≑ PROPN cana-4368 44 15 2(mod4);f 2(mod4);f PROPN cana-4368 44 16 βˆ—(vivi+1 βˆ—(vivi+1 PROPN cana-4368 44 17 ) ) PUNCT cana-4368 44 18 = = SYM cana-4368 45 1 hd([f(vi)]2 hd([f(vi)]2 NUM cana-4368 45 2 , , PUNCT cana-4368 45 3 [ [ X cana-4368 45 4 f(vi+1)]2 f(vi+1)]2 X cana-4368 45 5 ) ) PUNCT cana-4368 45 6 = = SYM cana-4368 46 1 hd([6]2 hd([6]2 X cana-4368 46 2 , , PUNCT cana-4368 46 3 [ [ X cana-4368 46 4 2]2 2]2 NUM cana-4368 46 5 ) ) PUNCT cana-4368 46 6 = = SYM cana-4368 46 7 1 1 NUM cana-4368 46 8 communications communication NOUN cana-4368 46 9 on on ADP cana-4368 46 10 applied apply VERB cana-4368 46 11 nonlinear nonlinear ADJ cana-4368 46 12 analysis analysis NOUN cana-4368 46 13 issn issn NOUN cana-4368 46 14 : : PUNCT cana-4368 46 15 1074 1074 NUM cana-4368 46 16 - - PUNCT cana-4368 46 17 133x 133x NUM cana-4368 46 18 vol vol NOUN cana-4368 46 19 32 32 NUM cana-4368 46 20 no no NOUN cana-4368 46 21 . . PUNCT cana-4368 47 1 9s 9s NUM cana-4368 47 2 ( ( PUNCT cana-4368 47 3 2025 2025 NUM cana-4368 47 4 ) ) PUNCT cana-4368 47 5 1930 1930 NUM cana-4368 48 1 https://internationalpubls.com https://internationalpubls.com X cana-4368 48 2 case case NOUN cana-4368 48 3 ( ( PUNCT cana-4368 48 4 iii iii NOUN cana-4368 48 5 ): ): PUNCT cana-4368 48 6 when when SCONJ cana-4368 48 7 i i PRON cana-4368 48 8 ≑ ≑ VERB cana-4368 49 1 3(mod 3(mod NUM cana-4368 50 1 4);f 4);f NUM cana-4368 50 2 βˆ—(vivi+1 βˆ—(vivi+1 X cana-4368 50 3 ) ) PUNCT cana-4368 50 4 = = SYM cana-4368 51 1 hd([f(vi)]2 hd([f(vi)]2 NUM cana-4368 51 2 , , PUNCT cana-4368 51 3 [ [ X cana-4368 51 4 f(vi+1)]2 f(vi+1)]2 X cana-4368 51 5 ) ) PUNCT cana-4368 51 6 = = PUNCT cana-4368 52 1 hd([2]2 hd([2]2 NOUN cana-4368 52 2 , , PUNCT cana-4368 52 3 [ [ X cana-4368 52 4 5]2 5]2 X cana-4368 52 5 ) ) PUNCT cana-4368 52 6 = = SYM cana-4368 52 7 3 3 NUM cana-4368 52 8 case case NOUN cana-4368 52 9 ( ( PUNCT cana-4368 52 10 iv iv NUM cana-4368 52 11 ): ): PUNCT cana-4368 52 12 when when SCONJ cana-4368 52 13 i i PRON cana-4368 52 14 ≑ ≑ PROPN cana-4368 52 15 0(mod4);f 0(mod4);f PUNCT cana-4368 52 16 βˆ—(vivi+1 βˆ—(vivi+1 NUM cana-4368 52 17 ) ) PUNCT cana-4368 52 18 = = SYM cana-4368 53 1 hd([f(vi)]2 hd([f(vi)]2 NUM cana-4368 53 2 , , PUNCT cana-4368 53 3 [ [ X cana-4368 53 4 f(vi+1)]2 f(vi+1)]2 X cana-4368 53 5 ) ) PUNCT cana-4368 53 6 = = SYM cana-4368 53 7 hd([5]2 hd([5]2 NOUN cana-4368 53 8 , , PUNCT cana-4368 53 9 [ [ X cana-4368 53 10 1]2 1]2 NUM cana-4368 53 11 ) ) PUNCT cana-4368 53 12 = = SYM cana-4368 53 13 1 1 NUM cana-4368 53 14 from from ADP cana-4368 53 15 all all DET cana-4368 53 16 the the DET cana-4368 53 17 above above ADJ cana-4368 53 18 cases case NOUN cana-4368 53 19 , , PUNCT cana-4368 53 20 all all DET cana-4368 53 21 the the DET cana-4368 53 22 adjacent adjacent ADJ cana-4368 53 23 edges edge NOUN cana-4368 53 24 receive receive VERB cana-4368 53 25 distinct distinct ADJ cana-4368 53 26 odd odd ADJ cana-4368 53 27 labels label NOUN cana-4368 53 28 . . PUNCT cana-4368 54 1 hence hence ADV cana-4368 54 2 the the DET cana-4368 54 3 path path NOUN cana-4368 54 4 graph graph NOUN cana-4368 54 5 pm pm NOUN cana-4368 54 6 admits admit VERB cana-4368 54 7 odd odd ADJ cana-4368 54 8 hamming hamming NOUN cana-4368 54 9 distance distance NOUN cana-4368 54 10 labeling labeling NOUN cana-4368 54 11 and and CCONJ cana-4368 54 12 the the DET cana-4368 54 13 odd odd ADJ cana-4368 54 14 hamming hamming NOUN cana-4368 54 15 distance distance NOUN cana-4368 54 16 number number NOUN cana-4368 54 17 is be AUX cana-4368 54 18 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 54 19 β€² β€² NUM cana-4368 54 20 ( ( PUNCT cana-4368 54 21 pm pm NOUN cana-4368 54 22 ) ) PUNCT cana-4368 54 23 = = PRON cana-4368 54 24 { { PUNCT cana-4368 54 25 1 1 NUM cana-4368 54 26 𝑖𝑓 𝑖𝑓 ADP cana-4368 54 27 π‘š π‘š NOUN cana-4368 54 28 = = SYM cana-4368 54 29 1 1 NUM cana-4368 54 30 3 3 NUM cana-4368 54 31 𝑖𝑓 𝑖𝑓 ADP cana-4368 54 32 π‘š π‘š X cana-4368 54 33 > > X cana-4368 54 34 1 1 X cana-4368 54 35 . . PUNCT cana-4368 55 1 figure figure VERB cana-4368 55 2 3.odd 3.odd NUM cana-4368 55 3 hamming hamming NOUN cana-4368 55 4 distance distance NOUN cana-4368 55 5 π‘·πŸπŸŽ π‘·πŸπŸŽ NOUN cana-4368 55 6 graph graph NOUN cana-4368 55 7 . . PUNCT cana-4368 56 1 3.3.3 3.3.3 X cana-4368 56 2 . . PUNCT cana-4368 57 1 algorithm algorithm NOUN cana-4368 57 2 : : PUNCT cana-4368 57 3 odd odd ADJ cana-4368 57 4 hamming hamming NOUN cana-4368 57 5 distance distance NOUN cana-4368 57 6 labeling labeling NOUN cana-4368 57 7 of of ADP cana-4368 57 8 𝑺𝒏 𝑺𝒏 PROPN cana-4368 57 9 graph graph NOUN cana-4368 57 10 . . PUNCT cana-4368 58 1 procedure procedure NOUN cana-4368 58 2 : : PUNCT cana-4368 58 3 vertex vertex NOUN cana-4368 58 4 labeling labeling NOUN cana-4368 58 5 of of ADP cana-4368 58 6 𝑆𝑛 𝑆𝑛 ADJ cana-4368 58 7 graph graph NOUN cana-4368 58 8 𝑛 𝑛 PRON cana-4368 58 9 β‰₯ β‰₯ NUM cana-4368 58 10 1 1 NUM cana-4368 58 11 . . PUNCT cana-4368 59 1 input input NOUN cana-4368 59 2 : : PUNCT cana-4368 59 3 star star NOUN cana-4368 59 4 graph graph NOUN cana-4368 59 5 𝑆𝑛 𝑆𝑛 PROPN cana-4368 59 6 𝑉 𝑉 PROPN cana-4368 59 7 ← ← PROPN cana-4368 59 8 { { PUNCT cana-4368 59 9 𝑣𝑖 𝑣𝑖 NOUN cana-4368 59 10 /0 /0 PUNCT cana-4368 59 11 ≀ ≀ NUM cana-4368 59 12 𝑖 𝑖 SYM cana-4368 59 13 ≀ ≀ NUM cana-4368 59 14 𝑛 𝑛 PRON cana-4368 59 15 } } PUNCT cana-4368 59 16 𝑣0 𝑣0 PROPN cana-4368 59 17 ← ← PROPN cana-4368 59 18 0 0 NUM cana-4368 59 19 ; ; PUNCT cana-4368 59 20 for for ADP cana-4368 59 21 𝑖 𝑖 SYM cana-4368 59 22 = = SYM cana-4368 59 23 1 1 NUM cana-4368 59 24 π‘‘π‘œ π‘‘π‘œ NOUN cana-4368 59 25 𝑛 𝑛 VERB cana-4368 59 26 do do AUX cana-4368 59 27 𝑣𝑖 𝑣𝑖 VERB cana-4368 59 28 ← ← PROPN cana-4368 59 29 22π‘–βˆ’1 22π‘–βˆ’1 PROPN cana-4368 59 30 βˆ’ βˆ’ PROPN cana-4368 59 31 1 1 NUM cana-4368 59 32 ; ; PUNCT cana-4368 59 33 end end VERB cana-4368 59 34 for for ADP cana-4368 59 35 end end NOUN cana-4368 59 36 procedure procedure NOUN cana-4368 59 37 output output NOUN cana-4368 59 38 : : PUNCT cana-4368 59 39 the the DET cana-4368 59 40 labeled label VERB cana-4368 59 41 vertices vertex NOUN cana-4368 59 42 of of ADP cana-4368 59 43 star star NOUN cana-4368 59 44 graph graph NOUN cana-4368 59 45 . . PUNCT cana-4368 60 1 𝑆𝑛. 𝑆𝑛. PROPN cana-4368 60 2 3.3.4.theorem 3.3.4.theorem NUM cana-4368 60 3 the the DET cana-4368 60 4 star star NOUN cana-4368 60 5 graph graph NOUN cana-4368 60 6 𝑆𝑛 𝑆𝑛 PROPN cana-4368 60 7 , , PUNCT cana-4368 60 8 𝑛 𝑛 DET cana-4368 60 9 β‰₯ β‰₯ NOUN cana-4368 60 10 1 1 NUM cana-4368 60 11 is be AUX cana-4368 60 12 an an DET cana-4368 60 13 odd odd ADJ cana-4368 60 14 hamming hamming NOUN cana-4368 60 15 distance distance NOUN cana-4368 60 16 graph graph NOUN cana-4368 60 17 and and CCONJ cana-4368 60 18 the the DET cana-4368 60 19 odd odd ADJ cana-4368 60 20 hamming hamming NOUN cana-4368 60 21 distance distance NOUN cana-4368 60 22 number number NOUN cana-4368 60 23 is be AUX cana-4368 60 24 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 60 25 β€² β€² NUM cana-4368 60 26 ( ( PUNCT cana-4368 60 27 𝑆𝑛 𝑆𝑛 PROPN cana-4368 60 28 ) ) PUNCT cana-4368 60 29 = = SYM cana-4368 60 30 2𝑛 2𝑛 PROPN cana-4368 61 1 βˆ’ βˆ’ PROPN cana-4368 61 2 1,where 1,where NUM cana-4368 61 3 n n PRON cana-4368 61 4 is be AUX cana-4368 61 5 the the DET cana-4368 61 6 number number NOUN cana-4368 61 7 of of ADP cana-4368 61 8 pendant pendant ADJ cana-4368 61 9 edges edge NOUN cana-4368 61 10 . . PUNCT cana-4368 62 1 proof proof NOUN cana-4368 62 2 : : PUNCT cana-4368 62 3 let let VERB cana-4368 62 4 us we PRON cana-4368 62 5 consider consider VERB cana-4368 62 6 the the DET cana-4368 62 7 star star NOUN cana-4368 62 8 graph graph NOUN cana-4368 62 9 𝑆𝑛 𝑆𝑛 PROPN cana-4368 62 10 with with ADP cana-4368 62 11 vertex vertex NOUN cana-4368 62 12 set set VERB cana-4368 62 13 𝑉 𝑉 PROPN cana-4368 62 14 = = PUNCT cana-4368 62 15 { { PUNCT cana-4368 62 16 𝑣𝑖 𝑣𝑖 NOUN cana-4368 62 17 /0 /0 NOUN cana-4368 62 18 ≀ ≀ NUM cana-4368 62 19 𝑖 𝑖 SYM cana-4368 62 20 ≀ ≀ NUM cana-4368 62 21 𝑛 𝑛 PRON cana-4368 62 22 } } PUNCT cana-4368 62 23 and and CCONJ cana-4368 62 24 edge edge NOUN cana-4368 62 25 set set VERB cana-4368 62 26 𝐸 𝐸 NOUN cana-4368 62 27 = = SYM cana-4368 62 28 { { PUNCT cana-4368 62 29 𝑣0𝑣𝑖 𝑣0𝑣𝑖 NOUN cana-4368 62 30 / / SYM cana-4368 62 31 1 1 NUM cana-4368 62 32 ≀ ≀ NUM cana-4368 62 33 𝑖 𝑖 SYM cana-4368 62 34 ≀ ≀ NUM cana-4368 62 35 𝑛 𝑛 NOUN cana-4368 62 36 } } PUNCT cana-4368 62 37 . . PUNCT cana-4368 63 1 define define VERB cana-4368 63 2 a a DET cana-4368 63 3 function function NOUN cana-4368 63 4 𝑓 𝑓 NOUN cana-4368 63 5 : : PUNCT cana-4368 63 6 𝑉 𝑉 PROPN cana-4368 63 7 ← ← NOUN cana-4368 63 8 𝑁 𝑁 PROPN cana-4368 63 9 βˆͺ βˆͺ ADJ cana-4368 63 10 { { PUNCT cana-4368 63 11 0 0 NUM cana-4368 63 12 } } PUNCT cana-4368 63 13 such such ADJ cana-4368 63 14 that that DET cana-4368 63 15 𝑓(𝑣0 𝑓(𝑣0 NOUN cana-4368 63 16 ) ) PUNCT cana-4368 64 1 β‰  β‰  PROPN cana-4368 64 2 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-4368 64 3 ) ) PUNCT cana-4368 64 4 , , PUNCT cana-4368 64 5 1 1 NUM cana-4368 64 6 ≀ ≀ NUM cana-4368 64 7 𝑖 𝑖 SYM cana-4368 64 8 ≀ ≀ NUM cana-4368 64 9 𝑛 𝑛 PRON cana-4368 64 10 as as SCONJ cana-4368 64 11 given give VERB cana-4368 64 12 in in ADP cana-4368 64 13 the the DET cana-4368 64 14 above above ADJ cana-4368 64 15 algorithm algorithm NOUN cana-4368 64 16 3.3.3 3.3.3 NUM cana-4368 64 17 . . PUNCT cana-4368 65 1 here here ADV cana-4368 65 2 all all DET cana-4368 65 3 the the DET cana-4368 65 4 adjacent adjacent ADJ cana-4368 65 5 vertices vertex NOUN cana-4368 65 6 receive receive VERB cana-4368 65 7 distinct distinct ADJ cana-4368 65 8 labels.the labels.the DET cana-4368 65 9 edge edge NOUN cana-4368 65 10 labels label NOUN cana-4368 65 11 are be AUX cana-4368 65 12 obtained obtain VERB cana-4368 65 13 as as SCONJ cana-4368 65 14 follows follow VERB cana-4368 65 15 : : PUNCT cana-4368 65 16 for for ADP cana-4368 65 17 1 1 NUM cana-4368 65 18 ≀ ≀ NUM cana-4368 65 19 𝑖 𝑖 SYM cana-4368 65 20 ≀ ≀ NOUN cana-4368 65 21 𝑛 𝑛 NOUN cana-4368 65 22 ; ; PUNCT cana-4368 65 23 π‘“βˆ—(𝑣0𝑣𝑖 π‘“βˆ—(𝑣0𝑣𝑖 ADJ cana-4368 65 24 ) ) PUNCT cana-4368 65 25 = = SYM cana-4368 65 26 β„Žπ‘‘([𝑓(𝑣0)]2 β„Žπ‘‘([𝑓(𝑣0)]2 ADV cana-4368 65 27 , , PUNCT cana-4368 65 28 [ [ X cana-4368 65 29 𝑓(𝑣𝑖)]2 𝑓(𝑣𝑖)]2 PROPN cana-4368 65 30 ) ) PUNCT cana-4368 65 31 = = PUNCT cana-4368 66 1 β„Žπ‘‘([0]2 β„Žπ‘‘([0]2 PROPN cana-4368 66 2 , , PUNCT cana-4368 66 3 [ [ X cana-4368 66 4 2 2 NUM cana-4368 66 5 2π‘–βˆ’1 2π‘–βˆ’1 NUM cana-4368 66 6 βˆ’ βˆ’ NOUN cana-4368 66 7 1 1 NUM cana-4368 66 8 ] ] SYM cana-4368 66 9 2 2 NUM cana-4368 66 10 ) ) PUNCT cana-4368 66 11 = = NOUN cana-4368 66 12 2𝑖 2𝑖 NOUN cana-4368 66 13 βˆ’ βˆ’ NOUN cana-4368 66 14 1 1 X cana-4368 66 15 . . PUNCT cana-4368 66 16 for for ADP cana-4368 66 17 each each DET cana-4368 66 18 𝑖 𝑖 NOUN cana-4368 66 19 , , PUNCT cana-4368 66 20 the the DET cana-4368 66 21 corresponding corresponding ADJ cana-4368 66 22 vertex vertex NOUN cana-4368 66 23 label label NOUN cana-4368 66 24 and and CCONJ cana-4368 66 25 edge edge NOUN cana-4368 66 26 label label NOUN cana-4368 66 27 are be AUX cana-4368 66 28 given give VERB cana-4368 66 29 in in ADP cana-4368 66 30 the the DET cana-4368 66 31 following follow VERB cana-4368 66 32 table table NOUN cana-4368 66 33 : : PUNCT cana-4368 66 34 𝑖 𝑖 SYM cana-4368 66 35 1 1 NUM cana-4368 66 36 2 2 NUM cana-4368 66 37 3 3 NUM cana-4368 66 38 4 4 NUM cana-4368 66 39 5 5 NUM cana-4368 66 40 6 6 NUM cana-4368 66 41 7 7 NUM cana-4368 66 42 … … PUNCT cana-4368 66 43 … … PUNCT cana-4368 66 44 … … PUNCT cana-4368 66 45 n n PRON cana-4368 66 46 communications communication NOUN cana-4368 66 47 on on ADP cana-4368 66 48 applied apply VERB cana-4368 66 49 nonlinear nonlinear ADJ cana-4368 66 50 analysis analysis NOUN cana-4368 66 51 issn issn NOUN cana-4368 66 52 : : PUNCT cana-4368 66 53 1074 1074 NUM cana-4368 66 54 - - PUNCT cana-4368 66 55 133x 133x NUM cana-4368 66 56 vol vol NOUN cana-4368 66 57 32 32 NUM cana-4368 66 58 no no NOUN cana-4368 66 59 . . PUNCT cana-4368 67 1 9s 9s NUM cana-4368 67 2 ( ( PUNCT cana-4368 67 3 2025 2025 NUM cana-4368 67 4 ) ) PUNCT cana-4368 67 5 1931 1931 NUM cana-4368 67 6 https://internationalpubls.com https://internationalpubls.com X cana-4368 67 7 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-4368 67 8 ) ) PUNCT cana-4368 67 9 1 1 NUM cana-4368 67 10 7 7 NUM cana-4368 67 11 31 31 NUM cana-4368 67 12 127 127 NUM cana-4368 67 13 511 511 NUM cana-4368 67 14 2047 2047 NUM cana-4368 67 15 8191 8191 NUM cana-4368 67 16 … … PUNCT cana-4368 67 17 … … PUNCT cana-4368 67 18 … … PUNCT cana-4368 68 1 22π‘›βˆ’1 22π‘›βˆ’1 NUM cana-4368 68 2 βˆ’ βˆ’ NOUN cana-4368 68 3 1 1 NUM cana-4368 68 4 e e NOUN cana-4368 68 5 = = PUNCT cana-4368 68 6 β„Žπ‘‘([𝑓(𝑣0)]2 β„Žπ‘‘([𝑓(𝑣0)]2 PRON cana-4368 68 7 , , PUNCT cana-4368 68 8 [ [ X cana-4368 68 9 𝑓(𝑣𝑖)]2 𝑓(𝑣𝑖)]2 PROPN cana-4368 68 10 ) ) PUNCT cana-4368 68 11 1 1 NUM cana-4368 68 12 3 3 NUM cana-4368 68 13 5 5 NUM cana-4368 68 14 7 7 NUM cana-4368 68 15 9 9 NUM cana-4368 68 16 11 11 NUM cana-4368 68 17 13 13 NUM cana-4368 68 18 .. .. PUNCT cana-4368 68 19 … … PUNCT cana-4368 68 20 … … PUNCT cana-4368 68 21 . . PUNCT cana-4368 69 1 2n 2n NUM cana-4368 70 1 βˆ’ βˆ’ NOUN cana-4368 70 2 1 1 NUM cana-4368 70 3 where where SCONJ cana-4368 70 4 𝑓(𝑣0 𝑓(𝑣0 ADV cana-4368 70 5 ) ) PUNCT cana-4368 71 1 = = VERB cana-4368 71 2 0.from 0.from NUM cana-4368 71 3 the the DET cana-4368 71 4 above above ADJ cana-4368 71 5 table table NOUN cana-4368 71 6 it it PRON cana-4368 71 7 is be AUX cana-4368 71 8 clear clear ADJ cana-4368 71 9 that that SCONJ cana-4368 71 10 all all DET cana-4368 71 11 the the DET cana-4368 71 12 adjacent adjacent ADJ cana-4368 71 13 edges edge NOUN cana-4368 71 14 receive receive VERB cana-4368 71 15 distinct distinct ADJ cana-4368 71 16 odd odd ADJ cana-4368 71 17 labels label NOUN cana-4368 71 18 . . PUNCT cana-4368 72 1 hence hence ADV cana-4368 72 2 the the DET cana-4368 72 3 star star NOUN cana-4368 72 4 graph graph NOUN cana-4368 72 5 𝑆𝑛 𝑆𝑛 PROPN cana-4368 72 6 admits admit VERB cana-4368 72 7 odd odd ADJ cana-4368 72 8 hamming hamming NOUN cana-4368 72 9 distance distance NOUN cana-4368 72 10 labeling labeling NOUN cana-4368 72 11 and and CCONJ cana-4368 72 12 the the DET cana-4368 72 13 odd odd ADJ cana-4368 72 14 hamming hamming NOUN cana-4368 72 15 distance distance NOUN cana-4368 72 16 number number NOUN cana-4368 72 17 is be AUX cana-4368 72 18 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 72 19 β€² β€² NUM cana-4368 72 20 ( ( PUNCT cana-4368 72 21 𝑆𝑛 𝑆𝑛 PROPN cana-4368 72 22 ) ) PUNCT cana-4368 72 23 = = SYM cana-4368 72 24 2𝑛 2𝑛 PROPN cana-4368 73 1 βˆ’ βˆ’ NOUN cana-4368 73 2 1 1 NUM cana-4368 73 3 for for ADP cana-4368 73 4 any any DET cana-4368 73 5 𝑛 𝑛 PRON cana-4368 73 6 β‰₯ β‰₯ NOUN cana-4368 73 7 1 1 NUM cana-4368 73 8 . . X cana-4368 73 9 3.3.5 3.3.5 X cana-4368 73 10 . . PUNCT cana-4368 74 1 algorithm algorithm NOUN cana-4368 74 2 : : PUNCT cana-4368 74 3 odd odd ADJ cana-4368 74 4 hamming hamming NOUN cana-4368 74 5 distance distance NOUN cana-4368 74 6 labeling labeling NOUN cana-4368 74 7 of of ADP cana-4368 74 8 π‘·π’Ž π‘·π’Ž PROPN cana-4368 74 9 𝒏 𝒏 PROPN cana-4368 74 10 graph graph NOUN cana-4368 74 11 procedure procedure NOUN cana-4368 74 12 : : PUNCT cana-4368 74 13 vertex vertex NOUN cana-4368 74 14 labeling labeling NOUN cana-4368 74 15 of of ADP cana-4368 74 16 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 74 17 𝑛 𝑛 PROPN cana-4368 74 18 , , PUNCT cana-4368 74 19 graph graph NOUN cana-4368 74 20 π‘š π‘š PROPN cana-4368 74 21 , , PUNCT cana-4368 74 22 𝑛 𝑛 DET cana-4368 74 23 β‰₯ β‰₯ NOUN cana-4368 74 24 2 2 NUM cana-4368 74 25 . . PUNCT cana-4368 75 1 input input NOUN cana-4368 75 2 : : PUNCT cana-4368 75 3 one one NUM cana-4368 75 4 point point NOUN cana-4368 75 5 union union NOUN cana-4368 75 6 of of ADP cana-4368 75 7 path path NOUN cana-4368 75 8 graph graph NOUN cana-4368 75 9 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 75 10 𝑛. 𝑛. NOUN cana-4368 75 11 𝑉 𝑉 X cana-4368 75 12 ← ← PROPN cana-4368 75 13 { { PUNCT cana-4368 75 14 𝑣𝑖𝑗 𝑣𝑖𝑗 NUM cana-4368 75 15 /1 /1 NOUN cana-4368 75 16 ≀ ≀ NUM cana-4368 75 17 𝑖 𝑖 SYM cana-4368 75 18 ≀ ≀ NUM cana-4368 75 19 𝑛 𝑛 NOUN cana-4368 75 20 , , PUNCT cana-4368 75 21 0 0 NUM cana-4368 75 22 ≀ ≀ NUM cana-4368 75 23 𝑗 𝑗 PRON cana-4368 75 24 ≀ ≀ NUM cana-4368 75 25 π‘š π‘š NOUN cana-4368 75 26 ; ; PUNCT cana-4368 75 27 𝑣10 𝑣10 NOUN cana-4368 75 28 = = SYM cana-4368 75 29 𝑣20 𝑣20 NOUN cana-4368 75 30 = = NOUN cana-4368 75 31 𝑣30 𝑣30 NOUN cana-4368 75 32 = = SYM cana-4368 75 33 β‹― β‹― NOUN cana-4368 75 34 = = SYM cana-4368 75 35 𝑣𝑛0 𝑣𝑛0 PROPN cana-4368 75 36 } } PUNCT cana-4368 75 37 𝑣0 𝑣0 PROPN cana-4368 75 38 ← ← PROPN cana-4368 75 39 0 0 NUM cana-4368 75 40 ; ; PUNCT cana-4368 75 41 𝑣0 𝑣0 PROPN cana-4368 75 42 = = SYM cana-4368 75 43 𝑣10 𝑣10 NOUN cana-4368 75 44 = = SYM cana-4368 75 45 𝑣20 𝑣20 NOUN cana-4368 75 46 = = NOUN cana-4368 75 47 𝑣30 𝑣30 NOUN cana-4368 75 48 = = SYM cana-4368 75 49 β‹― β‹― NOUN cana-4368 75 50 = = SYM cana-4368 75 51 𝑣𝑛0 𝑣𝑛0 PROPN cana-4368 75 52 for for ADP cana-4368 75 53 𝑗 𝑗 NOUN cana-4368 75 54 = = SYM cana-4368 75 55 1 1 NUM cana-4368 75 56 π‘‘π‘œ π‘‘π‘œ NOUN cana-4368 75 57 π‘š π‘š NOUN cana-4368 75 58 do do VERB cana-4368 75 59 𝑣1𝑗 𝑣1𝑗 PROPN cana-4368 75 60 ← ← PROPN cana-4368 75 61 { { PUNCT cana-4368 75 62 1 1 NUM cana-4368 75 63 if if SCONJ cana-4368 75 64 𝑗 𝑗 PRON cana-4368 75 65 ≑ ≑ PROPN cana-4368 75 66 1(π‘šπ‘œπ‘‘4 1(π‘šπ‘œπ‘‘4 NUM cana-4368 75 67 ) ) PUNCT cana-4368 75 68 6 6 NUM cana-4368 75 69 if if SCONJ cana-4368 75 70 𝑗 𝑗 PRON cana-4368 75 71 ≑ ≑ PROPN cana-4368 75 72 2(π‘šπ‘œπ‘‘4 2(π‘šπ‘œπ‘‘4 NUM cana-4368 75 73 ) ) PUNCT cana-4368 75 74 2 2 NUM cana-4368 75 75 if if SCONJ cana-4368 75 76 𝑗 𝑗 PRON cana-4368 75 77 ≑ ≑ PROPN cana-4368 75 78 3(π‘šπ‘œπ‘‘4 3(π‘šπ‘œπ‘‘4 NUM cana-4368 75 79 ) ) PUNCT cana-4368 75 80 5 5 NUM cana-4368 75 81 if if SCONJ cana-4368 75 82 𝑗 𝑗 PRON cana-4368 75 83 ≑ ≑ PROPN cana-4368 75 84 0(π‘šπ‘œπ‘‘4 0(π‘šπ‘œπ‘‘4 PROPN cana-4368 75 85 ) ) PUNCT cana-4368 75 86 ; ; PUNCT cana-4368 75 87 𝑣2𝑗 𝑣2𝑗 NUM cana-4368 75 88 ← ← PROPN cana-4368 75 89 { { PUNCT cana-4368 75 90 7 7 NUM cana-4368 75 91 if if SCONJ cana-4368 75 92 𝑗 𝑗 PRON cana-4368 75 93 ≑ ≑ PROPN cana-4368 75 94 1(π‘šπ‘œπ‘‘4 1(π‘šπ‘œπ‘‘4 NUM cana-4368 75 95 ) ) PUNCT cana-4368 75 96 3 3 NUM cana-4368 75 97 if if SCONJ cana-4368 75 98 𝑗 𝑗 PRON cana-4368 75 99 ≑ ≑ PROPN cana-4368 75 100 2(π‘šπ‘œπ‘‘4 2(π‘šπ‘œπ‘‘4 NUM cana-4368 75 101 ) ) PUNCT cana-4368 75 102 4 4 NUM cana-4368 75 103 if if SCONJ cana-4368 75 104 𝑗 𝑗 PRON cana-4368 75 105 ≑ ≑ PROPN cana-4368 75 106 3(π‘šπ‘œπ‘‘4 3(π‘šπ‘œπ‘‘4 NUM cana-4368 75 107 ) ) PUNCT cana-4368 75 108 0 0 PUNCT cana-4368 76 1 if if SCONJ cana-4368 76 2 𝑗 𝑗 PROPN cana-4368 76 3 ≑ ≑ PROPN cana-4368 76 4 0(π‘šπ‘œπ‘‘4 0(π‘šπ‘œπ‘‘4 PROPN cana-4368 76 5 ) ) PUNCT cana-4368 76 6 ; ; PUNCT cana-4368 76 7 end end VERB cana-4368 76 8 for for ADP cana-4368 76 9 for for ADP cana-4368 76 10 𝑖 𝑖 DET cana-4368 76 11 = = SYM cana-4368 76 12 3 3 NUM cana-4368 76 13 π‘‘π‘œ π‘‘π‘œ NOUN cana-4368 76 14 𝑛 𝑛 PROPN cana-4368 76 15 do do AUX cana-4368 76 16 𝑣𝑖1 𝑣𝑖1 VERB cana-4368 76 17 ← ← PROPN cana-4368 76 18 22π‘–βˆ’1 22π‘–βˆ’1 PROPN cana-4368 76 19 βˆ’ βˆ’ PROPN cana-4368 76 20 1 1 NUM cana-4368 76 21 ; ; PUNCT cana-4368 77 1 𝑣𝑖2 𝑣𝑖2 ADP cana-4368 77 2 ← ← PROPN cana-4368 77 3 22π‘–βˆ’2 22π‘–βˆ’2 NUM cana-4368 77 4 βˆ’ βˆ’ PROPN cana-4368 77 5 1 1 NUM cana-4368 77 6 ; ; PUNCT cana-4368 77 7 𝑣𝑖3 𝑣𝑖3 NOUN cana-4368 77 8 ← ← PROPN cana-4368 77 9 22π‘–βˆ’5 22π‘–βˆ’5 PROPN cana-4368 77 10 βˆ’ βˆ’ PROPN cana-4368 77 11 1 1 NUM cana-4368 77 12 ; ; PUNCT cana-4368 77 13 𝑣𝑖4 𝑣𝑖4 NOUN cana-4368 77 14 ← ← PROPN cana-4368 77 15 22π‘–βˆ’6 22π‘–βˆ’6 PROPN cana-4368 77 16 βˆ’ βˆ’ PROPN cana-4368 77 17 1 1 NUM cana-4368 77 18 ; ; PUNCT cana-4368 77 19 for for ADP cana-4368 77 20 𝑗 𝑗 NOUN cana-4368 77 21 = = SYM cana-4368 77 22 5 5 NUM cana-4368 77 23 π‘‘π‘œ π‘‘π‘œ NOUN cana-4368 77 24 π‘š π‘š AUX cana-4368 77 25 do do VERB cana-4368 77 26 if if SCONJ cana-4368 77 27 𝑖 𝑖 NOUN cana-4368 77 28 = = SYM cana-4368 77 29 3 3 NUM cana-4368 77 30 𝑣𝑖𝑗 𝑣𝑖𝑗 PROPN cana-4368 77 31 ← ← PROPN cana-4368 77 32 ( ( PUNCT cana-4368 77 33 𝑣(π‘–βˆ’1)(π‘—βˆ’4 𝑣(π‘–βˆ’1)(π‘—βˆ’4 NOUN cana-4368 77 34 ) ) PUNCT cana-4368 77 35 ) ) PUNCT cana-4368 77 36 ; ; PUNCT cana-4368 77 37 else else ADV cana-4368 77 38 𝑣𝑖𝑗 𝑣𝑖𝑗 PROPN cana-4368 77 39 ← ← PROPN cana-4368 77 40 ( ( PUNCT cana-4368 77 41 𝑣(π‘–βˆ’2)(π‘—βˆ’2 𝑣(π‘–βˆ’2)(π‘—βˆ’2 PROPN cana-4368 77 42 ) ) PUNCT cana-4368 77 43 ) ) PUNCT cana-4368 77 44 ; ; PUNCT cana-4368 77 45 end end VERB cana-4368 77 46 if if SCONJ cana-4368 77 47 end end NOUN cana-4368 77 48 for for ADP cana-4368 77 49 end end NOUN cana-4368 77 50 for for ADP cana-4368 77 51 end end NOUN cana-4368 77 52 procedure procedure NOUN cana-4368 77 53 output output NOUN cana-4368 77 54 : : PUNCT cana-4368 77 55 the the DET cana-4368 77 56 labeled label VERB cana-4368 77 57 vertices vertex NOUN cana-4368 77 58 of of ADP cana-4368 77 59 one one NUM cana-4368 77 60 point point NOUN cana-4368 77 61 union union NOUN cana-4368 77 62 of of ADP cana-4368 77 63 path path NOUN cana-4368 77 64 graph graph NOUN cana-4368 77 65 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 77 66 𝑛. 𝑛. NOUN cana-4368 77 67 communications communication NOUN cana-4368 77 68 on on ADP cana-4368 77 69 applied apply VERB cana-4368 77 70 nonlinear nonlinear ADJ cana-4368 77 71 analysis analysis NOUN cana-4368 77 72 issn issn NOUN cana-4368 77 73 : : PUNCT cana-4368 77 74 1074 1074 NUM cana-4368 77 75 - - PUNCT cana-4368 77 76 133x 133x NUM cana-4368 77 77 vol vol NOUN cana-4368 77 78 32 32 NUM cana-4368 78 1 no no NOUN cana-4368 78 2 . . PUNCT cana-4368 79 1 9s 9s NUM cana-4368 79 2 ( ( PUNCT cana-4368 79 3 2025 2025 NUM cana-4368 79 4 ) ) PUNCT cana-4368 79 5 1932 1932 NUM cana-4368 80 1 https://internationalpubls.com https://internationalpubls.com X cana-4368 80 2 3.3.6.theorem 3.3.6.theorem NUM cana-4368 80 3 the the DET cana-4368 80 4 one one NUM cana-4368 80 5 point point NOUN cana-4368 80 6 union union NOUN cana-4368 80 7 of of ADP cana-4368 80 8 path path NOUN cana-4368 80 9 graphs graph NOUN cana-4368 80 10 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 80 11 𝑛 𝑛 PROPN cana-4368 80 12 , , PUNCT cana-4368 80 13 π‘š π‘š PROPN cana-4368 80 14 , , PUNCT cana-4368 80 15 𝑛 𝑛 DET cana-4368 80 16 β‰₯ β‰₯ NUM cana-4368 80 17 2 2 NUM cana-4368 80 18 is be AUX cana-4368 80 19 an an DET cana-4368 80 20 odd odd ADJ cana-4368 80 21 hamming hamming NOUN cana-4368 80 22 distance distance NOUN cana-4368 80 23 graph graph NOUN cana-4368 80 24 and and CCONJ cana-4368 80 25 the the DET cana-4368 80 26 odd odd ADJ cana-4368 80 27 hamming hamming NOUN cana-4368 80 28 distance distance NOUN cana-4368 80 29 number number NOUN cana-4368 80 30 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 80 31 β€² β€² NUM cana-4368 81 1 ( ( PUNCT cana-4368 81 2 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 81 3 𝑛 𝑛 PROPN cana-4368 81 4 ) ) PUNCT cana-4368 82 1 = = SYM cana-4368 82 2 2𝑛 2𝑛 PROPN cana-4368 82 3 βˆ’ βˆ’ NOUN cana-4368 82 4 1 1 X cana-4368 82 5 . . PUNCT cana-4368 83 1 proof proof NOUN cana-4368 83 2 : : PUNCT cana-4368 83 3 let let VERB cana-4368 83 4 us we PRON cana-4368 83 5 consider consider VERB cana-4368 83 6 the the DET cana-4368 83 7 one one NUM cana-4368 83 8 point point NOUN cana-4368 83 9 union union NOUN cana-4368 83 10 of of ADP cana-4368 83 11 path path NOUN cana-4368 83 12 graphs graph NOUN cana-4368 83 13 π‘ƒπ‘š π‘ƒπ‘š PRON cana-4368 83 14 𝑛 𝑛 NOUN cana-4368 83 15 with with ADP cana-4368 83 16 vertex vertex NOUN cana-4368 83 17 set set VERB cana-4368 83 18 𝑉 𝑉 PROPN cana-4368 83 19 = = PROPN cana-4368 83 20 { { PUNCT cana-4368 83 21 𝑣𝑖𝑗 𝑣𝑖𝑗 NUM cana-4368 83 22 /1 /1 NOUN cana-4368 83 23 ≀ ≀ NUM cana-4368 83 24 𝑖 𝑖 SYM cana-4368 83 25 ≀ ≀ NUM cana-4368 83 26 𝑛 𝑛 NOUN cana-4368 83 27 , , PUNCT cana-4368 83 28 0 0 NUM cana-4368 83 29 ≀ ≀ NUM cana-4368 83 30 𝑗 𝑗 PRON cana-4368 83 31 ≀ ≀ NUM cana-4368 83 32 π‘š π‘š NOUN cana-4368 83 33 ; ; PUNCT cana-4368 83 34 𝑣10 𝑣10 NOUN cana-4368 83 35 = = SYM cana-4368 83 36 𝑣20 𝑣20 NOUN cana-4368 83 37 = = NOUN cana-4368 83 38 𝑣30 𝑣30 NOUN cana-4368 83 39 = = SYM cana-4368 83 40 β‹― β‹― NOUN cana-4368 83 41 = = SYM cana-4368 83 42 𝑣𝑛0 𝑣𝑛0 PROPN cana-4368 83 43 } } PUNCT cana-4368 83 44 , , PUNCT cana-4368 83 45 let let VERB cana-4368 83 46 𝑣0 𝑣0 PROPN cana-4368 83 47 = = SYM cana-4368 83 48 𝑣𝑖0 𝑣𝑖0 NOUN cana-4368 83 49 , , PUNCT cana-4368 83 50 1 1 NUM cana-4368 83 51 ≀ ≀ NUM cana-4368 83 52 𝑖 𝑖 SYM cana-4368 83 53 ≀ ≀ NOUN cana-4368 83 54 𝑛 𝑛 PRON cana-4368 83 55 and and CCONJ cana-4368 83 56 edge edge VERB cana-4368 83 57 set set VERB cana-4368 83 58 𝐸 𝐸 NOUN cana-4368 83 59 = = PUNCT cana-4368 83 60 { { PUNCT cana-4368 83 61 𝑣𝑖𝑗𝑣𝑖(𝑗+1 𝑣𝑖𝑗𝑣𝑖(𝑗+1 NOUN cana-4368 83 62 ) ) PUNCT cana-4368 83 63 / / SYM cana-4368 83 64 1 1 NUM cana-4368 83 65 ≀ ≀ NUM cana-4368 83 66 𝑖 𝑖 SYM cana-4368 83 67 ≀ ≀ NUM cana-4368 83 68 𝑛 𝑛 NOUN cana-4368 83 69 , , PUNCT cana-4368 83 70 0 0 NUM cana-4368 83 71 ≀ ≀ NUM cana-4368 83 72 𝑗 𝑗 PRON cana-4368 83 73 ≀ ≀ NOUN cana-4368 83 74 π‘š π‘š ADP cana-4368 83 75 βˆ’ βˆ’ PROPN cana-4368 83 76 1 1 NUM cana-4368 83 77 } } PUNCT cana-4368 83 78 . . PUNCT cana-4368 84 1 this this DET cana-4368 84 2 graph graph NOUN cana-4368 84 3 has have VERB cana-4368 84 4 mn+1 mn+1 NOUN cana-4368 84 5 vertices vertex NOUN cana-4368 84 6 and and CCONJ cana-4368 84 7 mn mn PROPN cana-4368 84 8 edges edge NOUN cana-4368 84 9 . . PUNCT cana-4368 85 1 define define VERB cana-4368 85 2 a a DET cana-4368 85 3 function function NOUN cana-4368 85 4 𝑓:𝑉 𝑓:𝑉 PROPN cana-4368 86 1 β†’ β†’ PUNCT cana-4368 86 2 𝑁 𝑁 PROPN cana-4368 86 3 βˆͺ βˆͺ ADJ cana-4368 86 4 { { PUNCT cana-4368 86 5 0 0 NUM cana-4368 86 6 } } PUNCT cana-4368 86 7 such such ADJ cana-4368 86 8 that that SCONJ cana-4368 86 9 𝑓(𝑣𝑖𝑗 𝑓(𝑣𝑖𝑗 NOUN cana-4368 86 10 ) ) PUNCT cana-4368 86 11 β‰  β‰  PROPN cana-4368 86 12 𝑓(𝑣𝑖(𝑗+1 𝑓(𝑣𝑖(𝑗+1 PROPN cana-4368 86 13 ) ) PUNCT cana-4368 86 14 ) ) PUNCT cana-4368 86 15 , , PUNCT cana-4368 86 16 1 1 NUM cana-4368 86 17 ≀ ≀ NUM cana-4368 86 18 𝑖 𝑖 SYM cana-4368 86 19 ≀ ≀ NUM cana-4368 86 20 𝑛 𝑛 PRON cana-4368 86 21 , , PUNCT cana-4368 86 22 0 0 NUM cana-4368 86 23 ≀ ≀ NUM cana-4368 86 24 𝑗 𝑗 PRON cana-4368 86 25 ≀ ≀ NOUN cana-4368 86 26 π‘š π‘š ADP cana-4368 86 27 βˆ’ βˆ’ PROPN cana-4368 86 28 1 1 NUM cana-4368 86 29 as as SCONJ cana-4368 86 30 given give VERB cana-4368 86 31 in in ADP cana-4368 86 32 the the DET cana-4368 86 33 above above ADJ cana-4368 86 34 algorithm algorithm NOUN cana-4368 86 35 3.3.6 3.3.6 NUM cana-4368 86 36 . . PUNCT cana-4368 87 1 hence hence ADV cana-4368 87 2 the the DET cana-4368 87 3 adjacent adjacent ADJ cana-4368 87 4 vertices vertex NOUN cana-4368 87 5 receive receive VERB cana-4368 87 6 distinct distinct ADJ cana-4368 87 7 labels.the labels.the DET cana-4368 87 8 edge edge NOUN cana-4368 87 9 labels label NOUN cana-4368 87 10 are be AUX cana-4368 87 11 obtained obtain VERB cana-4368 87 12 as as SCONJ cana-4368 87 13 follows follow VERB cana-4368 87 14 : : PUNCT cana-4368 87 15 for for ADP cana-4368 87 16 1 1 NUM cana-4368 87 17 ≀ ≀ NUM cana-4368 87 18 𝑖 𝑖 SYM cana-4368 87 19 ≀ ≀ NUM cana-4368 87 20 𝑛 𝑛 NOUN cana-4368 87 21 , , PUNCT cana-4368 87 22 β„Žπ‘‘([𝑓(𝑣0)]2 β„Žπ‘‘([𝑓(𝑣0)]2 PRON cana-4368 87 23 , , PUNCT cana-4368 87 24 [ [ X cana-4368 87 25 𝑓(𝑣𝑖1)]2 𝑓(𝑣𝑖1)]2 NOUN cana-4368 87 26 ) ) PUNCT cana-4368 87 27 = = SYM cana-4368 87 28 2𝑖 2𝑖 NOUN cana-4368 87 29 βˆ’ βˆ’ NOUN cana-4368 88 1 1 1 X cana-4368 88 2 . . NOUN cana-4368 88 3 which which PRON cana-4368 88 4 is be AUX cana-4368 88 5 given give VERB cana-4368 88 6 in in ADP cana-4368 88 7 the the DET cana-4368 88 8 following follow VERB cana-4368 88 9 table table NOUN cana-4368 88 10 , , PUNCT cana-4368 88 11 here here ADV cana-4368 88 12 𝑓(𝑣0 𝑓(𝑣0 ADV cana-4368 88 13 ) ) PUNCT cana-4368 89 1 = = SYM cana-4368 89 2 0 0 X cana-4368 89 3 . . PUNCT cana-4368 90 1 𝑖 𝑖 SYM cana-4368 90 2 1 1 NUM cana-4368 90 3 2 2 NUM cana-4368 90 4 3 3 NUM cana-4368 90 5 4 4 NUM cana-4368 90 6 5 5 NUM cana-4368 90 7 6 6 NUM cana-4368 90 8 7 7 NUM cana-4368 90 9 … … PUNCT cana-4368 90 10 … … PUNCT cana-4368 90 11 … … PUNCT cana-4368 90 12 n n CCONJ cana-4368 90 13 𝑓(𝑣𝑖1 𝑓(𝑣𝑖1 NOUN cana-4368 90 14 ) ) PUNCT cana-4368 90 15 1 1 NUM cana-4368 90 16 7 7 NUM cana-4368 90 17 31 31 NUM cana-4368 90 18 127 127 NUM cana-4368 90 19 511 511 NUM cana-4368 90 20 2047 2047 NUM cana-4368 90 21 8191 8191 NUM cana-4368 90 22 … … PUNCT cana-4368 90 23 … … PUNCT cana-4368 90 24 … … PUNCT cana-4368 91 1 22π‘›βˆ’1 22π‘›βˆ’1 NUM cana-4368 91 2 βˆ’ βˆ’ NOUN cana-4368 91 3 1 1 NUM cana-4368 91 4 e e NOUN cana-4368 91 5 = = PUNCT cana-4368 91 6 β„Žπ‘‘([𝑓(𝑣0)]2 β„Žπ‘‘([𝑓(𝑣0)]2 PRON cana-4368 91 7 , , PUNCT cana-4368 91 8 [ [ X cana-4368 91 9 𝑓(𝑣𝑖1)]2 𝑓(𝑣𝑖1)]2 NOUN cana-4368 91 10 ) ) PUNCT cana-4368 91 11 1 1 NUM cana-4368 91 12 3 3 NUM cana-4368 91 13 5 5 NUM cana-4368 91 14 7 7 NUM cana-4368 91 15 9 9 NUM cana-4368 91 16 11 11 NUM cana-4368 91 17 13 13 NUM cana-4368 91 18 … … PUNCT cana-4368 91 19 … … PUNCT cana-4368 91 20 … … PUNCT cana-4368 91 21 2𝑛 2𝑛 NOUN cana-4368 92 1 βˆ’ βˆ’ NOUN cana-4368 92 2 1 1 NUM cana-4368 92 3 for for ADP cana-4368 92 4 1 1 NUM cana-4368 92 5 ≀ ≀ NUM cana-4368 92 6 𝑖 𝑖 SYM cana-4368 92 7 ≀ ≀ NUM cana-4368 92 8 𝑛 𝑛 DET cana-4368 92 9 case case NOUN cana-4368 92 10 ( ( PUNCT cana-4368 92 11 i i NOUN cana-4368 92 12 ): ): PUNCT cana-4368 92 13 when when SCONJ cana-4368 92 14 𝑗 𝑗 PRON cana-4368 92 15 ≑ ≑ PROPN cana-4368 92 16 1(π‘šπ‘œπ‘‘4 1(π‘šπ‘œπ‘‘4 NUM cana-4368 92 17 ) ) PUNCT cana-4368 92 18 π‘“βˆ—(𝑣11𝑣12 π‘“βˆ—(𝑣11𝑣12 NOUN cana-4368 92 19 ) ) PUNCT cana-4368 92 20 = = SYM cana-4368 92 21 β„Žπ‘‘([𝑓(𝑣11)]2 β„Žπ‘‘([𝑓(𝑣11)]2 NOUN cana-4368 92 22 , , PUNCT cana-4368 92 23 [ [ X cana-4368 92 24 𝑓(𝑣12))]2 𝑓(𝑣12))]2 ADJ cana-4368 92 25 ) ) PUNCT cana-4368 92 26 = = SYM cana-4368 93 1 hd([1]2 hd([1]2 PROPN cana-4368 93 2 , , PUNCT cana-4368 93 3 [ [ X cana-4368 93 4 6]2 6]2 NOUN cana-4368 93 5 ) ) PUNCT cana-4368 93 6 = = SYM cana-4368 94 1 3 3 X cana-4368 94 2 . . X cana-4368 94 3 π‘“βˆ—(𝑣21𝑣22 π‘“βˆ—(𝑣21𝑣22 NOUN cana-4368 94 4 ) ) PUNCT cana-4368 95 1 = = SYM cana-4368 95 2 β„Žπ‘‘([𝑓(𝑣21)]2 β„Žπ‘‘([𝑓(𝑣21)]2 PROPN cana-4368 95 3 , , PUNCT cana-4368 95 4 [ [ X cana-4368 95 5 𝑓(𝑣22))]2 𝑓(𝑣22))]2 NOUN cana-4368 95 6 ) ) PUNCT cana-4368 95 7 = = SYM cana-4368 96 1 hd([7]2 hd([7]2 PROPN cana-4368 96 2 , , PUNCT cana-4368 96 3 [ [ X cana-4368 96 4 3]2 3]2 NUM cana-4368 96 5 ) ) PUNCT cana-4368 96 6 = = SYM cana-4368 96 7 1 1 NUM cana-4368 96 8 𝑖 𝑖 SYM cana-4368 96 9 3 3 NUM cana-4368 96 10 4 4 NUM cana-4368 96 11 5 5 NUM cana-4368 96 12 6 6 NUM cana-4368 96 13 7 7 NUM cana-4368 96 14 … … SYM cana-4368 96 15 …… …… NOUN cana-4368 96 16 n n X cana-4368 96 17 𝑓(𝑣i1 𝑓(𝑣i1 ADJ cana-4368 96 18 ) ) PUNCT cana-4368 96 19 31 31 NUM cana-4368 96 20 127 127 NUM cana-4368 96 21 511 511 NUM cana-4368 96 22 2047 2047 NUM cana-4368 96 23 8191 8191 NUM cana-4368 96 24 … … PUNCT cana-4368 96 25 … … PUNCT cana-4368 96 26 … … PUNCT cana-4368 96 27 . . PUNCT cana-4368 97 1 22π‘›βˆ’1 22π‘›βˆ’1 NUM cana-4368 97 2 βˆ’ βˆ’ NOUN cana-4368 97 3 1 1 NUM cana-4368 97 4 𝑓(𝑣𝑖2 𝑓(𝑣𝑖2 NOUN cana-4368 97 5 ) ) PUNCT cana-4368 97 6 15 15 NUM cana-4368 97 7 63 63 NUM cana-4368 97 8 255 255 NUM cana-4368 97 9 1023 1023 NUM cana-4368 97 10 4095 4095 NUM cana-4368 97 11 … … PUNCT cana-4368 97 12 … … PUNCT cana-4368 97 13 … … PUNCT cana-4368 97 14 . . PUNCT cana-4368 98 1 22π‘›βˆ’2 22π‘›βˆ’2 NOUN cana-4368 98 2 βˆ’ βˆ’ NOUN cana-4368 98 3 1 1 NUM cana-4368 98 4 e e NOUN cana-4368 98 5 = = SYM cana-4368 98 6 β„Žπ‘‘([𝑓(𝑣i1)]2 β„Žπ‘‘([𝑓(𝑣i1)]2 PROPN cana-4368 98 7 , , PUNCT cana-4368 98 8 [ [ X cana-4368 98 9 𝑓(𝑣𝑖2)]2 𝑓(𝑣𝑖2)]2 NOUN cana-4368 98 10 ) ) PUNCT cana-4368 98 11 1 1 NUM cana-4368 98 12 1 1 NUM cana-4368 98 13 1 1 NUM cana-4368 98 14 1 1 NUM cana-4368 98 15 1 1 NUM cana-4368 98 16 1 1 NUM cana-4368 98 17 1 1 NUM cana-4368 98 18 for for ADP cana-4368 98 19 1 1 NUM cana-4368 98 20 ≀ ≀ NUM cana-4368 98 21 𝑖 𝑖 SYM cana-4368 98 22 ≀ ≀ NUM cana-4368 98 23 𝑛 𝑛 NOUN cana-4368 98 24 , , PUNCT cana-4368 98 25 5 5 NUM cana-4368 98 26 ≀ ≀ NUM cana-4368 98 27 𝑗 𝑗 PRON cana-4368 98 28 ≀ ≀ NOUN cana-4368 98 29 π‘š π‘š X cana-4368 98 30 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 PROPN cana-4368 98 31 ) ) PUNCT cana-4368 98 32 ) ) PUNCT cana-4368 99 1 = = SYM cana-4368 99 2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 PROPN cana-4368 99 3 , , PUNCT cana-4368 99 4 [ [ X cana-4368 99 5 𝑓(𝑣𝑖(𝑗+1)))]2 𝑓(𝑣𝑖(𝑗+1)))]2 NOUN cana-4368 99 6 ) ) PUNCT cana-4368 99 7 = = SYM cana-4368 100 1 1 1 X cana-4368 100 2 . . X cana-4368 100 3 communications communication NOUN cana-4368 100 4 on on ADP cana-4368 100 5 applied apply VERB cana-4368 100 6 nonlinear nonlinear ADJ cana-4368 100 7 analysis analysis NOUN cana-4368 100 8 issn issn NOUN cana-4368 100 9 : : PUNCT cana-4368 100 10 1074 1074 NUM cana-4368 100 11 - - PUNCT cana-4368 100 12 133x 133x NUM cana-4368 100 13 vol vol NOUN cana-4368 100 14 32 32 NUM cana-4368 100 15 no no NOUN cana-4368 100 16 . . PUNCT cana-4368 101 1 9s 9s NUM cana-4368 101 2 ( ( PUNCT cana-4368 101 3 2025 2025 NUM cana-4368 101 4 ) ) PUNCT cana-4368 101 5 1933 1933 NUM cana-4368 102 1 https://internationalpubls.com https://internationalpubls.com X cana-4368 102 2 𝑖 𝑖 SYM cana-4368 102 3 1 1 NUM cana-4368 102 4 2 2 NUM cana-4368 102 5 3 3 NUM cana-4368 102 6 4 4 NUM cana-4368 102 7 5 5 NUM cana-4368 102 8 … … PUNCT cana-4368 102 9 n n CCONJ cana-4368 102 10 𝑓(𝑣ij 𝑓(𝑣ij PROPN cana-4368 102 11 ) ) PUNCT cana-4368 102 12 1 1 NUM cana-4368 102 13 7 7 NUM cana-4368 102 14 7=𝑣(3βˆ’1)(jβˆ’4 7=𝑣(3βˆ’1)(jβˆ’4 NUM cana-4368 102 15 ) ) PUNCT cana-4368 102 16 4= 4= NUM cana-4368 102 17 𝑣(4βˆ’2)(jβˆ’2 𝑣(4βˆ’2)(jβˆ’2 PROPN cana-4368 102 18 ) ) PUNCT cana-4368 102 19 1=𝑣(5βˆ’2)(jβˆ’2 1=𝑣(5βˆ’2)(jβˆ’2 NUM cana-4368 102 20 ) ) PUNCT cana-4368 102 21 … … PUNCT cana-4368 102 22 𝑣(nβˆ’2)(jβˆ’2 𝑣(nβˆ’2)(jβˆ’2 X cana-4368 102 23 ) ) PUNCT cana-4368 102 24 𝑓(𝑣𝑖(𝑗+1 𝑓(𝑣𝑖(𝑗+1 PROPN cana-4368 102 25 ) ) PUNCT cana-4368 102 26 ) ) PUNCT cana-4368 102 27 6 6 NUM cana-4368 102 28 3 3 NUM cana-4368 102 29 3=𝑣(3βˆ’1)(jβˆ’3 3=𝑣(3βˆ’1)(jβˆ’3 NUM cana-4368 102 30 ) ) PUNCT cana-4368 102 31 0= 0= NUM cana-4368 103 1 𝑣(4βˆ’2)(jβˆ’1 𝑣(4βˆ’2)(jβˆ’1 NOUN cana-4368 103 2 ) ) PUNCT cana-4368 103 3 0=𝑣(5βˆ’2)(jβˆ’1 0=𝑣(5βˆ’2)(jβˆ’1 NUM cana-4368 103 4 ) ) PUNCT cana-4368 103 5 … … PUNCT cana-4368 103 6 𝑣(nβˆ’2)(jβˆ’1 𝑣(nβˆ’2)(jβˆ’1 PRON cana-4368 103 7 ) ) PUNCT cana-4368 103 8 e e X cana-4368 103 9 = = SYM cana-4368 103 10 β„Žπ‘‘([𝑓(𝑣ij)]2 β„Žπ‘‘([𝑓(𝑣ij)]2 PROPN cana-4368 103 11 , , PUNCT cana-4368 103 12 [ [ X cana-4368 103 13 𝑓(𝑣𝑖(𝑗+1))]2 𝑓(𝑣𝑖(𝑗+1))]2 NOUN cana-4368 103 14 ) ) PUNCT cana-4368 103 15 3 3 NUM cana-4368 103 16 1 1 NUM cana-4368 103 17 1 1 NUM cana-4368 103 18 1 1 NUM cana-4368 103 19 1 1 NUM cana-4368 103 20 1 1 NUM cana-4368 103 21 case case NOUN cana-4368 103 22 ( ( PUNCT cana-4368 103 23 ii ii NOUN cana-4368 103 24 ): ): PUNCT cana-4368 103 25 when when SCONJ cana-4368 103 26 𝑗 𝑗 PROPN cana-4368 103 27 ≑ ≑ PROPN cana-4368 103 28 2(π‘šπ‘œπ‘‘4 2(π‘šπ‘œπ‘‘4 NUM cana-4368 103 29 ) ) PUNCT cana-4368 103 30 π‘“βˆ—(𝑣12𝑣13 π‘“βˆ—(𝑣12𝑣13 PROPN cana-4368 103 31 ) ) PUNCT cana-4368 103 32 = = PUNCT cana-4368 104 1 β„Žπ‘‘([𝑓(𝑣12)]2 β„Žπ‘‘([𝑓(𝑣12)]2 PROPN cana-4368 104 2 , , PUNCT cana-4368 104 3 [ [ X cana-4368 104 4 𝑓(𝑣13))]2 𝑓(𝑣13))]2 NUM cana-4368 104 5 ) ) PUNCT cana-4368 104 6 = = SYM cana-4368 105 1 hd([6]2 hd([6]2 X cana-4368 105 2 , , PUNCT cana-4368 105 3 [ [ X cana-4368 105 4 2]2 2]2 NUM cana-4368 105 5 ) ) PUNCT cana-4368 105 6 = = SYM cana-4368 105 7 1 1 X cana-4368 105 8 . . X cana-4368 105 9 π‘“βˆ—(𝑣22𝑣23 π‘“βˆ—(𝑣22𝑣23 NUM cana-4368 105 10 ) ) PUNCT cana-4368 105 11 = = SYM cana-4368 106 1 β„Žπ‘‘([𝑓(𝑣22)]2 β„Žπ‘‘([𝑓(𝑣22)]2 NOUN cana-4368 106 2 , , PUNCT cana-4368 106 3 [ [ X cana-4368 106 4 𝑓(𝑣23))]2 𝑓(𝑣23))]2 NOUN cana-4368 106 5 ) ) PUNCT cana-4368 106 6 = = SYM cana-4368 107 1 hd([3]2 hd([3]2 X cana-4368 107 2 , , PUNCT cana-4368 107 3 [ [ X cana-4368 107 4 4]2 4]2 NOUN cana-4368 107 5 ) ) PUNCT cana-4368 107 6 = = SYM cana-4368 108 1 3 3 X cana-4368 108 2 . . X cana-4368 108 3 𝑖 𝑖 SYM cana-4368 108 4 3 3 NUM cana-4368 108 5 4 4 NUM cana-4368 108 6 5 5 NUM cana-4368 108 7 6 6 NUM cana-4368 108 8 7 7 NUM cana-4368 108 9 … … SYM cana-4368 108 10 …… …… X cana-4368 108 11 n n X cana-4368 108 12 𝑓(𝑣i2 𝑓(𝑣i2 NOUN cana-4368 108 13 ) ) PUNCT cana-4368 108 14 15 15 NUM cana-4368 108 15 63 63 NUM cana-4368 108 16 255 255 NUM cana-4368 108 17 1023 1023 NUM cana-4368 108 18 4095 4095 NUM cana-4368 108 19 … … PUNCT cana-4368 108 20 … … PUNCT cana-4368 108 21 … … PUNCT cana-4368 108 22 . . PUNCT cana-4368 109 1 22π‘›βˆ’2 22π‘›βˆ’2 NOUN cana-4368 109 2 βˆ’ βˆ’ NOUN cana-4368 109 3 1 1 NUM cana-4368 109 4 𝑓(𝑣𝑖3 𝑓(𝑣𝑖3 NOUN cana-4368 109 5 ) ) PUNCT cana-4368 109 6 1 1 NUM cana-4368 109 7 7 7 NUM cana-4368 109 8 31 31 NUM cana-4368 109 9 127 127 NUM cana-4368 109 10 511 511 NUM cana-4368 109 11 … … SYM cana-4368 109 12 … … PUNCT cana-4368 109 13 … … PUNCT cana-4368 109 14 . . PUNCT cana-4368 110 1 22π‘›βˆ’5 22π‘›βˆ’5 NOUN cana-4368 110 2 βˆ’ βˆ’ NOUN cana-4368 110 3 1 1 NUM cana-4368 110 4 e e NOUN cana-4368 110 5 = = SYM cana-4368 110 6 β„Žπ‘‘([𝑓(𝑣i2)]2 β„Žπ‘‘([𝑓(𝑣i2)]2 PROPN cana-4368 110 7 , , PUNCT cana-4368 110 8 [ [ X cana-4368 110 9 𝑓(𝑣𝑖3)]2 𝑓(𝑣𝑖3)]2 NOUN cana-4368 110 10 ) ) PUNCT cana-4368 110 11 3 3 NUM cana-4368 110 12 3 3 NUM cana-4368 110 13 3 3 NUM cana-4368 110 14 3 3 NUM cana-4368 110 15 3 3 NUM cana-4368 110 16 3 3 NUM cana-4368 110 17 3 3 NUM cana-4368 110 18 for for ADP cana-4368 110 19 1 1 NUM cana-4368 110 20 ≀ ≀ NUM cana-4368 110 21 𝑖 𝑖 SYM cana-4368 110 22 ≀ ≀ NUM cana-4368 110 23 𝑛 𝑛 NOUN cana-4368 110 24 , , PUNCT cana-4368 110 25 6 6 NUM cana-4368 110 26 ≀ ≀ NUM cana-4368 110 27 𝑗 𝑗 PRON cana-4368 110 28 ≀ ≀ NOUN cana-4368 110 29 π‘š π‘š X cana-4368 110 30 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 PROPN cana-4368 110 31 ) ) PUNCT cana-4368 110 32 ) ) PUNCT cana-4368 111 1 = = SYM cana-4368 111 2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 PROPN cana-4368 111 3 , , PUNCT cana-4368 111 4 [ [ X cana-4368 111 5 𝑓(𝑣𝑖(𝑗+1)))]2 𝑓(𝑣𝑖(𝑗+1)))]2 NOUN cana-4368 111 6 ) ) PUNCT cana-4368 111 7 = = PUNCT cana-4368 112 1 3 3 X cana-4368 112 2 . . X cana-4368 112 3 𝑖 𝑖 SYM cana-4368 112 4 1 1 NUM cana-4368 112 5 2 2 NUM cana-4368 112 6 3 3 NUM cana-4368 112 7 4 4 NUM cana-4368 112 8 5 5 NUM cana-4368 112 9 … … PUNCT cana-4368 112 10 n n CCONJ cana-4368 112 11 𝑓(𝑣ij 𝑓(𝑣ij PROPN cana-4368 112 12 ) ) PUNCT cana-4368 112 13 6 6 NUM cana-4368 112 14 3 3 NUM cana-4368 112 15 3=𝑣(3βˆ’1)(jβˆ’4 3=𝑣(3βˆ’1)(jβˆ’4 NUM cana-4368 112 16 ) ) PUNCT cana-4368 112 17 0= 0= NUM cana-4368 113 1 𝑣(4βˆ’2)(jβˆ’2 𝑣(4βˆ’2)(jβˆ’2 PROPN cana-4368 113 2 ) ) PUNCT cana-4368 113 3 0= 0= NUM cana-4368 114 1 𝑣(5βˆ’2)(jβˆ’2 𝑣(5βˆ’2)(jβˆ’2 PROPN cana-4368 114 2 ) ) PUNCT cana-4368 114 3 … … PUNCT cana-4368 114 4 𝑣(nβˆ’2)(jβˆ’2 𝑣(nβˆ’2)(jβˆ’2 X cana-4368 114 5 ) ) PUNCT cana-4368 115 1 𝑓(𝑣𝑖(𝑗+1 𝑓(𝑣𝑖(𝑗+1 PROPN cana-4368 115 2 ) ) PUNCT cana-4368 115 3 ) ) PUNCT cana-4368 115 4 2 2 NUM cana-4368 115 5 4 4 NUM cana-4368 115 6 4=𝑣(3βˆ’1)(jβˆ’3 4=𝑣(3βˆ’1)(jβˆ’3 NUM cana-4368 115 7 ) ) PUNCT cana-4368 115 8 7= 7= NUM cana-4368 115 9 𝑣(4βˆ’2)(jβˆ’1 𝑣(4βˆ’2)(jβˆ’1 NOUN cana-4368 115 10 ) ) PUNCT cana-4368 115 11 7= 7= NUM cana-4368 115 12 𝑣(5βˆ’2)(jβˆ’1 𝑣(5βˆ’2)(jβˆ’1 NOUN cana-4368 115 13 ) ) PUNCT cana-4368 115 14 … … PUNCT cana-4368 115 15 𝑣(nβˆ’2)(jβˆ’1 𝑣(nβˆ’2)(jβˆ’1 NUM cana-4368 115 16 ) ) PUNCT cana-4368 115 17 e= e= NOUN cana-4368 115 18 β„Žπ‘‘([𝑓(𝑣ij)]2 β„Žπ‘‘([𝑓(𝑣ij)]2 PROPN cana-4368 115 19 , , PUNCT cana-4368 115 20 [ [ X cana-4368 115 21 𝑓(𝑣𝑖(𝑗+1))]2 𝑓(𝑣𝑖(𝑗+1))]2 NOUN cana-4368 115 22 ) ) PUNCT cana-4368 115 23 1 1 NUM cana-4368 115 24 3 3 NUM cana-4368 115 25 3 3 NUM cana-4368 115 26 3 3 NUM cana-4368 115 27 3 3 NUM cana-4368 115 28 3 3 NUM cana-4368 115 29 3 3 NUM cana-4368 115 30 case case NOUN cana-4368 115 31 ( ( PUNCT cana-4368 115 32 iii iii NOUN cana-4368 115 33 ): ): PUNCT cana-4368 115 34 when when SCONJ cana-4368 115 35 𝑗 𝑗 X cana-4368 115 36 ≑ ≑ PROPN cana-4368 115 37 3(π‘šπ‘œπ‘‘4 3(π‘šπ‘œπ‘‘4 NUM cana-4368 115 38 ) ) PUNCT cana-4368 115 39 communications communication NOUN cana-4368 115 40 on on ADP cana-4368 115 41 applied apply VERB cana-4368 115 42 nonlinear nonlinear ADJ cana-4368 115 43 analysis analysis NOUN cana-4368 115 44 issn issn NOUN cana-4368 115 45 : : PUNCT cana-4368 115 46 1074 1074 NUM cana-4368 115 47 - - PUNCT cana-4368 115 48 133x 133x NUM cana-4368 115 49 vol vol NOUN cana-4368 115 50 32 32 NUM cana-4368 115 51 no no NOUN cana-4368 115 52 . . PUNCT cana-4368 116 1 9s 9s NUM cana-4368 116 2 ( ( PUNCT cana-4368 116 3 2025 2025 NUM cana-4368 116 4 ) ) PUNCT cana-4368 116 5 1934 1934 NUM cana-4368 116 6 https://internationalpubls.com https://internationalpubls.com X cana-4368 116 7 for for ADP cana-4368 116 8 1 1 NUM cana-4368 116 9 ≀ ≀ NUM cana-4368 116 10 𝑖 𝑖 SYM cana-4368 116 11 ≀ ≀ NUM cana-4368 116 12 𝑛 𝑛 DET cana-4368 116 13 𝑖 𝑖 SYM cana-4368 116 14 1 1 NUM cana-4368 116 15 2 2 NUM cana-4368 116 16 3 3 NUM cana-4368 116 17 4 4 NUM cana-4368 116 18 5 5 NUM cana-4368 116 19 … … PUNCT cana-4368 116 20 n n PRON cana-4368 116 21 𝑓(𝑣i3 𝑓(𝑣i3 NOUN cana-4368 116 22 ) ) PUNCT cana-4368 116 23 2 2 NUM cana-4368 116 24 4 4 NUM cana-4368 116 25 1 1 NUM cana-4368 116 26 7=𝑣(4βˆ’2)(jβˆ’2 7=𝑣(4βˆ’2)(jβˆ’2 NUM cana-4368 116 27 ) ) PUNCT cana-4368 116 28 31=𝑣(5βˆ’2)(jβˆ’2 31=𝑣(5βˆ’2)(jβˆ’2 NUM cana-4368 116 29 ) ) PUNCT cana-4368 116 30 … … PUNCT cana-4368 116 31 𝑣(nβˆ’2)(jβˆ’2 𝑣(nβˆ’2)(jβˆ’2 NOUN cana-4368 116 32 ) ) PUNCT cana-4368 116 33 𝑓(𝑣𝑖4 𝑓(𝑣𝑖4 NOUN cana-4368 116 34 ) ) PUNCT cana-4368 116 35 5 5 NUM cana-4368 116 36 0 0 NUM cana-4368 116 37 0 0 NUM cana-4368 116 38 3=𝑣(4βˆ’2)(jβˆ’1 3=𝑣(4βˆ’2)(jβˆ’1 NUM cana-4368 116 39 ) ) PUNCT cana-4368 116 40 15=𝑣(5βˆ’2)(jβˆ’1 15=𝑣(5βˆ’2)(jβˆ’1 NUM cana-4368 116 41 ) ) PUNCT cana-4368 116 42 … … PUNCT cana-4368 116 43 𝑣(nβˆ’2)(jβˆ’1 𝑣(nβˆ’2)(jβˆ’1 PRON cana-4368 116 44 ) ) PUNCT cana-4368 116 45 e e X cana-4368 116 46 = = SYM cana-4368 116 47 β„Žπ‘‘([𝑓(𝑣i1)]2 β„Žπ‘‘([𝑓(𝑣i1)]2 PROPN cana-4368 116 48 , , PUNCT cana-4368 116 49 [ [ X cana-4368 116 50 𝑓(𝑣𝑖2)]2 𝑓(𝑣𝑖2)]2 NOUN cana-4368 116 51 ) ) PUNCT cana-4368 116 52 3 3 NUM cana-4368 116 53 1 1 NUM cana-4368 116 54 1 1 NUM cana-4368 116 55 1 1 NUM cana-4368 116 56 1 1 NUM cana-4368 116 57 1 1 NUM cana-4368 116 58 1 1 NUM cana-4368 116 59 for for ADP cana-4368 116 60 1 1 NUM cana-4368 116 61 ≀ ≀ NUM cana-4368 116 62 𝑖 𝑖 SYM cana-4368 116 63 ≀ ≀ NUM cana-4368 116 64 𝑛 𝑛 NOUN cana-4368 116 65 , , PUNCT cana-4368 116 66 7 7 NUM cana-4368 116 67 ≀ ≀ NUM cana-4368 116 68 𝑗 𝑗 PRON cana-4368 116 69 ≀ ≀ NOUN cana-4368 116 70 π‘š π‘š X cana-4368 116 71 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 PROPN cana-4368 116 72 ) ) PUNCT cana-4368 116 73 ) ) PUNCT cana-4368 117 1 = = SYM cana-4368 117 2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 PROPN cana-4368 117 3 , , PUNCT cana-4368 117 4 [ [ X cana-4368 117 5 𝑓(𝑣𝑖(𝑗+1)))]2 𝑓(𝑣𝑖(𝑗+1)))]2 NOUN cana-4368 117 6 ) ) PUNCT cana-4368 117 7 = = PUNCT cana-4368 118 1 1 1 X cana-4368 118 2 . . X cana-4368 118 3 𝑖 𝑖 SYM cana-4368 118 4 1 1 NUM cana-4368 118 5 2 2 NUM cana-4368 118 6 3 3 NUM cana-4368 118 7 4 4 NUM cana-4368 118 8 5 5 NUM cana-4368 118 9 … … PUNCT cana-4368 118 10 n n CCONJ cana-4368 118 11 𝑓(𝑣ij 𝑓(𝑣ij PROPN cana-4368 118 12 ) ) PUNCT cana-4368 118 13 2 2 NUM cana-4368 118 14 4 4 NUM cana-4368 118 15 4=𝑣(3βˆ’1)(jβˆ’4 4=𝑣(3βˆ’1)(jβˆ’4 NUM cana-4368 118 16 ) ) PUNCT cana-4368 118 17 7=𝑣(4βˆ’2)(jβˆ’2 7=𝑣(4βˆ’2)(jβˆ’2 NUM cana-4368 118 18 ) ) PUNCT cana-4368 118 19 7= 7= NUM cana-4368 118 20 𝑣(5βˆ’2)(jβˆ’2 𝑣(5βˆ’2)(jβˆ’2 NOUN cana-4368 118 21 ) ) PUNCT cana-4368 118 22 … … PUNCT cana-4368 118 23 𝑣(nβˆ’2)(jβˆ’2 𝑣(nβˆ’2)(jβˆ’2 X cana-4368 118 24 ) ) PUNCT cana-4368 118 25 𝑓(𝑣𝑖(𝑗+1 𝑓(𝑣𝑖(𝑗+1 PROPN cana-4368 118 26 ) ) PUNCT cana-4368 118 27 ) ) PUNCT cana-4368 119 1 5 5 NUM cana-4368 119 2 0 0 NUM cana-4368 119 3 0=𝑣(3βˆ’1)(jβˆ’3 0=𝑣(3βˆ’1)(jβˆ’3 NUM cana-4368 119 4 ) ) PUNCT cana-4368 119 5 3=𝑣(4βˆ’2)(jβˆ’1 3=𝑣(4βˆ’2)(jβˆ’1 NUM cana-4368 119 6 ) ) PUNCT cana-4368 119 7 3= 3= NUM cana-4368 119 8 𝑣(5βˆ’2)(jβˆ’1 𝑣(5βˆ’2)(jβˆ’1 NOUN cana-4368 119 9 ) ) PUNCT cana-4368 119 10 … … PUNCT cana-4368 119 11 𝑣(nβˆ’2)(jβˆ’1 𝑣(nβˆ’2)(jβˆ’1 PRON cana-4368 119 12 ) ) PUNCT cana-4368 119 13 e e X cana-4368 119 14 = = SYM cana-4368 119 15 β„Žπ‘‘([𝑓(𝑣ij)]2 β„Žπ‘‘([𝑓(𝑣ij)]2 PROPN cana-4368 119 16 , , PUNCT cana-4368 119 17 [ [ X cana-4368 119 18 𝑓(𝑣𝑖(𝑗+1))]2 𝑓(𝑣𝑖(𝑗+1))]2 NOUN cana-4368 119 19 ) ) PUNCT cana-4368 119 20 3 3 NUM cana-4368 119 21 1 1 NUM cana-4368 119 22 1 1 NUM cana-4368 119 23 1 1 NUM cana-4368 119 24 1 1 NUM cana-4368 119 25 1 1 NUM cana-4368 119 26 1 1 NUM cana-4368 119 27 case case NOUN cana-4368 119 28 ( ( PUNCT cana-4368 119 29 iv iv NUM cana-4368 119 30 ): ): PUNCT cana-4368 119 31 when when SCONJ cana-4368 119 32 𝑗 𝑗 PROPN cana-4368 119 33 ≑ ≑ PROPN cana-4368 119 34 0(π‘šπ‘œπ‘‘4 0(π‘šπ‘œπ‘‘4 NUM cana-4368 119 35 ) ) PUNCT cana-4368 119 36 𝑖 𝑖 ADP cana-4368 119 37 1 1 NUM cana-4368 119 38 2 2 NUM cana-4368 119 39 3 3 NUM cana-4368 119 40 4 4 NUM cana-4368 119 41 5 5 NUM cana-4368 119 42 … … PUNCT cana-4368 119 43 n n CCONJ cana-4368 119 44 𝑓(𝑣i4 𝑓(𝑣i4 NOUN cana-4368 119 45 ) ) PUNCT cana-4368 119 46 5 5 NUM cana-4368 119 47 0 0 NUM cana-4368 119 48 0 0 NUM cana-4368 119 49 3 3 NUM cana-4368 119 50 = = SYM cana-4368 119 51 𝑣(4βˆ’2)(jβˆ’2 𝑣(4βˆ’2)(jβˆ’2 PROPN cana-4368 119 52 ) ) PUNCT cana-4368 119 53 15 15 NUM cana-4368 119 54 = = SYM cana-4368 119 55 𝑣(5βˆ’2)(jβˆ’2 𝑣(5βˆ’2)(jβˆ’2 PROPN cana-4368 119 56 ) ) PUNCT cana-4368 119 57 … … PUNCT cana-4368 119 58 𝑣(nβˆ’2)(jβˆ’2 𝑣(nβˆ’2)(jβˆ’2 NUM cana-4368 119 59 ) ) PUNCT cana-4368 119 60 𝑓(𝑣𝑖5 𝑓(𝑣𝑖5 PROPN cana-4368 119 61 ) ) PUNCT cana-4368 119 62 3 3 NUM cana-4368 119 63 7 7 NUM cana-4368 119 64 7 7 NUM cana-4368 119 65 = = SYM cana-4368 119 66 𝑣(3βˆ’1)(jβˆ’3 𝑣(3βˆ’1)(jβˆ’3 NOUN cana-4368 119 67 ) ) PUNCT cana-4368 119 68 4 4 NUM cana-4368 119 69 = = SYM cana-4368 119 70 𝑣(4βˆ’2)(jβˆ’1 𝑣(4βˆ’2)(jβˆ’1 PROPN cana-4368 119 71 ) ) PUNCT cana-4368 119 72 1 1 NUM cana-4368 119 73 = = SYM cana-4368 119 74 𝑣(5βˆ’2)(jβˆ’1 𝑣(5βˆ’2)(jβˆ’1 NOUN cana-4368 119 75 ) ) PUNCT cana-4368 119 76 … … PUNCT cana-4368 119 77 𝑣(nβˆ’2)(jβˆ’1 𝑣(nβˆ’2)(jβˆ’1 NUM cana-4368 119 78 ) ) PUNCT cana-4368 119 79 𝑒 𝑒 PROPN cana-4368 119 80 = = SYM cana-4368 119 81 β„Žπ‘‘([𝑓(𝑣i2)]2 β„Žπ‘‘([𝑓(𝑣i2)]2 PROPN cana-4368 119 82 , , PUNCT cana-4368 119 83 [ [ X cana-4368 119 84 𝑓(𝑣𝑖3)]2 𝑓(𝑣𝑖3)]2 NOUN cana-4368 119 85 ) ) PUNCT cana-4368 119 86 3 3 NUM cana-4368 119 87 1 1 NUM cana-4368 119 88 1 1 NUM cana-4368 119 89 1 1 NUM cana-4368 119 90 1 1 NUM cana-4368 119 91 1 1 NUM cana-4368 119 92 1 1 NUM cana-4368 119 93 for for ADP cana-4368 119 94 1 1 NUM cana-4368 119 95 ≀ ≀ NUM cana-4368 119 96 𝑖 𝑖 SYM cana-4368 119 97 ≀ ≀ NUM cana-4368 119 98 𝑛 𝑛 NOUN cana-4368 119 99 , , PUNCT cana-4368 119 100 8 8 NUM cana-4368 119 101 ≀ ≀ NUM cana-4368 119 102 𝑗 𝑗 PRON cana-4368 119 103 ≀ ≀ NOUN cana-4368 119 104 π‘š π‘š X cana-4368 119 105 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 π‘“βˆ—(𝑣𝑖𝑗𝑣𝑖(𝑗+1 PROPN cana-4368 119 106 ) ) PUNCT cana-4368 119 107 ) ) PUNCT cana-4368 120 1 = = SYM cana-4368 120 2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 β„Žπ‘‘([𝑓(𝑣𝑖𝑗)]2 PROPN cana-4368 120 3 , , PUNCT cana-4368 120 4 [ [ X cana-4368 120 5 𝑓(𝑣𝑖(𝑗+1)))]2 𝑓(𝑣𝑖(𝑗+1)))]2 NOUN cana-4368 120 6 ) ) PUNCT cana-4368 120 7 = = SYM cana-4368 121 1 3 3 X cana-4368 121 2 . . X cana-4368 121 3 communications communication NOUN cana-4368 121 4 on on ADP cana-4368 121 5 applied apply VERB cana-4368 121 6 nonlinear nonlinear ADJ cana-4368 121 7 analysis analysis NOUN cana-4368 121 8 issn issn NOUN cana-4368 121 9 : : PUNCT cana-4368 121 10 1074 1074 NUM cana-4368 121 11 - - PUNCT cana-4368 121 12 133x 133x NUM cana-4368 121 13 vol vol NOUN cana-4368 121 14 32 32 NUM cana-4368 121 15 no no NOUN cana-4368 121 16 . . PUNCT cana-4368 122 1 9s 9s NUM cana-4368 122 2 ( ( PUNCT cana-4368 122 3 2025 2025 NUM cana-4368 122 4 ) ) PUNCT cana-4368 122 5 1935 1935 NUM cana-4368 123 1 https://internationalpubls.com https://internationalpubls.com X cana-4368 123 2 𝑖 𝑖 SYM cana-4368 123 3 1 1 NUM cana-4368 123 4 2 2 NUM cana-4368 123 5 3 3 NUM cana-4368 123 6 4 4 NUM cana-4368 123 7 5 5 NUM cana-4368 123 8 … … PUNCT cana-4368 123 9 n n CCONJ cana-4368 123 10 𝑓(𝑣ij 𝑓(𝑣ij PROPN cana-4368 123 11 ) ) PUNCT cana-4368 123 12 5 5 NUM cana-4368 123 13 0 0 NUM cana-4368 123 14 0=𝑣(3βˆ’1)(jβˆ’4 0=𝑣(3βˆ’1)(jβˆ’4 NUM cana-4368 123 15 ) ) PUNCT cana-4368 123 16 3=𝑣(4βˆ’2)(jβˆ’2 3=𝑣(4βˆ’2)(jβˆ’2 NUM cana-4368 123 17 ) ) PUNCT cana-4368 123 18 3=𝑣(5βˆ’2)(jβˆ’2 3=𝑣(5βˆ’2)(jβˆ’2 NUM cana-4368 123 19 ) ) PUNCT cana-4368 123 20 … … PUNCT cana-4368 123 21 𝑣(nβˆ’2)(jβˆ’2 𝑣(nβˆ’2)(jβˆ’2 X cana-4368 123 22 ) ) PUNCT cana-4368 123 23 𝑓(𝑣𝑖(𝑗+1 𝑓(𝑣𝑖(𝑗+1 PROPN cana-4368 123 24 ) ) PUNCT cana-4368 123 25 ) ) PUNCT cana-4368 123 26 1 1 NUM cana-4368 123 27 7 7 NUM cana-4368 123 28 7=𝑣(3βˆ’1)(jβˆ’3 7=𝑣(3βˆ’1)(jβˆ’3 NUM cana-4368 123 29 ) ) PUNCT cana-4368 123 30 4=𝑣(4βˆ’2)(jβˆ’1 4=𝑣(4βˆ’2)(jβˆ’1 NUM cana-4368 123 31 ) ) PUNCT cana-4368 123 32 4=𝑣(5βˆ’2)(jβˆ’1 4=𝑣(5βˆ’2)(jβˆ’1 NUM cana-4368 123 33 ) ) PUNCT cana-4368 123 34 … … PUNCT cana-4368 123 35 𝑣(nβˆ’2)(jβˆ’1 𝑣(nβˆ’2)(jβˆ’1 NUM cana-4368 123 36 ) ) PUNCT cana-4368 123 37 e= e= X cana-4368 123 38 β„Žπ‘‘([𝑓(𝑣ij)]2 β„Žπ‘‘([𝑓(𝑣ij)]2 PROPN cana-4368 123 39 , , PUNCT cana-4368 123 40 [ [ X cana-4368 123 41 𝑓(𝑣𝑖(𝑗+1))]2 𝑓(𝑣𝑖(𝑗+1))]2 NOUN cana-4368 123 42 ) ) PUNCT cana-4368 123 43 3 3 NUM cana-4368 123 44 1 1 NUM cana-4368 123 45 1 1 NUM cana-4368 123 46 1 1 NUM cana-4368 123 47 1 1 NUM cana-4368 123 48 1 1 NUM cana-4368 123 49 1 1 NUM cana-4368 123 50 from from ADP cana-4368 123 51 all all DET cana-4368 123 52 the the DET cana-4368 123 53 above above ADJ cana-4368 123 54 cases case NOUN cana-4368 123 55 , , PUNCT cana-4368 123 56 all all DET cana-4368 123 57 the the DET cana-4368 123 58 adjacent adjacent ADJ cana-4368 123 59 edges edge NOUN cana-4368 123 60 receive receive VERB cana-4368 123 61 distinct distinct ADJ cana-4368 123 62 odd odd ADJ cana-4368 123 63 labels label NOUN cana-4368 123 64 . . PUNCT cana-4368 124 1 hence hence ADV cana-4368 124 2 the the DET cana-4368 124 3 one one NUM cana-4368 124 4 point point NOUN cana-4368 124 5 union union NOUN cana-4368 124 6 of of ADP cana-4368 124 7 path path NOUN cana-4368 124 8 graphs graph NOUN cana-4368 124 9 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 124 10 𝑛 𝑛 PROPN cana-4368 124 11 , , PUNCT cana-4368 124 12 admits admit VERB cana-4368 124 13 odd odd ADJ cana-4368 124 14 hamming hamming NOUN cana-4368 124 15 distance distance NOUN cana-4368 124 16 labeling labeling NOUN cana-4368 124 17 and and CCONJ cana-4368 124 18 the the DET cana-4368 124 19 odd odd ADJ cana-4368 124 20 hamming hamming NOUN cana-4368 124 21 distance distance NOUN cana-4368 124 22 number number NOUN cana-4368 124 23 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 124 24 β€² β€² NUM cana-4368 125 1 ( ( PUNCT cana-4368 125 2 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-4368 125 3 𝑛 𝑛 PROPN cana-4368 125 4 ) ) PUNCT cana-4368 125 5 is be AUX cana-4368 125 6 2𝑛 2𝑛 PROPN cana-4368 125 7 βˆ’ βˆ’ PROPN cana-4368 125 8 1 1 X cana-4368 125 9 . . PUNCT cana-4368 125 10 figure figure VERB cana-4368 125 11 5 5 NUM cana-4368 125 12 odd odd ADJ cana-4368 125 13 hamming hamming NOUN cana-4368 125 14 distance distance NOUN cana-4368 125 15 π‘·πŸ“ π‘·πŸ“ X cana-4368 125 16 πŸ– πŸ– NUM cana-4368 125 17 graph graph NOUN cana-4368 125 18 3.3.7 3.3.7 NOUN cana-4368 125 19 . . PUNCT cana-4368 126 1 algorithm algorithm NOUN cana-4368 126 2 : : PUNCT cana-4368 126 3 odd odd ADJ cana-4368 126 4 hamming hamming NOUN cana-4368 126 5 distance distance NOUN cana-4368 126 6 labeling labeling NOUN cana-4368 126 7 of of ADP cana-4368 126 8 ct(n ct(n NOUN cana-4368 126 9 , , PUNCT cana-4368 126 10 m m NOUN cana-4368 126 11 ) ) PUNCT cana-4368 126 12 graph graph NOUN cana-4368 126 13 procedure procedure NOUN cana-4368 126 14 : : PUNCT cana-4368 126 15 vertex vertex NOUN cana-4368 126 16 labeling labeling NOUN cana-4368 126 17 of of ADP cana-4368 126 18 coconut coconut NOUN cana-4368 126 19 tree tree NOUN cana-4368 126 20 ct(n ct(n X cana-4368 126 21 , , PUNCT cana-4368 126 22 m m NOUN cana-4368 126 23 ) ) PUNCT cana-4368 126 24 input input NOUN cana-4368 126 25 : : PUNCT cana-4368 126 26 coconut coconut NOUN cana-4368 126 27 tree tree NOUN cana-4368 126 28 graph graph NOUN cana-4368 126 29 ct(n ct(n NOUN cana-4368 126 30 , , PUNCT cana-4368 126 31 m m NOUN cana-4368 126 32 ) ) PUNCT cana-4368 126 33 . . PUNCT cana-4368 127 1 v v X cana-4368 127 2 ← ← PROPN cana-4368 127 3 { { PUNCT cana-4368 127 4 { { PUNCT cana-4368 127 5 ui ui PROPN cana-4368 127 6 /0 /0 NOUN cana-4368 127 7 ≀ ≀ NUM cana-4368 128 1 i i PRON cana-4368 128 2 ≀ ≀ NOUN cana-4368 128 3 m m VERB cana-4368 128 4 } } PUNCT cana-4368 128 5 βˆͺ βˆͺ ADJ cana-4368 128 6 { { PUNCT cana-4368 128 7 vj vj INTJ cana-4368 128 8 /1 /1 PROPN cana-4368 128 9 ≀ ≀ PROPN cana-4368 128 10 j j PROPN cana-4368 128 11 ≀ ≀ PROPN cana-4368 128 12 n n CCONJ cana-4368 128 13 } } PUNCT cana-4368 128 14 } } PUNCT cana-4368 128 15 u0 u0 ADJ cana-4368 128 16 ← ← PROPN cana-4368 128 17 0 0 NUM cana-4368 128 18 ; ; PUNCT cana-4368 128 19 for for ADP cana-4368 128 20 i i PRON cana-4368 128 21 = = SYM cana-4368 128 22 1 1 NUM cana-4368 128 23 to to PART cana-4368 128 24 m m PROPN cana-4368 128 25 do do VERB cana-4368 128 26 ui ui PROPN cana-4368 128 27 ← ← PROPN cana-4368 128 28 { { PUNCT cana-4368 128 29 1 1 NUM cana-4368 128 30 if if SCONJ cana-4368 128 31 i i PRON cana-4368 128 32 ≑ ≑ PROPN cana-4368 128 33 1(mod4 1(mod4 NUM cana-4368 128 34 ) ) PUNCT cana-4368 128 35 6 6 NUM cana-4368 128 36 if if SCONJ cana-4368 128 37 i i PRON cana-4368 128 38 ≑ ≑ PROPN cana-4368 128 39 2(mod4 2(mod4 NUM cana-4368 128 40 ) ) PUNCT cana-4368 128 41 2 2 NUM cana-4368 128 42 if if SCONJ cana-4368 128 43 i i PRON cana-4368 128 44 ≑ ≑ PROPN cana-4368 128 45 3(mod4 3(mod4 NUM cana-4368 128 46 ) ) PUNCT cana-4368 128 47 5 5 NUM cana-4368 128 48 if if SCONJ cana-4368 128 49 i i PRON cana-4368 128 50 ≑ ≑ PROPN cana-4368 128 51 0(mod4 0(mod4 NUM cana-4368 128 52 ) ) PUNCT cana-4368 128 53 end end NOUN cana-4368 128 54 fo fo INTJ cana-4368 128 55 for for ADP cana-4368 128 56 j j PROPN cana-4368 128 57 = = SYM cana-4368 128 58 1 1 NUM cana-4368 128 59 to to PART cana-4368 128 60 n n PRON cana-4368 128 61 do do VERB cana-4368 128 62 vi vi PROPN cana-4368 128 63 ← ← PROPN cana-4368 128 64 { { PUNCT cana-4368 128 65 6 6 NUM cana-4368 128 66 if if SCONJ cana-4368 128 67 i i PRON cana-4368 128 68 ≑ ≑ PROPN cana-4368 128 69 1(mod4 1(mod4 NUM cana-4368 128 70 ) ) PUNCT cana-4368 128 71 2 2 NUM cana-4368 128 72 if if SCONJ cana-4368 128 73 i i PRON cana-4368 128 74 ≑ ≑ PROPN cana-4368 128 75 2(mod4 2(mod4 NUM cana-4368 128 76 ) ) PUNCT cana-4368 128 77 5 5 NUM cana-4368 128 78 if if SCONJ cana-4368 128 79 i i PRON cana-4368 128 80 ≑ ≑ PROPN cana-4368 128 81 3(mod4 3(mod4 NUM cana-4368 128 82 ) ) PUNCT cana-4368 128 83 1 1 NUM cana-4368 128 84 if if SCONJ cana-4368 128 85 i i PRON cana-4368 128 86 ≑ ≑ PROPN cana-4368 128 87 0(mod4 0(mod4 NUM cana-4368 128 88 ) ) PUNCT cana-4368 128 89 communications communication NOUN cana-4368 128 90 on on ADP cana-4368 128 91 applied apply VERB cana-4368 128 92 nonlinear nonlinear ADJ cana-4368 128 93 analysis analysis NOUN cana-4368 128 94 issn issn NOUN cana-4368 128 95 : : PUNCT cana-4368 128 96 1074 1074 NUM cana-4368 128 97 - - PUNCT cana-4368 128 98 133x 133x NUM cana-4368 128 99 vol vol NOUN cana-4368 128 100 32 32 NUM cana-4368 128 101 no no NOUN cana-4368 128 102 . . PUNCT cana-4368 129 1 9s 9s NUM cana-4368 129 2 ( ( PUNCT cana-4368 129 3 2025 2025 NUM cana-4368 129 4 ) ) PUNCT cana-4368 129 5 1936 1936 NUM cana-4368 129 6 https://internationalpubls.com https://internationalpubls.com X cana-4368 129 7 end end VERB cana-4368 129 8 for for ADP cana-4368 129 9 for for ADP cana-4368 129 10 j j PROPN cana-4368 129 11 = = SYM cana-4368 129 12 2 2 NUM cana-4368 129 13 to to PART cana-4368 129 14 n n PRON cana-4368 129 15 do do VERB cana-4368 129 16 vj vj PRON cana-4368 129 17 ← ← PROPN cana-4368 129 18 22j+1 22j+1 NUM cana-4368 129 19 βˆ’ βˆ’ PROPN cana-4368 130 1 um um INTJ cana-4368 130 2 + + CCONJ cana-4368 130 3 1 1 X cana-4368 130 4 ) ) PUNCT cana-4368 130 5 end end NOUN cana-4368 130 6 for for ADP cana-4368 130 7 end end NOUN cana-4368 130 8 procedure procedure NOUN cana-4368 130 9 output output NOUN cana-4368 130 10 : : PUNCT cana-4368 130 11 the the DET cana-4368 130 12 labeled label VERB cana-4368 130 13 vertices vertex NOUN cana-4368 130 14 of of ADP cana-4368 130 15 coconut coconut NOUN cana-4368 130 16 tree tree NOUN cana-4368 130 17 graph graph NOUN cana-4368 130 18 ct(n ct(n NOUN cana-4368 130 19 , , PUNCT cana-4368 130 20 m m PROPN cana-4368 130 21 ) ) PUNCT cana-4368 130 22 . . PUNCT cana-4368 131 1 3.3.8.theorem 3.3.8.theorem NUM cana-4368 131 2 the the DET cana-4368 131 3 coconut coconut NOUN cana-4368 131 4 tree tree NOUN cana-4368 131 5 graph graph NOUN cana-4368 131 6 ct(n ct(n NOUN cana-4368 131 7 , , PUNCT cana-4368 131 8 m m VERB cana-4368 131 9 ) ) PUNCT cana-4368 131 10 is be AUX cana-4368 131 11 an an DET cana-4368 131 12 odd odd ADJ cana-4368 131 13 hamming hamming NOUN cana-4368 131 14 distance distance NOUN cana-4368 131 15 graph graph NOUN cana-4368 131 16 and and CCONJ cana-4368 131 17 the the DET cana-4368 131 18 odd odd ADJ cana-4368 131 19 hamming hamming NOUN cana-4368 131 20 distance distance NOUN cana-4368 131 21 number number NOUN cana-4368 131 22 is be AUX cana-4368 131 23 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 131 24 β€² β€² NUM cana-4368 131 25 ( ( PUNCT cana-4368 131 26 ct(n ct(n X cana-4368 131 27 , , PUNCT cana-4368 131 28 m m NOUN cana-4368 131 29 ) ) PUNCT cana-4368 131 30 ) ) PUNCT cana-4368 132 1 = = SYM cana-4368 133 1 2𝑛 2𝑛 PROPN cana-4368 134 1 + + CCONJ cana-4368 134 2 1 1 X cana-4368 134 3 . . X cana-4368 134 4 proof proof NOUN cana-4368 134 5 : : PUNCT cana-4368 134 6 let let VERB cana-4368 134 7 us we PRON cana-4368 134 8 consider consider VERB cana-4368 134 9 the the DET cana-4368 134 10 coconut coconut NOUN cana-4368 134 11 tree tree NOUN cana-4368 134 12 graph graph NOUN cana-4368 134 13 ct(n ct(n NOUN cana-4368 134 14 , , PUNCT cana-4368 134 15 m m NOUN cana-4368 134 16 ) ) PUNCT cana-4368 134 17 with with ADP cana-4368 134 18 vertex vertex NOUN cana-4368 134 19 set set VERB cana-4368 134 20 𝑉 𝑉 PROPN cana-4368 134 21 = = PUNCT cana-4368 134 22 { { PUNCT cana-4368 134 23 { { PUNCT cana-4368 134 24 𝑒𝑖 𝑒𝑖 NOUN cana-4368 134 25 /0 /0 NOUN cana-4368 134 26 ≀ ≀ NUM cana-4368 135 1 𝑖 𝑖 SYM cana-4368 135 2 ≀ ≀ NUM cana-4368 135 3 π‘š π‘š ADP cana-4368 135 4 } } PUNCT cana-4368 135 5 βˆͺ βˆͺ X cana-4368 135 6 { { PUNCT cana-4368 135 7 𝑣𝑗 𝑣𝑗 ADP cana-4368 135 8 /1 /1 NOUN cana-4368 135 9 ≀ ≀ NUM cana-4368 135 10 𝑗 𝑗 PRON cana-4368 135 11 ≀ ≀ NUM cana-4368 135 12 𝑛 𝑛 NOUN cana-4368 135 13 } } PUNCT cana-4368 135 14 } } PUNCT cana-4368 135 15 and and CCONJ cana-4368 135 16 edge edge NOUN cana-4368 135 17 set set VERB cana-4368 135 18 𝐸 𝐸 NOUN cana-4368 135 19 = = SYM cana-4368 135 20 { { PUNCT cana-4368 135 21 { { PUNCT cana-4368 135 22 𝑒𝑖𝑒𝑖+1 𝑒𝑖𝑒𝑖+1 NOUN cana-4368 135 23 / / SYM cana-4368 135 24 0 0 NUM cana-4368 135 25 ≀ ≀ NUM cana-4368 135 26 𝑖 𝑖 SYM cana-4368 135 27 ≀ ≀ NOUN cana-4368 135 28 π‘š π‘š ADP cana-4368 135 29 βˆ’ βˆ’ PROPN cana-4368 135 30 1 1 NUM cana-4368 135 31 } } PUNCT cana-4368 135 32 βˆͺ βˆͺ ADJ cana-4368 135 33 { { PUNCT cana-4368 135 34 π‘’π‘šπ‘£π‘— π‘’π‘šπ‘£π‘— ADJ cana-4368 135 35 / / SYM cana-4368 135 36 1 1 NUM cana-4368 135 37 ≀ ≀ NUM cana-4368 135 38 𝑗 𝑗 PRON cana-4368 135 39 ≀ ≀ NOUN cana-4368 135 40 𝑛}}.this 𝑛}}.this PRON cana-4368 135 41 graph graph NOUN cana-4368 135 42 has have VERB cana-4368 135 43 m+n+1 m+n+1 NOUN cana-4368 135 44 vertices vertex NOUN cana-4368 135 45 and and CCONJ cana-4368 135 46 m+n m+n NUM cana-4368 135 47 edges edge NOUN cana-4368 135 48 . . PUNCT cana-4368 136 1 define define VERB cana-4368 136 2 a a DET cana-4368 136 3 function function NOUN cana-4368 136 4 𝑓 𝑓 NOUN cana-4368 136 5 : : PUNCT cana-4368 136 6 𝑉 𝑉 PROPN cana-4368 136 7 β†’ β†’ PUNCT cana-4368 136 8 𝑁 𝑁 PROPN cana-4368 136 9 βˆͺ βˆͺ ADJ cana-4368 136 10 { { PUNCT cana-4368 136 11 0 0 NUM cana-4368 136 12 } } PUNCT cana-4368 136 13 such such ADJ cana-4368 136 14 that that SCONJ cana-4368 136 15 𝑓(𝑒 𝑓(𝑒 PROPN cana-4368 136 16 ) ) PUNCT cana-4368 136 17 β‰  β‰  PROPN cana-4368 136 18 𝑓(𝑣 𝑓(𝑣 PROPN cana-4368 136 19 ) ) PUNCT cana-4368 136 20 for for ADP cana-4368 136 21 any any DET cana-4368 136 22 two two NUM cana-4368 136 23 adjacent adjacent ADJ cana-4368 136 24 vertices vertex NOUN cana-4368 136 25 𝑒 𝑒 NOUN cana-4368 136 26 and and CCONJ cana-4368 136 27 𝑣 𝑣 ADP cana-4368 136 28 as as SCONJ cana-4368 136 29 given give VERB cana-4368 136 30 in in ADP cana-4368 136 31 the the DET cana-4368 136 32 above above ADJ cana-4368 136 33 algorithm algorithm NOUN cana-4368 136 34 3.3.7.hence 3.3.7.hence NUM cana-4368 136 35 all all DET cana-4368 136 36 the the DET cana-4368 136 37 adjacent adjacent ADJ cana-4368 136 38 vertices vertex NOUN cana-4368 136 39 receive receive VERB cana-4368 136 40 distinct distinct ADJ cana-4368 136 41 labels.the labels.the DET cana-4368 136 42 edge edge NOUN cana-4368 136 43 labels label NOUN cana-4368 136 44 are be AUX cana-4368 136 45 obtained obtain VERB cana-4368 136 46 as as SCONJ cana-4368 136 47 follows follow VERB cana-4368 136 48 : : PUNCT cana-4368 136 49 f f PROPN cana-4368 136 50 βˆ—(𝑒0𝑒1 βˆ—(𝑒0𝑒1 PROPN cana-4368 136 51 ) ) PUNCT cana-4368 136 52 = = PUNCT cana-4368 137 1 hd([f(𝑒0)]2 hd([f(𝑒0)]2 NUM cana-4368 137 2 , , PUNCT cana-4368 137 3 [ [ X cana-4368 137 4 f(𝑒1)]2 f(𝑒1)]2 X cana-4368 137 5 ) ) PUNCT cana-4368 137 6 = = SYM cana-4368 138 1 hd([0]2 hd([0]2 NOUN cana-4368 138 2 , , PUNCT cana-4368 138 3 [ [ X cana-4368 138 4 1]2 1]2 NUM cana-4368 138 5 ) ) PUNCT cana-4368 138 6 = = NOUN cana-4368 138 7 1 1 X cana-4368 138 8 . . PUNCT cana-4368 139 1 for for ADP cana-4368 139 2 1 1 NUM cana-4368 139 3 ≀ ≀ NUM cana-4368 139 4 𝑖 𝑖 PUNCT cana-4368 139 5 ≀ ≀ NOUN cana-4368 139 6 π‘š π‘š ADP cana-4368 139 7 , , PUNCT cana-4368 139 8 where where SCONJ cana-4368 139 9 π‘’π‘š+1 π‘’π‘š+1 PROPN cana-4368 139 10 = = PROPN cana-4368 139 11 𝑣1 𝑣1 NOUN cana-4368 139 12 case case NOUN cana-4368 139 13 ( ( PUNCT cana-4368 139 14 i i NOUN cana-4368 139 15 ): ): PUNCT cana-4368 139 16 if if SCONJ cana-4368 139 17 𝑖 𝑖 ADP cana-4368 139 18 ≑ ≑ PROPN cana-4368 139 19 1(π‘šπ‘œπ‘‘ 1(π‘šπ‘œπ‘‘ ADJ cana-4368 139 20 4 4 NUM cana-4368 139 21 ) ) PUNCT cana-4368 139 22 ; ; PUNCT cana-4368 139 23 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 X cana-4368 139 24 ) ) PUNCT cana-4368 139 25 = = SYM cana-4368 139 26 β„Žπ‘‘([𝑓(𝑒𝑖)]2 β„Žπ‘‘([𝑓(𝑒𝑖)]2 PROPN cana-4368 139 27 , , PUNCT cana-4368 139 28 [ [ X cana-4368 139 29 𝑓(𝑒𝑖+1)]2 𝑓(𝑒𝑖+1)]2 PUNCT cana-4368 139 30 ) ) PUNCT cana-4368 139 31 = = SYM cana-4368 139 32 hd([1]2 hd([1]2 NOUN cana-4368 139 33 , , PUNCT cana-4368 139 34 [ [ X cana-4368 139 35 6]2 6]2 NOUN cana-4368 139 36 ) ) PUNCT cana-4368 139 37 = = SYM cana-4368 139 38 3 3 X cana-4368 139 39 . . X cana-4368 139 40 case case NOUN cana-4368 139 41 ( ( PUNCT cana-4368 139 42 ii ii NOUN cana-4368 139 43 ): ): PUNCT cana-4368 139 44 if if SCONJ cana-4368 139 45 𝑖 𝑖 PRON cana-4368 139 46 ≑ ≑ PROPN cana-4368 139 47 2(π‘šπ‘œπ‘‘ 2(π‘šπ‘œπ‘‘ NOUN cana-4368 139 48 4 4 NUM cana-4368 139 49 ) ) PUNCT cana-4368 139 50 ; ; PUNCT cana-4368 139 51 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 X cana-4368 139 52 ) ) PUNCT cana-4368 139 53 = = SYM cana-4368 140 1 β„Žπ‘‘([𝑓(𝑒𝑖)]2 β„Žπ‘‘([𝑓(𝑒𝑖)]2 PROPN cana-4368 140 2 , , PUNCT cana-4368 140 3 [ [ X cana-4368 140 4 𝑓(𝑒𝑖+1)]2 𝑓(𝑒𝑖+1)]2 PUNCT cana-4368 140 5 ) ) PUNCT cana-4368 140 6 = = SYM cana-4368 141 1 hd([6]2 hd([6]2 X cana-4368 141 2 , , PUNCT cana-4368 141 3 [ [ X cana-4368 141 4 2]2 2]2 NUM cana-4368 141 5 ) ) PUNCT cana-4368 141 6 = = SYM cana-4368 141 7 1 1 X cana-4368 141 8 . . X cana-4368 142 1 case case NOUN cana-4368 142 2 ( ( PUNCT cana-4368 142 3 iii iii NOUN cana-4368 142 4 ): ): PUNCT cana-4368 142 5 if if SCONJ cana-4368 142 6 𝑖 𝑖 ADP cana-4368 142 7 ≑ ≑ PROPN cana-4368 142 8 3(π‘šπ‘œπ‘‘ 3(π‘šπ‘œπ‘‘ NUM cana-4368 142 9 4 4 NUM cana-4368 142 10 ) ) PUNCT cana-4368 142 11 ; ; PUNCT cana-4368 142 12 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 X cana-4368 142 13 ) ) PUNCT cana-4368 142 14 = = SYM cana-4368 143 1 β„Žπ‘‘([𝑓(𝑒𝑖)]2 β„Žπ‘‘([𝑓(𝑒𝑖)]2 PROPN cana-4368 143 2 , , PUNCT cana-4368 143 3 [ [ X cana-4368 143 4 𝑓(𝑒𝑖+1)]2 𝑓(𝑒𝑖+1)]2 PUNCT cana-4368 143 5 ) ) PUNCT cana-4368 143 6 = = SYM cana-4368 143 7 hd([2]2 hd([2]2 X cana-4368 143 8 , , PUNCT cana-4368 143 9 [ [ X cana-4368 143 10 5]2 5]2 X cana-4368 143 11 ) ) PUNCT cana-4368 143 12 = = SYM cana-4368 143 13 3 3 X cana-4368 143 14 . . X cana-4368 143 15 case case NOUN cana-4368 143 16 ( ( PUNCT cana-4368 143 17 iv iv NUM cana-4368 143 18 ): ): PUNCT cana-4368 143 19 if if SCONJ cana-4368 143 20 𝑖 𝑖 ADP cana-4368 143 21 ≑ ≑ PROPN cana-4368 143 22 0(π‘šπ‘œπ‘‘ 0(π‘šπ‘œπ‘‘ NOUN cana-4368 143 23 4 4 NUM cana-4368 143 24 ) ) PUNCT cana-4368 143 25 ; ; PUNCT cana-4368 143 26 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 π‘“βˆ—(𝑒𝑖𝑒𝑖+1 X cana-4368 143 27 ) ) PUNCT cana-4368 143 28 = = SYM cana-4368 144 1 β„Žπ‘‘([𝑓(𝑒𝑖)]2 β„Žπ‘‘([𝑓(𝑒𝑖)]2 PROPN cana-4368 144 2 , , PUNCT cana-4368 144 3 [ [ X cana-4368 144 4 𝑓(𝑒𝑖+1)]2 𝑓(𝑒𝑖+1)]2 PUNCT cana-4368 144 5 ) ) PUNCT cana-4368 144 6 = = SYM cana-4368 144 7 hd([5]2 hd([5]2 NOUN cana-4368 144 8 , , PUNCT cana-4368 144 9 [ [ X cana-4368 144 10 1]2 1]2 NUM cana-4368 144 11 ) ) PUNCT cana-4368 144 12 = = SYM cana-4368 145 1 1 1 X cana-4368 145 2 . . X cana-4368 145 3 for for ADP cana-4368 145 4 2 2 NUM cana-4368 145 5 ≀ ≀ NOUN cana-4368 145 6 𝑗 𝑗 PRON cana-4368 145 7 ≀ ≀ NUM cana-4368 145 8 𝑛 𝑛 DET cana-4368 145 9 case case NOUN cana-4368 145 10 ( ( PUNCT cana-4368 145 11 i i NOUN cana-4368 145 12 ): ): PUNCT cana-4368 145 13 if if SCONJ cana-4368 145 14 π‘š π‘š PROPN cana-4368 145 15 ≑ ≑ PROPN cana-4368 145 16 1(π‘šπ‘œπ‘‘ 1(π‘šπ‘œπ‘‘ NUM cana-4368 145 17 4 4 NUM cana-4368 145 18 ) ) PUNCT cana-4368 145 19 𝑗 𝑗 NOUN cana-4368 145 20 2 2 NUM cana-4368 145 21 3 3 NUM cana-4368 145 22 4 4 NUM cana-4368 145 23 5 5 NUM cana-4368 145 24 6 6 NUM cana-4368 145 25 … … SYM cana-4368 145 26 …… …… X cana-4368 145 27 n n PRON cana-4368 145 28 𝑓(𝑒m 𝑓(𝑒m NUM cana-4368 145 29 ) ) PUNCT cana-4368 145 30 1 1 NUM cana-4368 145 31 1 1 NUM cana-4368 145 32 1 1 NUM cana-4368 145 33 1 1 NUM cana-4368 145 34 1 1 NUM cana-4368 145 35 … … PUNCT cana-4368 145 36 … … PUNCT cana-4368 145 37 … … SYM cana-4368 145 38 . . NOUN cana-4368 145 39 1 1 NUM cana-4368 145 40 𝑓(𝑣𝑗 𝑓(𝑣𝑗 NOUN cana-4368 145 41 ) ) PUNCT cana-4368 145 42 30 30 NUM cana-4368 145 43 126 126 NUM cana-4368 145 44 510 510 NUM cana-4368 145 45 2046 2046 NUM cana-4368 145 46 8190 8190 NUM cana-4368 145 47 … … PUNCT cana-4368 145 48 … … PUNCT cana-4368 145 49 … … PUNCT cana-4368 145 50 . . PUNCT cana-4368 146 1 22j+1 22j+1 NUM cana-4368 146 2 βˆ’ βˆ’ NOUN cana-4368 146 3 2 2 NUM cana-4368 146 4 e e NOUN cana-4368 146 5 = = SYM cana-4368 146 6 β„Žπ‘‘([𝑓(𝑒m)]2 β„Žπ‘‘([𝑓(𝑒m)]2 PROPN cana-4368 146 7 , , PUNCT cana-4368 146 8 [ [ X cana-4368 146 9 𝑓(𝑣𝑗)]2 𝑓(𝑣𝑗)]2 NOUN cana-4368 146 10 ) ) PUNCT cana-4368 146 11 5 5 NUM cana-4368 146 12 7 7 NUM cana-4368 146 13 9 9 NUM cana-4368 146 14 11 11 NUM cana-4368 146 15 13 13 NUM cana-4368 146 16 19 19 NUM cana-4368 146 17 2n+1 2n+1 PROPN cana-4368 146 18 case case NOUN cana-4368 146 19 ( ( PUNCT cana-4368 146 20 ii ii NOUN cana-4368 146 21 ): ): PUNCT cana-4368 146 22 if if SCONJ cana-4368 146 23 π‘š π‘š PROPN cana-4368 146 24 ≑ ≑ PROPN cana-4368 146 25 2(π‘šπ‘œπ‘‘ 2(π‘šπ‘œπ‘‘ NOUN cana-4368 146 26 4 4 NUM cana-4368 146 27 ) ) PUNCT cana-4368 146 28 𝑗 𝑗 NOUN cana-4368 146 29 2 2 NUM cana-4368 146 30 3 3 NUM cana-4368 146 31 4 4 NUM cana-4368 146 32 5 5 NUM cana-4368 146 33 6 6 NUM cana-4368 146 34 … … PUNCT cana-4368 146 35 …… …… NOUN cana-4368 146 36 n n PRON cana-4368 146 37 communications communication NOUN cana-4368 146 38 on on ADP cana-4368 146 39 applied apply VERB cana-4368 146 40 nonlinear nonlinear ADJ cana-4368 146 41 analysis analysis NOUN cana-4368 146 42 issn issn NOUN cana-4368 146 43 : : PUNCT cana-4368 146 44 1074 1074 NUM cana-4368 146 45 - - PUNCT cana-4368 146 46 133x 133x NUM cana-4368 146 47 vol vol NOUN cana-4368 146 48 32 32 NUM cana-4368 146 49 no no NOUN cana-4368 146 50 . . PUNCT cana-4368 147 1 9s 9s NUM cana-4368 147 2 ( ( PUNCT cana-4368 147 3 2025 2025 NUM cana-4368 147 4 ) ) PUNCT cana-4368 147 5 1937 1937 NUM cana-4368 147 6 https://internationalpubls.com https://internationalpubls.com X cana-4368 147 7 𝑓(𝑒m 𝑓(𝑒m PROPN cana-4368 147 8 ) ) PUNCT cana-4368 147 9 6 6 NUM cana-4368 147 10 6 6 NUM cana-4368 147 11 6 6 NUM cana-4368 147 12 6 6 NUM cana-4368 147 13 6 6 NUM cana-4368 147 14 … … PUNCT cana-4368 147 15 … … PUNCT cana-4368 147 16 … … PUNCT cana-4368 147 17 . . NOUN cana-4368 147 18 6 6 NUM cana-4368 147 19 𝑓(𝑣𝑗 𝑓(𝑣𝑗 NOUN cana-4368 147 20 ) ) PUNCT cana-4368 148 1 25 25 NUM cana-4368 148 2 121 121 NUM cana-4368 148 3 505 505 NUM cana-4368 148 4 2041 2041 NUM cana-4368 148 5 8185 8185 NUM cana-4368 148 6 … … PUNCT cana-4368 148 7 … … PUNCT cana-4368 148 8 … … PUNCT cana-4368 148 9 . . PUNCT cana-4368 149 1 22j+1 22j+1 NUM cana-4368 149 2 βˆ’ βˆ’ NOUN cana-4368 149 3 7 7 NUM cana-4368 149 4 e e NOUN cana-4368 149 5 = = SYM cana-4368 149 6 β„Žπ‘‘([𝑓(𝑒m)]2 β„Žπ‘‘([𝑓(𝑒m)]2 PROPN cana-4368 149 7 , , PUNCT cana-4368 149 8 [ [ X cana-4368 149 9 𝑓(𝑣𝑗)]2 𝑓(𝑣𝑗)]2 NOUN cana-4368 149 10 ) ) PUNCT cana-4368 149 11 5 5 NUM cana-4368 149 12 7 7 NUM cana-4368 149 13 9 9 NUM cana-4368 149 14 11 11 NUM cana-4368 149 15 13 13 NUM cana-4368 149 16 19 19 NUM cana-4368 149 17 2n+1 2n+1 PROPN cana-4368 149 18 case case NOUN cana-4368 149 19 ( ( PUNCT cana-4368 149 20 iii iii NOUN cana-4368 149 21 ): ): PUNCT cana-4368 149 22 if if SCONJ cana-4368 149 23 π‘š π‘š PROPN cana-4368 149 24 ≑ ≑ PROPN cana-4368 149 25 3(π‘šπ‘œπ‘‘ 3(π‘šπ‘œπ‘‘ NUM cana-4368 149 26 4 4 NUM cana-4368 149 27 ) ) PUNCT cana-4368 149 28 𝑗 𝑗 NOUN cana-4368 149 29 2 2 NUM cana-4368 149 30 3 3 NUM cana-4368 149 31 4 4 NUM cana-4368 149 32 5 5 NUM cana-4368 149 33 6 6 NUM cana-4368 149 34 … … SYM cana-4368 149 35 …… …… X cana-4368 149 36 n n PRON cana-4368 149 37 𝑓(𝑒m 𝑓(𝑒m NUM cana-4368 149 38 ) ) PUNCT cana-4368 149 39 2 2 NUM cana-4368 149 40 2 2 NUM cana-4368 149 41 2 2 NUM cana-4368 149 42 2 2 NUM cana-4368 149 43 2 2 NUM cana-4368 149 44 … … PUNCT cana-4368 149 45 … … PUNCT cana-4368 149 46 … … SYM cana-4368 149 47 . . NOUN cana-4368 149 48 2 2 NUM cana-4368 149 49 𝑓(𝑣𝑗 𝑓(𝑣𝑗 NOUN cana-4368 149 50 ) ) PUNCT cana-4368 149 51 29 29 NUM cana-4368 149 52 125 125 NUM cana-4368 149 53 509 509 NUM cana-4368 149 54 2045 2045 NUM cana-4368 149 55 8189 8189 NUM cana-4368 149 56 … … PUNCT cana-4368 149 57 … … PUNCT cana-4368 149 58 … … PUNCT cana-4368 149 59 . . PUNCT cana-4368 150 1 22j+1 22j+1 NUM cana-4368 150 2 βˆ’ βˆ’ NOUN cana-4368 150 3 3 3 NUM cana-4368 150 4 e e NOUN cana-4368 150 5 = = SYM cana-4368 150 6 β„Žπ‘‘([𝑓(𝑒m)]2 β„Žπ‘‘([𝑓(𝑒m)]2 PROPN cana-4368 150 7 , , PUNCT cana-4368 150 8 [ [ X cana-4368 150 9 𝑓(𝑣𝑗)]2 𝑓(𝑣𝑗)]2 NOUN cana-4368 150 10 ) ) PUNCT cana-4368 150 11 5 5 NUM cana-4368 150 12 7 7 NUM cana-4368 150 13 9 9 NUM cana-4368 150 14 11 11 NUM cana-4368 150 15 13 13 NUM cana-4368 150 16 19 19 NUM cana-4368 150 17 2n+1 2n+1 PROPN cana-4368 150 18 case case NOUN cana-4368 150 19 ( ( PUNCT cana-4368 150 20 iv iv NUM cana-4368 150 21 ): ): PUNCT cana-4368 150 22 if if SCONJ cana-4368 150 23 π‘š π‘š PROPN cana-4368 150 24 ≑ ≑ PROPN cana-4368 150 25 0(π‘šπ‘œπ‘‘ 0(π‘šπ‘œπ‘‘ NOUN cana-4368 150 26 4 4 NUM cana-4368 150 27 ) ) PUNCT cana-4368 150 28 𝑗 𝑗 NOUN cana-4368 150 29 2 2 NUM cana-4368 150 30 3 3 NUM cana-4368 150 31 4 4 NUM cana-4368 150 32 5 5 NUM cana-4368 150 33 6 6 NUM cana-4368 150 34 … … SYM cana-4368 150 35 …… …… X cana-4368 150 36 n n PRON cana-4368 150 37 𝑓(𝑒m 𝑓(𝑒m NUM cana-4368 150 38 ) ) PUNCT cana-4368 150 39 5 5 NUM cana-4368 150 40 5 5 NUM cana-4368 150 41 5 5 NUM cana-4368 150 42 5 5 NUM cana-4368 150 43 5 5 NUM cana-4368 150 44 … … PUNCT cana-4368 150 45 … … PUNCT cana-4368 150 46 … … SYM cana-4368 150 47 . . X cana-4368 150 48 5 5 NUM cana-4368 150 49 𝑓(𝑣𝑗 𝑓(𝑣𝑗 NOUN cana-4368 150 50 ) ) PUNCT cana-4368 150 51 26 26 NUM cana-4368 150 52 122 122 NUM cana-4368 150 53 506 506 NUM cana-4368 150 54 2042 2042 NUM cana-4368 150 55 8186 8186 NUM cana-4368 150 56 … … SYM cana-4368 150 57 … … PUNCT cana-4368 150 58 … … PUNCT cana-4368 150 59 . . PUNCT cana-4368 151 1 22j+1 22j+1 NUM cana-4368 151 2 βˆ’ βˆ’ NOUN cana-4368 151 3 6 6 NUM cana-4368 151 4 e e NOUN cana-4368 151 5 = = SYM cana-4368 151 6 β„Žπ‘‘([𝑓(𝑒m)]2 β„Žπ‘‘([𝑓(𝑒m)]2 PROPN cana-4368 151 7 , , PUNCT cana-4368 151 8 [ [ X cana-4368 151 9 𝑓(𝑣𝑗)]2 𝑓(𝑣𝑗)]2 NOUN cana-4368 151 10 ) ) PUNCT cana-4368 151 11 5 5 NUM cana-4368 151 12 7 7 NUM cana-4368 151 13 9 9 NUM cana-4368 151 14 11 11 NUM cana-4368 151 15 13 13 NUM cana-4368 151 16 19 19 NUM cana-4368 151 17 2n+1 2n+1 PROPN cana-4368 151 18 from from ADP cana-4368 151 19 all all DET cana-4368 151 20 the the DET cana-4368 151 21 above above ADJ cana-4368 151 22 cases case NOUN cana-4368 151 23 , , PUNCT cana-4368 151 24 all all DET cana-4368 151 25 the the DET cana-4368 151 26 adjacent adjacent ADJ cana-4368 151 27 edges edge NOUN cana-4368 151 28 receive receive VERB cana-4368 151 29 distinct distinct ADJ cana-4368 151 30 odd odd ADJ cana-4368 151 31 labels label NOUN cana-4368 151 32 . . PUNCT cana-4368 152 1 hence hence ADV cana-4368 152 2 the the DET cana-4368 152 3 coconut coconut NOUN cana-4368 152 4 tree tree NOUN cana-4368 152 5 graph graph NOUN cana-4368 152 6 ct(n ct(n NOUN cana-4368 152 7 , , PUNCT cana-4368 152 8 m m NOUN cana-4368 152 9 ) ) PUNCT cana-4368 152 10 admits admit VERB cana-4368 152 11 odd odd ADJ cana-4368 152 12 hamming hamming NOUN cana-4368 152 13 distance distance NOUN cana-4368 152 14 labeling labeling NOUN cana-4368 152 15 and and CCONJ cana-4368 152 16 the the DET cana-4368 152 17 odd odd ADJ cana-4368 152 18 hamming hamming NOUN cana-4368 152 19 distance distance NOUN cana-4368 152 20 number number NOUN cana-4368 152 21 Ξ·β„Žπ‘‘ Ξ·β„Žπ‘‘ NOUN cana-4368 152 22 β€² β€² NUM cana-4368 152 23 ( ( PUNCT cana-4368 152 24 ct(n ct(n X cana-4368 152 25 , , PUNCT cana-4368 152 26 m m NOUN cana-4368 152 27 ) ) PUNCT cana-4368 152 28 ) ) PUNCT cana-4368 153 1 is be AUX cana-4368 153 2 2𝑛 2𝑛 PROPN cana-4368 154 1 + + CCONJ cana-4368 154 2 1 1 NUM cana-4368 154 3 . . NOUN cana-4368 154 4 3 3 NUM cana-4368 154 5 . . X cana-4368 154 6 conclusion conclusion NOUN cana-4368 154 7 in in ADP cana-4368 154 8 this this DET cana-4368 154 9 paper paper NOUN cana-4368 154 10 , , PUNCT cana-4368 154 11 the the DET cana-4368 154 12 odd odd ADJ cana-4368 154 13 hamming hamming NOUN cana-4368 154 14 distance distance NOUN cana-4368 154 15 number number NOUN cana-4368 154 16 of of ADP cana-4368 154 17 some some DET cana-4368 154 18 path path NOUN cana-4368 154 19 related relate VERB cana-4368 154 20 graphs graph NOUN cana-4368 154 21 were be AUX cana-4368 154 22 obtained obtain VERB cana-4368 154 23 . . PUNCT cana-4368 155 1 references reference NOUN cana-4368 155 2 [ [ X cana-4368 155 3 1 1 NUM cana-4368 155 4 ] ] X cana-4368 155 5 esakkiammal esakkiammal ADJ cana-4368 155 6 e e NOUN cana-4368 155 7 , , PUNCT cana-4368 155 8 thirusangu thirusangu PROPN cana-4368 155 9 k k PROPN cana-4368 155 10 , , PUNCT cana-4368 155 11 & & CCONJ cana-4368 155 12 seethalakshmi seethalakshmi PROPN cana-4368 155 13 s. s. PROPN cana-4368 155 14 d d PROPN cana-4368 155 15 - - PUNCT cana-4368 155 16 lucky lucky ADJ cana-4368 155 17 labeling labeling NOUN cana-4368 155 18 of of ADP cana-4368 155 19 arbitrary arbitrary ADJ cana-4368 155 20 super super ADJ cana-4368 155 21 subdivision subdivision NOUN cana-4368 155 22 of of ADP cana-4368 155 23 some some DET cana-4368 155 24 graphs graph NOUN cana-4368 155 25 , , PUNCT cana-4368 155 26 international international ADJ cana-4368 155 27 journal journal NOUN cana-4368 155 28 of of ADP cana-4368 155 29 pure pure ADJ cana-4368 155 30 and and CCONJ cana-4368 155 31 applied apply VERB cana-4368 155 32 mathematics.2017 mathematics.2017 PRON cana-4368 155 33 ; ; PUNCT cana-4368 155 34 113(7 113(7 NUM cana-4368 155 35 ): ): PUNCT cana-4368 155 36 93 93 NUM cana-4368 155 37 - - SYM cana-4368 155 38 101 101 NUM cana-4368 155 39 . . PUNCT cana-4368 156 1 [ [ X cana-4368 156 2 2 2 X cana-4368 156 3 ] ] X cana-4368 156 4 esakkiammal esakkiammal ADJ cana-4368 156 5 e e NOUN cana-4368 156 6 , , PUNCT cana-4368 156 7 thirusangu thirusangu PROPN cana-4368 156 8 k k PROPN cana-4368 156 9 , , PUNCT cana-4368 156 10 & & CCONJ cana-4368 156 11 seethalakshmi seethalakshmi PROPN cana-4368 156 12 s. s. PROPN cana-4368 156 13 lucky lucky PROPN cana-4368 156 14 edge edge PROPN cana-4368 156 15 labeling labeling NOUN cana-4368 156 16 of of ADP cana-4368 156 17 hsuper hsuper ADJ cana-4368 156 18 subdivision subdivision NOUN cana-4368 156 19 of of ADP cana-4368 156 20 some some DET cana-4368 156 21 graphs graph NOUN cana-4368 156 22 , , PUNCT cana-4368 156 23 annuals annual NOUN cana-4368 156 24 of of ADP cana-4368 156 25 pure pure ADJ cana-4368 156 26 and and CCONJ cana-4368 156 27 applied applied ADJ cana-4368 156 28 mathematics mathematic NOUN cana-4368 156 29 . . PUNCT cana-4368 157 1 2017 2017 NUM cana-4368 157 2 ; ; PUNCT cana-4368 157 3 14(3 14(3 NUM cana-4368 157 4 ): ): PUNCT cana-4368 157 5 601 601 NUM cana-4368 157 6 - - SYM cana-4368 157 7 610 610 NUM cana-4368 157 8 . . PUNCT cana-4368 158 1 communications communication NOUN cana-4368 158 2 on on ADP cana-4368 158 3 applied apply VERB cana-4368 158 4 nonlinear nonlinear ADJ cana-4368 158 5 analysis analysis NOUN cana-4368 158 6 issn issn NOUN cana-4368 158 7 : : PUNCT cana-4368 158 8 1074 1074 NUM cana-4368 158 9 - - PUNCT cana-4368 158 10 133x 133x NUM cana-4368 158 11 vol vol NOUN cana-4368 158 12 32 32 NUM cana-4368 158 13 no no NOUN cana-4368 158 14 . . PUNCT cana-4368 159 1 9s 9s NUM cana-4368 159 2 ( ( PUNCT cana-4368 159 3 2025 2025 NUM cana-4368 159 4 ) ) PUNCT cana-4368 159 5 1938 1938 NUM cana-4368 159 6 https://internationalpubls.com https://internationalpubls.com X cana-4368 160 1 [ [ X cana-4368 160 2 3 3 NUM cana-4368 160 3 ] ] PUNCT cana-4368 160 4 gallian gallian PROPN cana-4368 160 5 j.a j.a PROPN cana-4368 160 6 . . PROPN cana-4368 161 1 a a DET cana-4368 161 2 dynamic dynamic ADJ cana-4368 161 3 survey survey NOUN cana-4368 161 4 of of ADP cana-4368 161 5 graph graph NOUN cana-4368 161 6 labeling labeling NOUN cana-4368 161 7 . . PUNCT cana-4368 162 1 the the DET cana-4368 162 2 electronic electronic ADJ cana-4368 162 3 journal journal NOUN cana-4368 162 4 of of ADP cana-4368 162 5 combinatorics combinatoric NOUN cana-4368 162 6 . . PUNCT cana-4368 162 7 2019 2019 NUM cana-4368 162 8 . . PUNCT cana-4368 163 1 ds6 ds6 NOUN cana-4368 163 2 . . PUNCT cana-4368 164 1 [ [ X cana-4368 164 2 4 4 NUM cana-4368 164 3 ] ] X cana-4368 164 4 harary harary PROPN cana-4368 164 5 f. f. PROPN cana-4368 164 6 graph graph PROPN cana-4368 164 7 theory theory NOUN cana-4368 164 8 , , PUNCT cana-4368 164 9 addisonwesley addisonwesley ADJ cana-4368 164 10 publishing publishing NOUN cana-4368 164 11 company company NOUN cana-4368 164 12 , , PUNCT cana-4368 164 13 reading reading NOUN cana-4368 164 14 , , PUNCT cana-4368 164 15 massachussetts massachussetts PROPN cana-4368 164 16 ; ; PUNCT cana-4368 164 17 1972 1972 NUM cana-4368 164 18 . . PUNCT cana-4368 165 1 [ [ X cana-4368 165 2 5 5 NUM cana-4368 165 3 ] ] X cana-4368 165 4 kaneria kaneria PROPN cana-4368 165 5 v.j v.j PROPN cana-4368 165 6 , , PUNCT cana-4368 165 7 meera meera PROPN cana-4368 165 8 meghpara meghpara PROPN cana-4368 165 9 , , PUNCT cana-4368 165 10 graceful graceful ADJ cana-4368 165 11 labeling labeling NOUN cana-4368 165 12 for for ADP cana-4368 165 13 one one NUM cana-4368 165 14 point point NOUN cana-4368 165 15 union union NOUN cana-4368 165 16 for for ADP cana-4368 165 17 path path NOUN cana-4368 165 18 of of ADP cana-4368 165 19 graphs graph NOUN cana-4368 165 20 , , PUNCT cana-4368 165 21 international international ADJ cana-4368 165 22 journal journal NOUN cana-4368 165 23 of of ADP cana-4368 165 24 mathematics mathematic NOUN cana-4368 165 25 and and CCONJ cana-4368 165 26 its its PRON cana-4368 165 27 applications.2015 applications.2015 PROPN cana-4368 165 28 ; ; PUNCT cana-4368 165 29 3(1 3(1 NUM cana-4368 165 30 ): ): PUNCT cana-4368 165 31 49 49 NUM cana-4368 165 32 - - SYM cana-4368 165 33 54 54 NUM cana-4368 165 34 . . PUNCT cana-4368 166 1 [ [ X cana-4368 166 2 6 6 NUM cana-4368 166 3 ] ] PUNCT cana-4368 166 4 ramachandran ramachandran PROPN cana-4368 166 5 v v PROPN cana-4368 166 6 , , PUNCT cana-4368 166 7 sekar.c sekar.c PROPN cana-4368 166 8 , , PUNCT cana-4368 166 9 one one NUM cana-4368 166 10 modulo modulo NOUN cana-4368 166 11 n n DET cana-4368 166 12 gracefulness gracefulness NOUN cana-4368 166 13 of of ADP cana-4368 166 14 regular regular ADJ cana-4368 166 15 bamboo bamboo NOUN cana-4368 166 16 tree tree NOUN cana-4368 166 17 and and CCONJ cana-4368 166 18 coconut coconut NOUN cana-4368 166 19 tree tree NOUN cana-4368 166 20 . . PUNCT cana-4368 167 1 international international ADJ cana-4368 167 2 journal journal PROPN cana-4368 167 3 of of ADP cana-4368 167 4 applications application NOUN cana-4368 167 5 of of ADP cana-4368 167 6 graph graph NOUN cana-4368 167 7 theory theory NOUN cana-4368 167 8 in in ADP cana-4368 167 9 wireless wireless ADJ cana-4368 167 10 and and CCONJ cana-4368 167 11 hoc hoc X cana-4368 167 12 networks network NOUN cana-4368 167 13 and and CCONJ cana-4368 167 14 sensor sensor NOUN cana-4368 167 15 networks network NOUN cana-4368 167 16 . . PUNCT cana-4368 168 1 2014 2014 NUM cana-4368 168 2 ; ; PUNCT cana-4368 168 3 6(2 6(2 NUM cana-4368 168 4 ) ) PUNCT cana-4368 168 5 ; ; PUNCT cana-4368 169 1 1 1 NUM cana-4368 169 2 - - SYM cana-4368 169 3 10 10 NUM cana-4368 169 4 . . PUNCT cana-4368 169 5 doi:10.5121 doi:10.5121 VERB cana-4368 169 6 / / SYM cana-4368 169 7 jgraphoc.2014.6201 jgraphoc.2014.6201 NOUN cana-4368 169 8 . . PUNCT cana-4368 170 1 [ [ X cana-4368 170 2 7 7 X cana-4368 170 3 ] ] PUNCT cana-4368 170 4 shenbagadevi shenbagadevi PROPN cana-4368 170 5 s s PROPN cana-4368 170 6 , , PUNCT cana-4368 170 7 nagarajan nagarajan NOUN cana-4368 170 8 a a DET cana-4368 170 9 , , PUNCT cana-4368 170 10 near near ADP cana-4368 170 11 skolem skolem NOUN cana-4368 170 12 difference difference NOUN cana-4368 170 13 mean mean NOUN cana-4368 170 14 labeling labeling NOUN cana-4368 170 15 of of ADP cana-4368 170 16 special special ADJ cana-4368 170 17 types type NOUN cana-4368 170 18 of of ADP cana-4368 170 19 trees tree NOUN cana-4368 170 20 , , PUNCT cana-4368 170 21 international international ADJ cana-4368 170 22 journal journal NOUN cana-4368 170 23 of of ADP cana-4368 170 24 mathematics mathematics NOUN cana-4368 170 25 trends trend NOUN cana-4368 170 26 and and CCONJ cana-4368 170 27 technology technology NOUN cana-4368 170 28 .2017 .2017 NOUN cana-4368 170 29 ; ; PUNCT cana-4368 170 30 52(7 52(7 NUM cana-4368 170 31 ): ): PUNCT cana-4368 170 32 474 474 NUM cana-4368 170 33 - - SYM cana-4368 170 34 478 478 NUM cana-4368 170 35 . . PUNCT cana-4368 171 1 [ [ X cana-4368 171 2 8 8 NUM cana-4368 171 3 ] ] X cana-4368 171 4 seethalakshmi seethalakshmi PROPN cana-4368 171 5 s s NOUN cana-4368 171 6 , , PUNCT cana-4368 171 7 thirusangu thirusangu NOUN cana-4368 171 8 k k NOUN cana-4368 171 9 , , PUNCT cana-4368 171 10 esakkiammal esakkiammal ADJ cana-4368 171 11 e e NOUN cana-4368 171 12 , , PUNCT cana-4368 171 13 hamming ham VERB cana-4368 171 14 distance distance NOUN cana-4368 171 15 labeling labeling NOUN cana-4368 171 16 of of ADP cana-4368 171 17 certain certain ADJ cana-4368 171 18 graphs graph NOUN cana-4368 171 19 . . PUNCT cana-4368 172 1 journal journal PROPN cana-4368 172 2 of of ADP cana-4368 172 3 tianjin tianjin PROPN cana-4368 172 4 university university PROPN cana-4368 172 5 science science NOUN cana-4368 172 6 and and CCONJ cana-4368 172 7 technology technology NOUN cana-4368 172 8 . . PUNCT cana-4368 173 1 2021;54(10 2021;54(10 NUM cana-4368 173 2 ): ): PUNCT cana-4368 173 3 106 106 NUM cana-4368 173 4 - - SYM cana-4368 173 5 113 113 NUM cana-4368 173 6 . . PUNCT cana-4368 173 7 doi:10.17605 doi:10.17605 NOUN cana-4368 173 8 / / SYM cana-4368 173 9 osf.io osf.io NUM cana-4368 173 10 / / SYM cana-4368 173 11 fma2q fma2q PROPN cana-4368 173 12 . . PUNCT