id sid tid token lemma pos cana-6055 1 1 communications communication NOUN cana-6055 1 2 on on ADP cana-6055 1 3 applied apply VERB cana-6055 1 4 nonlinear nonlinear ADJ cana-6055 1 5 analysis analysis NOUN cana-6055 1 6 issn issn NOUN cana-6055 1 7 : : PUNCT cana-6055 1 8 1074 1074 NUM cana-6055 1 9 - - PUNCT cana-6055 1 10 133x 133x NUM cana-6055 1 11 vol vol NOUN cana-6055 1 12 31 31 NUM cana-6055 1 13 no no NOUN cana-6055 1 14 . . PUNCT cana-6055 2 1 8s 8s PROPN cana-6055 2 2 ( ( PUNCT cana-6055 2 3 2024 2024 NUM cana-6055 2 4 ) ) PUNCT cana-6055 2 5 1155 1155 NUM cana-6055 2 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 2 7 distance distance NOUN cana-6055 2 8 pair pair NOUN cana-6055 2 9 antimagic antimagic ADJ cana-6055 2 10 labeling labeling NOUN cana-6055 2 11 on on ADP cana-6055 2 12 cycle cycle NOUN cana-6055 2 13 related relate VERB cana-6055 2 14 graphs graph NOUN cana-6055 2 15 m. m. PROPN cana-6055 2 16 bala1 bala1 PROPN cana-6055 2 17 , , PUNCT cana-6055 2 18 * * PROPN cana-6055 2 19 , , PUNCT cana-6055 2 20 t. t. PROPN cana-6055 2 21 saratha saratha PROPN cana-6055 2 22 devi2 devi2 PROPN cana-6055 3 1 1research 1research NUM cana-6055 3 2 scholar scholar NOUN cana-6055 3 3 ( ( PUNCT cana-6055 3 4 reg reg NOUN cana-6055 3 5 . . PUNCT cana-6055 4 1 no no INTJ cana-6055 4 2 . . NOUN cana-6055 4 3 20222052091003 20222052091003 NUM cana-6055 4 4 ) ) PUNCT cana-6055 4 5 , , PUNCT cana-6055 4 6 manonmaniam manonmaniam PROPN cana-6055 4 7 sundaranar sundaranar PROPN cana-6055 4 8 university university PROPN cana-6055 4 9 , , PUNCT cana-6055 4 10 abishekapatti abishekapatti VERB cana-6055 4 11 627 627 NUM cana-6055 4 12 012 012 NUM cana-6055 4 13 , , PUNCT cana-6055 4 14 tirunelveli tirunelveli PROPN cana-6055 4 15 , , PUNCT cana-6055 4 16 tamil tamil PROPN cana-6055 4 17 nadu nadu PROPN cana-6055 4 18 , , PUNCT cana-6055 4 19 india india PROPN cana-6055 4 20 . . PUNCT cana-6055 5 1 2department 2department NUM cana-6055 5 2 of of ADP cana-6055 5 3 mathematics mathematic NOUN cana-6055 5 4 , , PUNCT cana-6055 5 5 research research NOUN cana-6055 5 6 center center NOUN cana-6055 5 7 , , PUNCT cana-6055 5 8 g. g. PROPN cana-6055 5 9 venkataswamy venkataswamy PROPN cana-6055 5 10 naidu naidu PROPN cana-6055 5 11 college college PROPN cana-6055 5 12 , , PUNCT cana-6055 5 13 kovilpatti-628 kovilpatti-628 PROPN cana-6055 5 14 502 502 NUM cana-6055 5 15 , , PUNCT cana-6055 5 16 tamil tamil PROPN cana-6055 5 17 nadu nadu PROPN cana-6055 5 18 , , PUNCT cana-6055 5 19 india india PROPN cana-6055 5 20 . . PUNCT cana-6055 6 1 e e X cana-6055 6 2 - - NOUN cana-6055 6 3 mails mail NOUN cana-6055 6 4 : : PUNCT cana-6055 6 5 1 1 NUM cana-6055 6 6 balamaths27@gmail.com balamaths27@gmail.com NOUN cana-6055 6 7 , , PUNCT cana-6055 6 8 2 2 NUM cana-6055 6 9 rajanvino03@gmail.com rajanvino03@gmail.com PROPN cana-6055 6 10 * * PUNCT cana-6055 6 11 corresponding correspond VERB cana-6055 6 12 author author NOUN cana-6055 6 13 : : PUNCT cana-6055 6 14 m. m. NOUN cana-6055 6 15 bala bala PROPN cana-6055 6 16 article article PROPN cana-6055 6 17 history history NOUN cana-6055 6 18 : : PUNCT cana-6055 6 19 received receive VERB cana-6055 6 20 : : PUNCT cana-6055 6 21 10/09/2024 10/09/2024 NUM cana-6055 6 22 revised revise VERB cana-6055 6 23 : : PUNCT cana-6055 6 24 27/10/2024 27/10/2024 NUM cana-6055 6 25 published publish VERB cana-6055 6 26 : : PUNCT cana-6055 6 27 14/11/2024 14/11/2024 NUM cana-6055 6 28 abstract abstract NOUN cana-6055 6 29 : : PUNCT cana-6055 6 30 a a DET cana-6055 6 31 distance distance NOUN cana-6055 6 32 pair pair NOUN cana-6055 6 33 antimagic antimagic ADJ cana-6055 6 34 labeling labeling NOUN cana-6055 6 35 of of ADP cana-6055 6 36 a a DET cana-6055 6 37 graph graph NOUN cana-6055 6 38 𝐺 𝐺 NOUN cana-6055 6 39 with with ADP cana-6055 6 40 𝑝 𝑝 PROPN cana-6055 6 41 vertices vertex NOUN cana-6055 6 42 is be AUX cana-6055 6 43 a a DET cana-6055 6 44 bijection bijection NOUN cana-6055 6 45 𝑓 𝑓 PRON cana-6055 6 46 : : PUNCT cana-6055 6 47 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 6 48 ) ) PUNCT cana-6055 6 49 ⟢ ⟢ NOUN cana-6055 6 50 𝑃 𝑃 NOUN cana-6055 6 51 where where SCONJ cana-6055 6 52 𝑃 𝑃 NOUN cana-6055 6 53 = = SYM cana-6055 6 54 { { PUNCT cana-6055 6 55 Β±1,Β±2,β‹― Β±1,Β±2,β‹― X cana-6055 6 56 , , PUNCT cana-6055 6 57 Β± Β± PROPN cana-6055 6 58 𝑝 𝑝 NOUN cana-6055 6 59 2 2 NUM cana-6055 6 60 , , PUNCT cana-6055 6 61 𝑖𝑓 𝑖𝑓 PROPN cana-6055 6 62 𝑝 𝑝 NOUN cana-6055 6 63 𝑖𝑠 𝑖𝑠 NOUN cana-6055 6 64 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-6055 6 65 0 0 NUM cana-6055 6 66 , , PUNCT cana-6055 6 67 Β±1 Β±1 VERB cana-6055 6 68 , , PUNCT cana-6055 6 69 Β±2,β‹― Β±2,β‹― NUM cana-6055 6 70 , , PUNCT cana-6055 6 71 Β± Β± NUM cana-6055 6 72 π‘βˆ’1 π‘βˆ’1 PROPN cana-6055 6 73 2 2 NUM cana-6055 6 74 , , PUNCT cana-6055 6 75 𝑖𝑓 𝑖𝑓 PROPN cana-6055 6 76 𝑝 𝑝 NOUN cana-6055 6 77 𝑖𝑠 𝑖𝑠 NOUN cana-6055 6 78 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-6055 6 79 such such ADJ cana-6055 6 80 that that SCONJ cana-6055 6 81 the the DET cana-6055 6 82 induced induced ADJ cana-6055 6 83 weight weight NOUN cana-6055 6 84 function function NOUN cana-6055 6 85 𝑀 𝑀 ADP cana-6055 6 86 : : PUNCT cana-6055 6 87 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 6 88 ) ) PUNCT cana-6055 6 89 ⟢ ⟢ NOUN cana-6055 6 90 π‘Š π‘Š NOUN cana-6055 6 91 defined define VERB cana-6055 6 92 by by ADP cana-6055 6 93 𝑀(𝑣 𝑀(𝑣 PROPN cana-6055 6 94 ) ) PUNCT cana-6055 6 95 = = PUNCT cana-6055 6 96 βˆ‘ βˆ‘ PUNCT cana-6055 6 97 𝑓(𝑒 𝑓(𝑒 PROPN cana-6055 6 98 ) ) PUNCT cana-6055 6 99 = = SYM cana-6055 6 100 π‘˜π‘–π‘’βˆˆπ‘(𝑣 π‘˜π‘–π‘’βˆˆπ‘(𝑣 X cana-6055 6 101 ) ) PUNCT cana-6055 6 102 is be AUX cana-6055 6 103 one one NUM cana-6055 6 104 - - PUNCT cana-6055 6 105 one one NUM cana-6055 6 106 , , PUNCT cana-6055 6 107 where where SCONJ cana-6055 6 108 𝑁(𝑣 𝑁(𝑣 X cana-6055 6 109 ) ) PUNCT cana-6055 6 110 = = SYM cana-6055 6 111 { { PUNCT cana-6055 6 112 𝑒 𝑒 PRON cana-6055 6 113 ∈ ∈ PROPN cana-6055 6 114 𝑉 𝑉 PROPN cana-6055 6 115 : : PUNCT cana-6055 6 116 𝑒𝑣 𝑒𝑣 NOUN cana-6055 6 117 ∈ ∈ PROPN cana-6055 6 118 𝐸 𝐸 PROPN cana-6055 6 119 } } PUNCT cana-6055 6 120 is be AUX cana-6055 6 121 the the DET cana-6055 6 122 open open ADJ cana-6055 6 123 neighborhood neighborhood NOUN cana-6055 6 124 of of ADP cana-6055 6 125 𝑣 𝑣 PROPN cana-6055 6 126 and and CCONJ cana-6055 6 127 the the DET cana-6055 6 128 set set NOUN cana-6055 6 129 of of ADP cana-6055 6 130 all all DET cana-6055 6 131 weights weight NOUN cana-6055 6 132 π‘Š π‘Š NOUN cana-6055 6 133 is be AUX cana-6055 6 134 either either PRON cana-6055 6 135 of of ADP cana-6055 6 136 the the DET cana-6055 6 137 form form NOUN cana-6055 6 138 { { PUNCT cana-6055 6 139 Β±π‘˜1 Β±π‘˜1 NOUN cana-6055 6 140 , , PUNCT cana-6055 6 141 Β±π‘˜2 Β±π‘˜2 PROPN cana-6055 6 142 , , PUNCT cana-6055 6 143 Β±π‘˜3 Β±π‘˜3 PROPN cana-6055 6 144 , , PUNCT cana-6055 6 145 β‹― β‹― PROPN cana-6055 6 146 , , PUNCT cana-6055 6 147 Β±π‘˜π‘ Β±π‘˜π‘ NOUN cana-6055 6 148 2 2 NUM cana-6055 6 149 } } PUNCT cana-6055 6 150 or or CCONJ cana-6055 6 151 { { PUNCT cana-6055 6 152 0 0 NUM cana-6055 6 153 , , PUNCT cana-6055 6 154 Β±π‘˜1 Β±π‘˜1 NOUN cana-6055 6 155 , , PUNCT cana-6055 6 156 Β±π‘˜2 Β±π‘˜2 NOUN cana-6055 6 157 , , PUNCT cana-6055 6 158 Β±π‘˜3 Β±π‘˜3 PROPN cana-6055 6 159 , , PUNCT cana-6055 6 160 β‹― β‹― PROPN cana-6055 6 161 , , PUNCT cana-6055 6 162 Β±π‘˜π‘βˆ’1 Β±π‘˜π‘βˆ’1 PROPN cana-6055 6 163 2 2 NUM cana-6055 6 164 } } PUNCT cana-6055 6 165 according accord VERB cana-6055 6 166 as as SCONJ cana-6055 6 167 𝑝 𝑝 NOUN cana-6055 6 168 is be AUX cana-6055 6 169 even even ADV cana-6055 6 170 or or CCONJ cana-6055 6 171 odd odd ADJ cana-6055 6 172 . . PUNCT cana-6055 7 1 in in ADP cana-6055 7 2 this this DET cana-6055 7 3 paper paper NOUN cana-6055 7 4 , , PUNCT cana-6055 7 5 we we PRON cana-6055 7 6 explored explore VERB cana-6055 7 7 the the DET cana-6055 7 8 result result NOUN cana-6055 7 9 on on ADP cana-6055 7 10 distance distance NOUN cana-6055 7 11 pair pair NOUN cana-6055 7 12 antimagic antimagic ADJ cana-6055 7 13 labeling labeling NOUN cana-6055 7 14 of of ADP cana-6055 7 15 cycle cycle NOUN cana-6055 7 16 related relate VERB cana-6055 7 17 graphs graph NOUN cana-6055 7 18 . . PUNCT cana-6055 8 1 also also ADV cana-6055 8 2 we we PRON cana-6055 8 3 investigated investigate VERB cana-6055 8 4 the the DET cana-6055 8 5 closed closed ADJ cana-6055 8 6 distance distance NOUN cana-6055 8 7 magic magic ADJ cana-6055 8 8 labeling labeling NOUN cana-6055 8 9 of of ADP cana-6055 8 10 circulant circulant ADJ cana-6055 8 11 graph graph NOUN cana-6055 8 12 and and CCONJ cana-6055 8 13 its its PRON cana-6055 8 14 complement complement NOUN cana-6055 8 15 . . PUNCT cana-6055 9 1 keywords keyword NOUN cana-6055 9 2 : : PUNCT cana-6055 9 3 graph graph NOUN cana-6055 9 4 labeling labeling NOUN cana-6055 9 5 , , PUNCT cana-6055 9 6 distance distance NOUN cana-6055 9 7 antimagic antimagic NOUN cana-6055 9 8 , , PUNCT cana-6055 9 9 pair pair NOUN cana-6055 9 10 sum sum NOUN cana-6055 9 11 labeling labeling NOUN cana-6055 9 12 , , PUNCT cana-6055 9 13 distance distance NOUN cana-6055 9 14 pair pair NOUN cana-6055 9 15 antimagic antimagic ADJ cana-6055 9 16 labeling labeling NOUN cana-6055 9 17 . . PUNCT cana-6055 10 1 ams am NOUN cana-6055 10 2 subject subject PROPN cana-6055 10 3 classification(2010 classification(2010 PROPN cana-6055 10 4 ): ): PUNCT cana-6055 10 5 05c12 05c12 NOUN cana-6055 10 6 , , PUNCT cana-6055 10 7 05c78 05c78 NUM cana-6055 10 8 . . PROPN cana-6055 11 1 1 1 X cana-6055 11 2 . . X cana-6055 11 3 introduction introduction NOUN cana-6055 11 4 a a DET cana-6055 11 5 magic magic ADJ cana-6055 11 6 square square NOUN cana-6055 11 7 of of ADP cana-6055 11 8 order order NOUN cana-6055 11 9 𝑛 𝑛 NOUN cana-6055 11 10 is be AUX cana-6055 11 11 an an DET cana-6055 11 12 𝑛 𝑛 ADJ cana-6055 11 13 Γ— Γ— NOUN cana-6055 11 14 𝑛 𝑛 PROPN cana-6055 11 15 array array NOUN cana-6055 11 16 whose whose DET cana-6055 11 17 entries entry NOUN cana-6055 11 18 are be AUX cana-6055 11 19 an an DET cana-6055 11 20 arrangement arrangement NOUN cana-6055 11 21 of of ADP cana-6055 11 22 the the DET cana-6055 11 23 integers integer NOUN cana-6055 11 24 1,2,3 1,2,3 NUM cana-6055 11 25 , , PUNCT cana-6055 11 26 … … PUNCT cana-6055 11 27 , , PUNCT cana-6055 11 28 𝑛2 𝑛2 NOUN cana-6055 11 29 in in ADP cana-6055 11 30 which which PRON cana-6055 11 31 all all DET cana-6055 11 32 elements element NOUN cana-6055 11 33 in in ADP cana-6055 11 34 any any DET cana-6055 11 35 row row NOUN cana-6055 11 36 , , PUNCT cana-6055 11 37 any any DET cana-6055 11 38 column column NOUN cana-6055 11 39 , , PUNCT cana-6055 11 40 the the DET cana-6055 11 41 main main ADJ cana-6055 11 42 diagonal diagonal NOUN cana-6055 11 43 or or CCONJ cana-6055 11 44 the the DET cana-6055 11 45 main main ADJ cana-6055 11 46 back back NOUN cana-6055 11 47 diagonal diagonal ADJ cana-6055 11 48 add add VERB cana-6055 11 49 to to ADP cana-6055 11 50 the the DET cana-6055 11 51 same same ADJ cana-6055 11 52 sum sum NOUN cana-6055 11 53 π‘Ÿ. π‘Ÿ. NOUN cana-6055 11 54 vilfred vilfre VERB cana-6055 11 55 [ [ X cana-6055 11 56 9 9 NUM cana-6055 11 57 ] ] PUNCT cana-6055 11 58 in in ADP cana-6055 11 59 his his PRON cana-6055 11 60 doctoral doctoral ADJ cana-6055 11 61 thesis thesis NOUN cana-6055 11 62 introduced introduce VERB cana-6055 11 63 the the DET cana-6055 11 64 concept concept NOUN cana-6055 11 65 of of ADP cana-6055 11 66 Οƒ Οƒ NOUN cana-6055 11 67 - - NOUN cana-6055 11 68 labeling labeling NOUN cana-6055 11 69 . . PUNCT cana-6055 12 1 following follow VERB cana-6055 12 2 this this PRON cana-6055 12 3 , , PUNCT cana-6055 12 4 miller miller PROPN cana-6055 12 5 et et PROPN cana-6055 12 6 al al PROPN cana-6055 12 7 . . PUNCT cana-6055 13 1 [ [ X cana-6055 13 2 5 5 NUM cana-6055 13 3 ] ] PUNCT cana-6055 13 4 and and CCONJ cana-6055 13 5 b.d b.d PROPN cana-6055 13 6 . . PROPN cana-6055 13 7 acharya acharya PROPN cana-6055 13 8 et et PROPN cana-6055 13 9 al al PROPN cana-6055 14 1 [ [ X cana-6055 14 2 1 1 X cana-6055 14 3 ] ] PUNCT cana-6055 14 4 studied study VERB cana-6055 14 5 the the DET cana-6055 14 6 concepts concept NOUN cana-6055 14 7 under under ADP cana-6055 14 8 the the DET cana-6055 14 9 name name NOUN cana-6055 14 10 of of ADP cana-6055 14 11 neighborhood neighborhood NOUN cana-6055 14 12 magic magic NOUN cana-6055 14 13 graphs graph NOUN cana-6055 14 14 . . PUNCT cana-6055 15 1 further far ADV cana-6055 15 2 , , PUNCT cana-6055 15 3 sugeng sugeng PROPN cana-6055 15 4 et et PROPN cana-6055 15 5 al al PROPN cana-6055 15 6 . . PUNCT cana-6055 16 1 [ [ X cana-6055 16 2 7 7 X cana-6055 16 3 ] ] PUNCT cana-6055 16 4 used use VERB cana-6055 16 5 the the DET cana-6055 16 6 term term NOUN cana-6055 16 7 distance distance NOUN cana-6055 16 8 magic magic ADJ cana-6055 16 9 labeling labeling NOUN cana-6055 16 10 for for ADP cana-6055 16 11 the the DET cana-6055 16 12 same same ADJ cana-6055 16 13 concept concept NOUN cana-6055 16 14 . . PUNCT cana-6055 17 1 a a DET cana-6055 17 2 distance distance NOUN cana-6055 17 3 magic magic ADJ cana-6055 17 4 labeling labeling NOUN cana-6055 17 5 of of ADP cana-6055 17 6 a a DET cana-6055 17 7 graph graph NOUN cana-6055 17 8 𝐺 𝐺 NOUN cana-6055 17 9 = = SYM cana-6055 17 10 ( ( PUNCT cana-6055 17 11 𝑉 𝑉 PROPN cana-6055 17 12 , , PUNCT cana-6055 17 13 𝐸 𝐸 PROPN cana-6055 17 14 ) ) PUNCT cana-6055 17 15 of of ADP cana-6055 17 16 order order NOUN cana-6055 17 17 𝑛 𝑛 NOUN cana-6055 17 18 is be AUX cana-6055 17 19 a a DET cana-6055 17 20 bijection bijection NOUN cana-6055 17 21 𝑓 𝑓 PRON cana-6055 17 22 : : PUNCT cana-6055 17 23 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 17 24 ) ) PUNCT cana-6055 17 25 β†’ β†’ SYM cana-6055 17 26 { { PUNCT cana-6055 17 27 1,2 1,2 NUM cana-6055 17 28 , , PUNCT cana-6055 17 29 … … PUNCT cana-6055 17 30 , , PUNCT cana-6055 17 31 𝑛 𝑛 X cana-6055 17 32 } } PUNCT cana-6055 17 33 such such ADJ cana-6055 17 34 that that SCONJ cana-6055 17 35 οƒ₯ οƒ₯ X cana-6055 17 36 οƒŽ οƒŽ NOUN cana-6055 17 37 = = PUNCT cana-6055 17 38 ) ) PUNCT cana-6055 17 39 ( ( PUNCT cana-6055 17 40 ) ) PUNCT cana-6055 17 41 ( ( PUNCT cana-6055 17 42 vnu vnu ADV cana-6055 17 43 kuf kuf VERB cana-6055 17 44 for for ADP cana-6055 17 45 all all PRON cana-6055 17 46 𝑣 𝑣 PRON cana-6055 17 47 ∈ ∈ PROPN cana-6055 17 48 𝑉. 𝑉. NOUN cana-6055 17 49 the the DET cana-6055 17 50 constant constant ADJ cana-6055 17 51 π‘˜ π‘˜ NOUN cana-6055 17 52 is be AUX cana-6055 17 53 called call VERB cana-6055 17 54 the the DET cana-6055 17 55 magic magic ADJ cana-6055 17 56 constant constant NOUN cana-6055 17 57 of of ADP cana-6055 17 58 the the DET cana-6055 17 59 labeling labeling NOUN cana-6055 17 60 𝑓. 𝑓. NOUN cana-6055 17 61 a a DET cana-6055 17 62 graph graph NOUN cana-6055 17 63 which which PRON cana-6055 17 64 admits admit VERB cana-6055 17 65 a a DET cana-6055 17 66 distance distance NOUN cana-6055 17 67 magic magic ADJ cana-6055 17 68 labeling labeling NOUN cana-6055 17 69 is be AUX cana-6055 17 70 called call VERB cana-6055 17 71 a a DET cana-6055 17 72 distance distance NOUN cana-6055 17 73 magic magic NOUN cana-6055 17 74 graph graph NOUN cana-6055 17 75 . . PUNCT cana-6055 18 1 in in ADP cana-6055 18 2 2013 2013 NUM cana-6055 18 3 , , PUNCT cana-6055 18 4 kamatchi kamatchi PROPN cana-6055 18 5 and and CCONJ cana-6055 18 6 arumugam arumugam ADJ cana-6055 18 7 [ [ X cana-6055 18 8 3 3 X cana-6055 18 9 ] ] PUNCT cana-6055 18 10 introduced introduce VERB cana-6055 18 11 the the DET cana-6055 18 12 concept concept NOUN cana-6055 18 13 of of ADP cana-6055 18 14 a a DET cana-6055 18 15 distance distance NOUN cana-6055 18 16 antimagic antimagic NOUN cana-6055 18 17 graph graph NOUN cana-6055 18 18 . . PUNCT cana-6055 19 1 motivated motivate VERB cana-6055 19 2 by by ADP cana-6055 19 3 these these DET cana-6055 19 4 works work NOUN cana-6055 19 5 and and CCONJ cana-6055 19 6 pair pair NOUN cana-6055 19 7 sum sum NOUN cana-6055 19 8 labeling labeling NOUN cana-6055 19 9 [ [ X cana-6055 19 10 6 6 NUM cana-6055 19 11 ] ] PUNCT cana-6055 19 12 we we PRON cana-6055 19 13 introduced introduce VERB cana-6055 19 14 the the DET cana-6055 19 15 concept concept NOUN cana-6055 19 16 of of ADP cana-6055 19 17 distance distance NOUN cana-6055 19 18 pair pair NOUN cana-6055 19 19 antimagic antimagic ADJ cana-6055 19 20 labeling labeling NOUN cana-6055 19 21 [ [ X cana-6055 19 22 4 4 NUM cana-6055 19 23 ] ] PUNCT cana-6055 19 24 . . PUNCT cana-6055 20 1 in in ADP cana-6055 20 2 this this DET cana-6055 20 3 paper paper NOUN cana-6055 20 4 , , PUNCT cana-6055 20 5 we we PRON cana-6055 20 6 present present VERB cana-6055 20 7 several several ADJ cana-6055 20 8 results result NOUN cana-6055 20 9 of of ADP cana-6055 20 10 distance distance NOUN cana-6055 20 11 pair pair NOUN cana-6055 20 12 antimagic antimagic ADJ cana-6055 20 13 labeling labeling NOUN cana-6055 20 14 on on ADP cana-6055 20 15 cycle cycle NOUN cana-6055 20 16 related relate VERB cana-6055 20 17 graphs graph NOUN cana-6055 20 18 and and CCONJ cana-6055 20 19 complement complement NOUN cana-6055 20 20 of of ADP cana-6055 20 21 circulant circulant ADJ cana-6055 20 22 graph graph NOUN cana-6055 20 23 . . PUNCT cana-6055 21 1 2 2 X cana-6055 21 2 . . X cana-6055 21 3 preliminaries preliminary NOUN cana-6055 21 4 definition definition NOUN cana-6055 21 5 2.1 2.1 NUM cana-6055 21 6 a a DET cana-6055 21 7 graph graph NOUN cana-6055 21 8 g g NOUN cana-6055 21 9 is be AUX cana-6055 21 10 said say VERB cana-6055 21 11 to to PART cana-6055 21 12 be be AUX cana-6055 21 13 distance distance NOUN cana-6055 21 14 antimagic antimagic NOUN cana-6055 21 15 if if SCONJ cana-6055 21 16 there there PRON cana-6055 21 17 is be VERB cana-6055 21 18 a a DET cana-6055 21 19 bijection bijection NOUN cana-6055 21 20 𝑓 𝑓 PRON cana-6055 21 21 : : PUNCT cana-6055 21 22 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 21 23 ) ) PUNCT cana-6055 21 24 β†’ β†’ SYM cana-6055 21 25 { { PUNCT cana-6055 21 26 1,2 1,2 NUM cana-6055 21 27 , , PUNCT cana-6055 21 28 . . PUNCT cana-6055 21 29 . . PUNCT cana-6055 22 1 . . PUNCT cana-6055 23 1 , , PUNCT cana-6055 23 2 𝑝 𝑝 X cana-6055 23 3 } } PUNCT cana-6055 23 4 such such ADJ cana-6055 23 5 that that PRON cana-6055 23 6 for for ADP cana-6055 23 7 every every DET cana-6055 23 8 pair pair NOUN cana-6055 23 9 of of ADP cana-6055 23 10 distinct distinct ADJ cana-6055 23 11 vertices vertex NOUN cana-6055 23 12 π‘₯ π‘₯ PROPN cana-6055 23 13 and and CCONJ cana-6055 23 14 𝑦 𝑦 NOUN cana-6055 23 15 applies apply VERB cana-6055 23 16 𝑀(π‘₯ 𝑀(π‘₯ NOUN cana-6055 23 17 ) ) PUNCT cana-6055 23 18 β‰  β‰  PROPN cana-6055 23 19 𝑀(𝑦 𝑀(𝑦 ADV cana-6055 23 20 ) ) PUNCT cana-6055 23 21 . . PUNCT cana-6055 24 1 mailto:rajanvino03@gmail.com mailto:rajanvino03@gmail.com PROPN cana-6055 24 2 communications communication NOUN cana-6055 24 3 on on ADP cana-6055 24 4 applied apply VERB cana-6055 24 5 nonlinear nonlinear ADJ cana-6055 24 6 analysis analysis NOUN cana-6055 24 7 issn issn NOUN cana-6055 24 8 : : PUNCT cana-6055 24 9 1074 1074 NUM cana-6055 24 10 - - PUNCT cana-6055 24 11 133x 133x NUM cana-6055 24 12 vol vol NOUN cana-6055 24 13 31 31 NUM cana-6055 24 14 no no NOUN cana-6055 24 15 . . PUNCT cana-6055 25 1 8s 8s PROPN cana-6055 25 2 ( ( PUNCT cana-6055 25 3 2024 2024 NUM cana-6055 25 4 ) ) PUNCT cana-6055 25 5 1156 1156 NUM cana-6055 25 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 25 7 definition definition NOUN cana-6055 25 8 2.2 2.2 NUM cana-6055 25 9 a a DET cana-6055 25 10 injective injective ADJ cana-6055 25 11 map map NOUN cana-6055 25 12 𝑓 𝑓 NOUN cana-6055 25 13 : : PUNCT cana-6055 25 14 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 25 15 ) ) PUNCT cana-6055 25 16 β†’ β†’ SYM cana-6055 25 17 { { PUNCT cana-6055 25 18 Β±1,Β±2,β‹― Β±1,Β±2,β‹― X cana-6055 25 19 , , PUNCT cana-6055 25 20 ±𝑝 ±𝑝 NUM cana-6055 25 21 } } PUNCT cana-6055 25 22 is be AUX cana-6055 25 23 said say VERB cana-6055 25 24 to to PART cana-6055 25 25 be be AUX cana-6055 25 26 pair pair NOUN cana-6055 25 27 sum sum NOUN cana-6055 25 28 labeling labeling NOUN cana-6055 25 29 if if SCONJ cana-6055 25 30 the the DET cana-6055 25 31 induced induced ADJ cana-6055 25 32 edge edge NOUN cana-6055 25 33 function function NOUN cana-6055 25 34 𝑓𝑒 𝑓𝑒 PROPN cana-6055 25 35 : : PUNCT cana-6055 25 36 𝐸(𝐺 𝐸(𝐺 PROPN cana-6055 25 37 ) ) PUNCT cana-6055 25 38 β†’ β†’ SYM cana-6055 25 39 𝑍\{0 𝑍\{0 PROPN cana-6055 25 40 } } PUNCT cana-6055 25 41 defined define VERB cana-6055 25 42 by by ADP cana-6055 25 43 𝑓𝑒(𝑒𝑣 𝑓𝑒(𝑒𝑣 ADJ cana-6055 25 44 ) ) PUNCT cana-6055 25 45 = = PUNCT cana-6055 25 46 𝑓(𝑒 𝑓(𝑒 PROPN cana-6055 25 47 ) ) PUNCT cana-6055 26 1 + + CCONJ cana-6055 26 2 𝑓(𝑣 𝑓(𝑣 NOUN cana-6055 26 3 ) ) PUNCT cana-6055 26 4 is be AUX cana-6055 26 5 one one NUM cana-6055 26 6 - - PUNCT cana-6055 26 7 one one NUM cana-6055 26 8 and and CCONJ cana-6055 26 9 𝑓𝑒(𝐸(𝐺 𝑓𝑒(𝐸(𝐺 NOUN cana-6055 26 10 ) ) PUNCT cana-6055 26 11 ) ) PUNCT cana-6055 27 1 is be AUX cana-6055 27 2 either either PRON cana-6055 27 3 of of ADP cana-6055 27 4 the the DET cana-6055 27 5 form form NOUN cana-6055 27 6 { { PUNCT cana-6055 27 7 Β±π‘˜1 Β±π‘˜1 NOUN cana-6055 27 8 , , PUNCT cana-6055 27 9 Β±π‘˜2 Β±π‘˜2 PROPN cana-6055 27 10 , , PUNCT cana-6055 27 11 Β±π‘˜3 Β±π‘˜3 PROPN cana-6055 27 12 , , PUNCT cana-6055 27 13 β‹― β‹― PROPN cana-6055 27 14 , , PUNCT cana-6055 27 15 Β±π‘˜π‘ž Β±π‘˜π‘ž NOUN cana-6055 27 16 2 2 NUM cana-6055 27 17 } } PUNCT cana-6055 27 18 or or CCONJ cana-6055 27 19 { { PUNCT cana-6055 27 20 Β±π‘˜1 Β±π‘˜1 NOUN cana-6055 27 21 , , PUNCT cana-6055 27 22 Β±π‘˜2 Β±π‘˜2 PROPN cana-6055 27 23 , , PUNCT cana-6055 27 24 Β±π‘˜3 Β±π‘˜3 PROPN cana-6055 27 25 , , PUNCT cana-6055 27 26 β‹― β‹― PROPN cana-6055 27 27 , , PUNCT cana-6055 27 28 Β±π‘˜π‘žβˆ’1 Β±π‘˜π‘žβˆ’1 PROPN cana-6055 27 29 2 2 NUM cana-6055 27 30 } } PUNCT cana-6055 27 31 βˆͺ βˆͺ ADJ cana-6055 27 32 { { PUNCT cana-6055 27 33 Β±π‘˜π‘ž+1 Β±π‘˜π‘ž+1 ADV cana-6055 27 34 2 2 NUM cana-6055 27 35 } } PUNCT cana-6055 27 36 according accord VERB cana-6055 27 37 as as SCONJ cana-6055 27 38 q q NOUN cana-6055 27 39 is be AUX cana-6055 27 40 even even ADV cana-6055 27 41 or or CCONJ cana-6055 27 42 odd odd ADJ cana-6055 27 43 . . PUNCT cana-6055 28 1 definition definition NOUN cana-6055 28 2 2.3 2.3 NUM cana-6055 28 3 let let VERB cana-6055 28 4 𝐺 𝐺 PROPN cana-6055 28 5 be be AUX cana-6055 28 6 a a DET cana-6055 28 7 ( ( PUNCT cana-6055 28 8 𝑝 𝑝 PROPN cana-6055 28 9 , , PUNCT cana-6055 28 10 π‘ž π‘ž NOUN cana-6055 28 11 ) ) PUNCT cana-6055 28 12 graph graph NOUN cana-6055 28 13 . . PUNCT cana-6055 29 1 let let VERB cana-6055 29 2 𝑓 𝑓 PRON cana-6055 29 3 : : PUNCT cana-6055 29 4 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 29 5 ) ) PUNCT cana-6055 29 6 ⟢ ⟢ NOUN cana-6055 29 7 𝑃 𝑃 NOUN cana-6055 29 8 be be VERB cana-6055 29 9 a a DET cana-6055 29 10 bijection bijection NOUN cana-6055 29 11 where where SCONJ cana-6055 29 12 𝑃 𝑃 NOUN cana-6055 29 13 = = SYM cana-6055 29 14 { { PUNCT cana-6055 29 15 Β±1 Β±1 PROPN cana-6055 29 16 , , PUNCT cana-6055 29 17 Β±2,β‹― Β±2,β‹― NUM cana-6055 29 18 , , PUNCT cana-6055 29 19 Β± Β± PROPN cana-6055 29 20 𝑝 𝑝 NOUN cana-6055 29 21 2 2 NUM cana-6055 29 22 , , PUNCT cana-6055 29 23 𝑖𝑓 𝑖𝑓 PROPN cana-6055 29 24 𝑝 𝑝 NOUN cana-6055 29 25 𝑖𝑠 𝑖𝑠 NOUN cana-6055 29 26 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-6055 29 27 0 0 NUM cana-6055 29 28 , , PUNCT cana-6055 29 29 Β±1 Β±1 VERB cana-6055 29 30 , , PUNCT cana-6055 29 31 Β±2,β‹― Β±2,β‹― NUM cana-6055 29 32 , , PUNCT cana-6055 29 33 Β± Β± NUM cana-6055 29 34 π‘βˆ’1 π‘βˆ’1 PROPN cana-6055 29 35 2 2 NUM cana-6055 29 36 , , PUNCT cana-6055 29 37 𝑖𝑓 𝑖𝑓 PROPN cana-6055 29 38 𝑝 𝑝 NOUN cana-6055 29 39 𝑖𝑠 𝑖𝑠 PROPN cana-6055 29 40 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-6055 30 1 then then ADV cana-6055 30 2 𝑓 𝑓 PRON cana-6055 30 3 is be AUX cana-6055 30 4 called call VERB cana-6055 30 5 a a DET cana-6055 30 6 distance distance NOUN cana-6055 30 7 pair pair NOUN cana-6055 30 8 antimagic antimagic NOUN cana-6055 30 9 ( ( PUNCT cana-6055 30 10 dpam dpam NOUN cana-6055 30 11 ) ) PUNCT cana-6055 30 12 labeling labeling NOUN cana-6055 30 13 if if SCONJ cana-6055 30 14 the the DET cana-6055 30 15 induced induce VERB cana-6055 30 16 weight weight NOUN cana-6055 30 17 function function NOUN cana-6055 30 18 𝑀 𝑀 ADP cana-6055 30 19 : : PUNCT cana-6055 30 20 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 30 21 ) ) PUNCT cana-6055 30 22 ⟢ ⟢ NOUN cana-6055 30 23 π‘Š π‘Š NOUN cana-6055 30 24 defined define VERB cana-6055 30 25 by by ADP cana-6055 30 26 𝑀(𝑣 𝑀(𝑣 PROPN cana-6055 30 27 ) ) PUNCT cana-6055 31 1 = = SYM cana-6055 31 2 βˆ‘π‘’βˆˆπ‘(𝑣 βˆ‘π‘’βˆˆπ‘(𝑣 PROPN cana-6055 31 3 ) ) PUNCT cana-6055 31 4 𝑓(𝑣 𝑓(𝑣 NOUN cana-6055 31 5 ) ) PUNCT cana-6055 31 6 = = PRON cana-6055 31 7 π‘˜π‘– π‘˜π‘– NOUN cana-6055 31 8 is be AUX cana-6055 31 9 one one NUM cana-6055 31 10 - - PUNCT cana-6055 31 11 one one NUM cana-6055 31 12 , , PUNCT cana-6055 31 13 where where SCONJ cana-6055 31 14 𝑁(𝑣 𝑁(𝑣 X cana-6055 31 15 ) ) PUNCT cana-6055 31 16 = = SYM cana-6055 31 17 { { PUNCT cana-6055 31 18 𝑒 𝑒 PRON cana-6055 31 19 ∈ ∈ PROPN cana-6055 31 20 𝑉 𝑉 PROPN cana-6055 31 21 : : PUNCT cana-6055 31 22 𝑒𝑣 𝑒𝑣 NOUN cana-6055 31 23 ∈ ∈ PROPN cana-6055 31 24 𝐸 𝐸 PROPN cana-6055 31 25 } } PUNCT cana-6055 31 26 is be AUX cana-6055 31 27 the the DET cana-6055 31 28 open open ADJ cana-6055 31 29 neighborhood neighborhood NOUN cana-6055 31 30 of of ADP cana-6055 31 31 𝑣 𝑣 PROPN cana-6055 31 32 and and CCONJ cana-6055 31 33 the the DET cana-6055 31 34 set set NOUN cana-6055 31 35 of of ADP cana-6055 31 36 all all DET cana-6055 31 37 weights weight NOUN cana-6055 31 38 π‘Š π‘Š NOUN cana-6055 31 39 is be AUX cana-6055 31 40 either either PRON cana-6055 31 41 of of ADP cana-6055 31 42 the the DET cana-6055 31 43 form form NOUN cana-6055 31 44 { { PUNCT cana-6055 31 45 Β±π‘˜1 Β±π‘˜1 NOUN cana-6055 31 46 , , PUNCT cana-6055 31 47 Β±π‘˜2 Β±π‘˜2 PROPN cana-6055 31 48 , , PUNCT cana-6055 31 49 Β±π‘˜3 Β±π‘˜3 PROPN cana-6055 31 50 , , PUNCT cana-6055 31 51 β‹― β‹― PROPN cana-6055 31 52 , , PUNCT cana-6055 31 53 Β±π‘˜π‘ Β±π‘˜π‘ NOUN cana-6055 31 54 2 2 NUM cana-6055 31 55 } } PUNCT cana-6055 31 56 or or CCONJ cana-6055 31 57 { { PUNCT cana-6055 31 58 0,Β±π‘˜1 0,Β±π‘˜1 ADJ cana-6055 31 59 , , PUNCT cana-6055 31 60 Β±π‘˜2 Β±π‘˜2 PROPN cana-6055 31 61 , , PUNCT cana-6055 31 62 Β±π‘˜3 Β±π‘˜3 PROPN cana-6055 31 63 , , PUNCT cana-6055 31 64 β‹― β‹― PROPN cana-6055 31 65 , , PUNCT cana-6055 31 66 Β±π‘˜π‘βˆ’1 Β±π‘˜π‘βˆ’1 PROPN cana-6055 31 67 2 2 NUM cana-6055 31 68 } } PUNCT cana-6055 31 69 according accord VERB cana-6055 31 70 as as SCONJ cana-6055 31 71 𝑝 𝑝 NOUN cana-6055 31 72 is be AUX cana-6055 31 73 even even ADV cana-6055 31 74 or or CCONJ cana-6055 31 75 odd odd ADJ cana-6055 31 76 . . PUNCT cana-6055 32 1 a a DET cana-6055 32 2 graph graph NOUN cana-6055 32 3 which which PRON cana-6055 32 4 admits admit VERB cana-6055 32 5 distance distance NOUN cana-6055 32 6 pair pair NOUN cana-6055 32 7 antimagic antimagic ADJ cana-6055 32 8 labeling labeling NOUN cana-6055 32 9 is be AUX cana-6055 32 10 called call VERB cana-6055 32 11 a a DET cana-6055 32 12 distance distance NOUN cana-6055 32 13 pair pair NOUN cana-6055 32 14 antimagic antimagic ADJ cana-6055 32 15 graph graph NOUN cana-6055 32 16 . . PUNCT cana-6055 33 1 definition definition NOUN cana-6055 33 2 2.4 2.4 NUM cana-6055 33 3 a a DET cana-6055 33 4 function function NOUN cana-6055 33 5 𝑓 𝑓 NOUN cana-6055 33 6 is be AUX cana-6055 33 7 called call VERB cana-6055 33 8 closed closed ADJ cana-6055 33 9 distance distance NOUN cana-6055 33 10 pair pair NOUN cana-6055 33 11 antimagic antimagic ADJ cana-6055 33 12 labeling labeling NOUN cana-6055 33 13 , , PUNCT cana-6055 33 14 if if SCONJ cana-6055 33 15 we we PRON cana-6055 33 16 take take VERB cana-6055 33 17 closed closed ADJ cana-6055 33 18 neighborhood neighborhood NOUN cana-6055 33 19 𝑛[𝑣 𝑛[𝑣 NOUN cana-6055 33 20 ] ] PUNCT cana-6055 33 21 instead instead ADV cana-6055 33 22 of of ADP cana-6055 33 23 open open VERB cana-6055 33 24 neighborhood neighborhood NOUN cana-6055 33 25 𝑁(𝑣 𝑁(𝑣 NUM cana-6055 33 26 ) ) PUNCT cana-6055 33 27 in in ADP cana-6055 33 28 the the DET cana-6055 33 29 previous previous ADJ cana-6055 33 30 definition definition NOUN cana-6055 33 31 . . PUNCT cana-6055 34 1 definition definition NOUN cana-6055 34 2 2.5 2.5 NUM cana-6055 34 3 the the DET cana-6055 34 4 𝑛-sunlet 𝑛-sunlet NOUN cana-6055 34 5 graph graph NOUN cana-6055 34 6 πΆπ‘›βŠ™πΎ1 πΆπ‘›βŠ™πΎ1 NOUN cana-6055 34 7 is be AUX cana-6055 34 8 the the DET cana-6055 34 9 graph graph NOUN cana-6055 34 10 on on ADP cana-6055 34 11 2𝑛 2𝑛 PROPN cana-6055 34 12 vertices vertex NOUN cana-6055 34 13 obtained obtain VERB cana-6055 34 14 by by ADP cana-6055 34 15 attaching attach VERB cana-6055 34 16 a a DET cana-6055 34 17 pendant pendant ADJ cana-6055 34 18 edge edge NOUN cana-6055 34 19 to to ADP cana-6055 34 20 each each DET cana-6055 34 21 vertices vertex NOUN cana-6055 34 22 of of ADP cana-6055 34 23 a a DET cana-6055 34 24 cycle cycle NOUN cana-6055 35 1 𝐢𝑛 𝐢𝑛 PROPN cana-6055 35 2 and and CCONJ cana-6055 35 3 it it PRON cana-6055 35 4 is be AUX cana-6055 35 5 denoted denote VERB cana-6055 35 6 by by ADP cana-6055 35 7 𝑆𝑛. 𝑆𝑛. PROPN cana-6055 35 8 definition definition NOUN cana-6055 35 9 2.6 2.6 NUM cana-6055 35 10 the the DET cana-6055 35 11 helm helm NOUN cana-6055 35 12 graph graph NOUN cana-6055 36 1 𝐻𝑛 𝐻𝑛 PROPN cana-6055 36 2 is be AUX cana-6055 36 3 the the DET cana-6055 36 4 graph graph NOUN cana-6055 36 5 obtained obtain VERB cana-6055 36 6 from from ADP cana-6055 36 7 an an DET cana-6055 36 8 wheel wheel NOUN cana-6055 36 9 graph graph NOUN cana-6055 36 10 π‘Šπ‘› π‘Šπ‘› PROPN cana-6055 36 11 by by ADP cana-6055 36 12 adjoining adjoin VERB cana-6055 36 13 a a DET cana-6055 36 14 pendant pendant ADJ cana-6055 36 15 edge edge NOUN cana-6055 36 16 at at ADP cana-6055 36 17 each each DET cana-6055 36 18 vertex vertex NOUN cana-6055 36 19 of of ADP cana-6055 36 20 the the DET cana-6055 36 21 cycle cycle NOUN cana-6055 36 22 . . PUNCT cana-6055 37 1 definition definition NOUN cana-6055 37 2 2.7 2.7 NUM cana-6055 37 3 the the DET cana-6055 37 4 triangular triangular NOUN cana-6055 37 5 snake snake NOUN cana-6055 38 1 𝑇𝑛 𝑇𝑛 PROPN cana-6055 38 2 is be AUX cana-6055 38 3 obtained obtain VERB cana-6055 38 4 from from ADP cana-6055 38 5 a a DET cana-6055 38 6 path path NOUN cana-6055 38 7 𝑃𝑛 𝑃𝑛 NOUN cana-6055 38 8 by by ADP cana-6055 38 9 replacing replace VERB cana-6055 38 10 each each DET cana-6055 38 11 edge edge NOUN cana-6055 38 12 of of ADP cana-6055 38 13 the the DET cana-6055 38 14 path path NOUN cana-6055 38 15 by by ADP cana-6055 38 16 a a DET cana-6055 38 17 triangle triangle NOUN cana-6055 38 18 𝐢3 𝐢3 NOUN cana-6055 38 19 . . PUNCT cana-6055 39 1 definition definition NOUN cana-6055 39 2 2.8 2.8 NUM cana-6055 39 3 the the DET cana-6055 39 4 gear gear NOUN cana-6055 39 5 graph graph NOUN cana-6055 39 6 𝐺𝑛 𝐺𝑛 PROPN cana-6055 39 7 is be AUX cana-6055 39 8 formed form VERB cana-6055 39 9 by by ADP cana-6055 39 10 adding add VERB cana-6055 39 11 a a DET cana-6055 39 12 vertex vertex NOUN cana-6055 39 13 between between ADP cana-6055 39 14 each each DET cana-6055 39 15 pair pair NOUN cana-6055 39 16 of of ADP cana-6055 39 17 adjacent adjacent ADJ cana-6055 39 18 vertices vertex NOUN cana-6055 39 19 of of ADP cana-6055 39 20 a a DET cana-6055 39 21 wheel wheel NOUN cana-6055 39 22 graph graph NOUN cana-6055 39 23 π‘Šπ‘›. π‘Šπ‘›. PROPN cana-6055 39 24 definition definition NOUN cana-6055 39 25 2.9 2.9 NUM cana-6055 39 26 the the DET cana-6055 39 27 n n NUM cana-6055 39 28 - - PUNCT cana-6055 39 29 book book NOUN cana-6055 39 30 graph graph NOUN cana-6055 39 31 is be AUX cana-6055 39 32 defined define VERB cana-6055 39 33 as as ADP cana-6055 39 34 the the DET cana-6055 39 35 graph graph NOUN cana-6055 39 36 cartesian cartesian ADJ cana-6055 39 37 product product NOUN cana-6055 39 38 𝐡𝑛 𝐡𝑛 PROPN cana-6055 39 39 = = PUNCT cana-6055 39 40 𝑆𝑛+1 𝑆𝑛+1 PROPN cana-6055 39 41 Γ— Γ— PROPN cana-6055 39 42 𝑃2 𝑃2 NOUN cana-6055 39 43 , , PUNCT cana-6055 39 44 where where SCONJ cana-6055 39 45 𝑆𝑛+1 𝑆𝑛+1 ADV cana-6055 39 46 is be AUX cana-6055 39 47 a a DET cana-6055 39 48 star star NOUN cana-6055 39 49 graph graph NOUN cana-6055 39 50 and and CCONJ cana-6055 39 51 𝑃2 𝑃2 NOUN cana-6055 39 52 is be AUX cana-6055 39 53 the the DET cana-6055 39 54 path path NOUN cana-6055 39 55 graph graph NOUN cana-6055 39 56 on on ADP cana-6055 39 57 two two NUM cana-6055 39 58 vertices vertex NOUN cana-6055 39 59 . . PUNCT cana-6055 40 1 definition definition NOUN cana-6055 40 2 2.10 2.10 NUM cana-6055 40 3 the the DET cana-6055 40 4 friendship friendship NOUN cana-6055 40 5 graph graph NOUN cana-6055 40 6 is be AUX cana-6055 40 7 defined define VERB cana-6055 40 8 as as ADP cana-6055 40 9 the the DET cana-6055 40 10 graph graph NOUN cana-6055 40 11 𝐹𝑛 𝐹𝑛 PROPN cana-6055 40 12 which which PRON cana-6055 40 13 consisting consist VERB cana-6055 40 14 of of ADP cana-6055 40 15 n n NOUN cana-6055 40 16 triangles triangle NOUN cana-6055 40 17 with with ADP cana-6055 40 18 a a DET cana-6055 40 19 common common ADJ cana-6055 40 20 vertex vertex NOUN cana-6055 40 21 . . PUNCT cana-6055 41 1 definition definition NOUN cana-6055 41 2 2.11 2.11 NUM cana-6055 41 3 the the DET cana-6055 41 4 prism prism NOUN cana-6055 41 5 graph graph NOUN cana-6055 42 1 𝐢𝑛 𝐢𝑛 PROPN cana-6055 42 2 Γ— Γ— NOUN cana-6055 42 3 𝐾2 𝐾2 NOUN cana-6055 42 4 is be AUX cana-6055 42 5 constructed construct VERB cana-6055 42 6 by by ADP cana-6055 42 7 the the DET cana-6055 42 8 cartesian cartesian ADJ cana-6055 42 9 product product NOUN cana-6055 42 10 of of ADP cana-6055 42 11 a a DET cana-6055 42 12 cycle cycle NOUN cana-6055 42 13 of of ADP cana-6055 42 14 length length NOUN cana-6055 42 15 𝑛 𝑛 PROPN cana-6055 42 16 β‰₯ β‰₯ NUM cana-6055 42 17 3 3 NUM cana-6055 42 18 and and CCONJ cana-6055 42 19 an an DET cana-6055 42 20 edge edge NOUN cana-6055 42 21 . . PUNCT cana-6055 43 1 definition definition NOUN cana-6055 43 2 2.12 2.12 NUM cana-6055 43 3 resty resty NOUN cana-6055 43 4 [ [ X cana-6055 43 5 8 8 NUM cana-6055 43 6 ] ] PUNCT cana-6055 43 7 defined define VERB cana-6055 43 8 the the DET cana-6055 43 9 following follow VERB cana-6055 43 10 graph graph NOUN cana-6055 43 11 . . PUNCT cana-6055 44 1 for for ADP cana-6055 44 2 𝑛 𝑛 PRON cana-6055 44 3 β‰₯ β‰₯ NUM cana-6055 44 4 4 4 NUM cana-6055 44 5 , , PUNCT cana-6055 44 6 the the DET cana-6055 44 7 n n ADV cana-6055 44 8 - - PUNCT cana-6055 44 9 crossed cross VERB cana-6055 44 10 prism prism NOUN cana-6055 44 11 graph graph NOUN cana-6055 44 12 obtained obtain VERB cana-6055 44 13 by by ADP cana-6055 44 14 taking take VERB cana-6055 44 15 two two NUM cana-6055 44 16 disjoint disjoint ADJ cana-6055 44 17 cycle cycle NOUN cana-6055 44 18 graphs graph NOUN cana-6055 44 19 namely namely ADV cana-6055 44 20 𝐢𝑛 𝐢𝑛 NOUN cana-6055 44 21 1 1 NUM cana-6055 44 22 and and CCONJ cana-6055 44 23 𝐢𝑛 𝐢𝑛 PROPN cana-6055 44 24 2 2 NUM cana-6055 44 25 , , PUNCT cana-6055 44 26 where where SCONJ cana-6055 44 27 𝑉(𝐢𝑛 𝑉(𝐢𝑛 ADJ cana-6055 44 28 1 1 NUM cana-6055 44 29 ) ) PUNCT cana-6055 44 30 = = PRON cana-6055 44 31 { { PUNCT cana-6055 44 32 𝑣1 𝑣1 PROPN cana-6055 44 33 , , PUNCT cana-6055 44 34 𝑣2 𝑣2 PROPN cana-6055 44 35 , , PUNCT cana-6055 44 36 . . PUNCT cana-6055 44 37 . . PUNCT cana-6055 45 1 . . PUNCT cana-6055 46 1 , , PUNCT cana-6055 46 2 𝑣𝑛 𝑣𝑛 NOUN cana-6055 46 3 } } PUNCT cana-6055 46 4 and and CCONJ cana-6055 46 5 𝑉(𝐢𝑛 𝑉(𝐢𝑛 ADJ cana-6055 46 6 2 2 NUM cana-6055 46 7 ) ) PUNCT cana-6055 46 8 = = PRON cana-6055 46 9 { { PUNCT cana-6055 46 10 𝑒1 𝑒1 PROPN cana-6055 46 11 , , PUNCT cana-6055 46 12 𝑒2 𝑒2 PROPN cana-6055 46 13 , , PUNCT cana-6055 46 14 . . PUNCT cana-6055 46 15 . . PUNCT cana-6055 47 1 . . PUNCT cana-6055 48 1 , , PUNCT cana-6055 48 2 𝑒𝑛 𝑒𝑛 X cana-6055 48 3 } } PUNCT cana-6055 48 4 , , PUNCT cana-6055 48 5 and and CCONJ cana-6055 48 6 adding add VERB cana-6055 48 7 edges edge NOUN cana-6055 48 8 𝑒𝑠𝑣𝑠+1 𝑒𝑠𝑣𝑠+1 NOUN cana-6055 48 9 for for ADP cana-6055 48 10 𝑠 𝑠 PROPN cana-6055 48 11 ∈ ∈ PROPN cana-6055 48 12 { { PUNCT cana-6055 48 13 1,3 1,3 NUM cana-6055 48 14 , , PUNCT cana-6055 48 15 . . PUNCT cana-6055 48 16 . . PUNCT cana-6055 49 1 . . PUNCT cana-6055 50 1 , , PUNCT cana-6055 50 2 𝑛 𝑛 DET cana-6055 50 3 βˆ’ βˆ’ NOUN cana-6055 50 4 1 1 NUM cana-6055 50 5 } } PUNCT cana-6055 50 6 and and CCONJ cana-6055 50 7 π‘’π‘‘π‘£π‘‘βˆ’1 π‘’π‘‘π‘£π‘‘βˆ’1 PROPN cana-6055 50 8 for for ADP cana-6055 50 9 𝑑 𝑑 PROPN cana-6055 50 10 ∈ ∈ PROPN cana-6055 50 11 { { PUNCT cana-6055 50 12 2,4 2,4 NUM cana-6055 50 13 , , PUNCT cana-6055 50 14 . . PUNCT cana-6055 50 15 . . PUNCT cana-6055 50 16 . . PUNCT cana-6055 51 1 , , PUNCT cana-6055 51 2 𝑛 𝑛 X cana-6055 51 3 } } PUNCT cana-6055 51 4 . . PUNCT cana-6055 52 1 the the DET cana-6055 52 2 n n ADV cana-6055 52 3 - - PUNCT cana-6055 52 4 crossed cross VERB cana-6055 52 5 prism prism NOUN cana-6055 52 6 graph graph NOUN cana-6055 52 7 denoted denote VERB cana-6055 52 8 by by ADP cana-6055 52 9 𝐢𝑃𝑛 𝐢𝑃𝑛 PROPN cana-6055 52 10 . . PUNCT cana-6055 53 1 definition definition NOUN cana-6055 53 2 2.13 2.13 NUM cana-6055 53 3 the the DET cana-6055 53 4 circulant circulant ADJ cana-6055 53 5 graph graph NOUN cana-6055 53 6 𝐢(𝑛 𝐢(𝑛 NOUN cana-6055 53 7 ; ; PUNCT cana-6055 53 8 𝐷 𝐷 NOUN cana-6055 53 9 ) ) PUNCT cana-6055 53 10 , , PUNCT cana-6055 53 11 where where SCONJ cana-6055 53 12 𝐷 𝐷 PROPN cana-6055 53 13 βŠ† βŠ† NUM cana-6055 53 14 { { PUNCT cana-6055 53 15 1,2 1,2 NUM cana-6055 53 16 , , PUNCT cana-6055 53 17 … … PUNCT cana-6055 53 18 , , PUNCT cana-6055 53 19 ⌊ ⌊ VERB cana-6055 53 20 𝑛 𝑛 DET cana-6055 53 21 2 2 NUM cana-6055 53 22 βŒ‹ βŒ‹ NOUN cana-6055 53 23 } } PUNCT cana-6055 53 24 is be AUX cana-6055 53 25 the the DET cana-6055 53 26 graph graph NOUN cana-6055 53 27 with with ADP cana-6055 53 28 vertex vertex NOUN cana-6055 53 29 set set NOUN cana-6055 53 30 { { PUNCT cana-6055 53 31 𝑣1 𝑣1 PROPN cana-6055 53 32 , , PUNCT cana-6055 53 33 𝑣2 𝑣2 PROPN cana-6055 53 34 , , PUNCT cana-6055 53 35 … … PUNCT cana-6055 53 36 , , PUNCT cana-6055 53 37 𝑣𝑛 𝑣𝑛 NOUN cana-6055 53 38 } } PUNCT cana-6055 53 39 , , PUNCT cana-6055 53 40 where where SCONJ cana-6055 53 41 𝑣𝑖 𝑣𝑖 NOUN cana-6055 53 42 and and CCONJ cana-6055 53 43 𝑣𝑗 𝑣𝑗 ADP cana-6055 53 44 are be AUX cana-6055 53 45 adjacent adjacent ADJ cana-6055 53 46 if if SCONJ cana-6055 54 1 and and CCONJ cana-6055 54 2 only only ADV cana-6055 54 3 if if SCONJ cana-6055 54 4 there there PRON cana-6055 54 5 is be VERB cana-6055 54 6 a a DET cana-6055 54 7 number number NOUN cana-6055 54 8 𝑑 𝑑 PROPN cana-6055 54 9 ∈ ∈ PROPN cana-6055 54 10 𝐷 𝐷 NOUN cana-6055 54 11 such such ADJ cana-6055 54 12 that that SCONJ cana-6055 54 13 𝑖 𝑖 NOUN cana-6055 55 1 + + NUM cana-6055 55 2 𝑑 𝑑 PROPN cana-6055 55 3 ≑ ≑ PROPN cana-6055 55 4 𝑗(π‘šπ‘œπ‘‘ 𝑗(π‘šπ‘œπ‘‘ PUNCT cana-6055 55 5 𝑛 𝑛 X cana-6055 55 6 ) ) PUNCT cana-6055 55 7 or or CCONJ cana-6055 55 8 𝑗 𝑗 PRON cana-6055 55 9 + + CCONJ cana-6055 55 10 𝑑 𝑑 PROPN cana-6055 55 11 ≑ ≑ PROPN cana-6055 55 12 𝑖(π‘šπ‘œπ‘‘ 𝑖(π‘šπ‘œπ‘‘ X cana-6055 55 13 𝑛 𝑛 PROPN cana-6055 55 14 ) ) PUNCT cana-6055 55 15 . . PUNCT cana-6055 56 1 3 3 X cana-6055 56 2 . . X cana-6055 56 3 main main ADJ cana-6055 56 4 results result NOUN cana-6055 56 5 theorem theorem VERB cana-6055 56 6 3.1 3.1 NUM cana-6055 56 7 the the DET cana-6055 56 8 sunlet sunlet NOUN cana-6055 56 9 graph graph NOUN cana-6055 56 10 𝑆𝑛 𝑆𝑛 PROPN cana-6055 56 11 is be AUX cana-6055 56 12 a a DET cana-6055 56 13 distance distance NOUN cana-6055 56 14 pair pair NOUN cana-6055 56 15 antimagic antimagic ADJ cana-6055 56 16 graph graph NOUN cana-6055 56 17 if if SCONJ cana-6055 56 18 𝑛 𝑛 PROPN cana-6055 56 19 = = SYM cana-6055 56 20 4,6,8 4,6,8 NUM cana-6055 56 21 , , PUNCT cana-6055 56 22 … … PUNCT cana-6055 56 23 proof proof NOUN cana-6055 56 24 . . PUNCT cana-6055 57 1 let let VERB cana-6055 57 2 𝑉(𝑆𝑛 𝑉(𝑆𝑛 NOUN cana-6055 57 3 ) ) PUNCT cana-6055 57 4 = = PRON cana-6055 57 5 { { PUNCT cana-6055 57 6 𝑣𝑖 𝑣𝑖 NOUN cana-6055 57 7 , , PUNCT cana-6055 57 8 𝑒𝑖 𝑒𝑖 ADP cana-6055 57 9 : : PUNCT cana-6055 57 10 1 1 NUM cana-6055 57 11 ≀ ≀ NUM cana-6055 57 12 𝑖 𝑖 SYM cana-6055 57 13 ≀ ≀ NUM cana-6055 57 14 𝑛 𝑛 DET cana-6055 57 15 } } PUNCT cana-6055 57 16 be be VERB cana-6055 57 17 the the DET cana-6055 57 18 vertex vertex NOUN cana-6055 57 19 set set NOUN cana-6055 57 20 and and CCONJ cana-6055 57 21 communications communication NOUN cana-6055 57 22 on on ADP cana-6055 57 23 applied apply VERB cana-6055 57 24 nonlinear nonlinear ADJ cana-6055 57 25 analysis analysis NOUN cana-6055 57 26 issn issn NOUN cana-6055 57 27 : : PUNCT cana-6055 57 28 1074 1074 NUM cana-6055 57 29 - - PUNCT cana-6055 57 30 133x 133x NUM cana-6055 57 31 vol vol NOUN cana-6055 57 32 31 31 NUM cana-6055 57 33 no no NOUN cana-6055 57 34 . . PUNCT cana-6055 58 1 8s 8s PROPN cana-6055 58 2 ( ( PUNCT cana-6055 58 3 2024 2024 NUM cana-6055 58 4 ) ) PUNCT cana-6055 58 5 1157 1157 NUM cana-6055 58 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 58 7 𝐸(𝑆𝑛 𝐸(𝑆𝑛 NOUN cana-6055 58 8 ) ) PUNCT cana-6055 58 9 = = PRON cana-6055 58 10 { { PUNCT cana-6055 58 11 𝑣𝑖𝑒𝑖 𝑣𝑖𝑒𝑖 ADV cana-6055 58 12 : : PUNCT cana-6055 58 13 1 1 NUM cana-6055 58 14 ≀ ≀ NUM cana-6055 58 15 𝑖 𝑖 SYM cana-6055 58 16 ≀ ≀ NUM cana-6055 58 17 𝑛 𝑛 NOUN cana-6055 58 18 } } PUNCT cana-6055 58 19 βˆͺ βˆͺ ADJ cana-6055 58 20 { { PUNCT cana-6055 58 21 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 NOUN cana-6055 58 22 : : PUNCT cana-6055 58 23 1 1 NUM cana-6055 58 24 ≀ ≀ NUM cana-6055 58 25 𝑖 𝑖 SYM cana-6055 58 26 ≀ ≀ NUM cana-6055 58 27 𝑛 𝑛 PRON cana-6055 58 28 βˆ’ βˆ’ PROPN cana-6055 58 29 1 1 NUM cana-6055 58 30 } } PUNCT cana-6055 58 31 βˆͺ βˆͺ ADJ cana-6055 58 32 { { PUNCT cana-6055 58 33 𝑣1𝑣𝑛 𝑣1𝑣𝑛 NOUN cana-6055 58 34 } } PUNCT cana-6055 58 35 be be AUX cana-6055 58 36 the the DET cana-6055 58 37 edge edge NOUN cana-6055 58 38 set set NOUN cana-6055 58 39 of of ADP cana-6055 58 40 𝑆𝑛. 𝑆𝑛. PROPN cana-6055 58 41 define define VERB cana-6055 58 42 𝑓 𝑓 PRON cana-6055 58 43 : : PUNCT cana-6055 58 44 𝑉(𝑆𝑛 𝑉(𝑆𝑛 NUM cana-6055 58 45 ) ) PUNCT cana-6055 58 46 ⟢ ⟢ NOUN cana-6055 58 47 { { PUNCT cana-6055 58 48 Β±1,Β±2,β‹― Β±1,Β±2,β‹― NOUN cana-6055 58 49 , , PUNCT cana-6055 58 50 ±𝑛 ±𝑛 PROPN cana-6055 58 51 } } PUNCT cana-6055 58 52 by by ADP cana-6055 58 53 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 58 54 ) ) PUNCT cana-6055 58 55 = = PRON cana-6055 58 56 { { PUNCT cana-6055 58 57 ( ( PUNCT cana-6055 58 58 βˆ’1)𝑖(𝑖 βˆ’1)𝑖(𝑖 NOUN cana-6055 58 59 ) ) PUNCT cana-6055 58 60 , , PUNCT cana-6055 58 61 𝑖𝑓 𝑖𝑓 ADP cana-6055 58 62 1 1 NUM cana-6055 58 63 ≀ ≀ NUM cana-6055 58 64 𝑖 𝑖 SYM cana-6055 58 65 ≀ ≀ NUM cana-6055 58 66 𝑛 𝑛 DET cana-6055 58 67 2 2 NUM cana-6055 58 68 ( ( PUNCT cana-6055 58 69 βˆ’1)𝑖(𝑛 βˆ’1)𝑖(𝑛 PRON cana-6055 58 70 + + NOUN cana-6055 58 71 1 1 NUM cana-6055 58 72 βˆ’ βˆ’ NOUN cana-6055 58 73 𝑖 𝑖 NOUN cana-6055 58 74 ) ) PUNCT cana-6055 58 75 , , PUNCT cana-6055 58 76 𝑖𝑓 𝑖𝑓 ADP cana-6055 58 77 𝑛 𝑛 PRON cana-6055 58 78 2 2 NUM cana-6055 58 79 < < X cana-6055 58 80 𝑖 𝑖 PRON cana-6055 58 81 ≀ ≀ X cana-6055 58 82 𝑛 𝑛 DET cana-6055 58 83 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 58 84 ) ) PUNCT cana-6055 58 85 = = SYM cana-6055 58 86 { { PUNCT cana-6055 58 87 ( ( PUNCT cana-6055 58 88 βˆ’1 βˆ’1 NOUN cana-6055 58 89 ) ) PUNCT cana-6055 58 90 𝑖+1 𝑖+1 PUNCT cana-6055 59 1 ( ( PUNCT cana-6055 59 2 𝑛 𝑛 PROPN cana-6055 59 3 2 2 NUM cana-6055 59 4 + + NUM cana-6055 59 5 𝑖 𝑖 NOUN cana-6055 59 6 ) ) PUNCT cana-6055 59 7 , , PUNCT cana-6055 59 8 𝑖𝑓 𝑖𝑓 ADP cana-6055 59 9 1 1 NUM cana-6055 59 10 ≀ ≀ NUM cana-6055 59 11 𝑖 𝑖 SYM cana-6055 59 12 ≀ ≀ NUM cana-6055 59 13 𝑛 𝑛 DET cana-6055 59 14 2 2 NUM cana-6055 59 15 ( ( PUNCT cana-6055 59 16 βˆ’1)𝑖+1 βˆ’1)𝑖+1 PROPN cana-6055 59 17 ( ( PUNCT cana-6055 59 18 3𝑛 3𝑛 NUM cana-6055 59 19 2 2 NUM cana-6055 59 20 + + CCONJ cana-6055 59 21 1 1 NUM cana-6055 59 22 βˆ’ βˆ’ NUM cana-6055 59 23 𝑖 𝑖 NUM cana-6055 59 24 ) ) PUNCT cana-6055 59 25 , , PUNCT cana-6055 59 26 𝑖𝑓 𝑖𝑓 ADP cana-6055 59 27 𝑛 𝑛 PRON cana-6055 59 28 2 2 NUM cana-6055 59 29 < < X cana-6055 59 30 𝑖 𝑖 SYM cana-6055 59 31 ≀ ≀ NOUN cana-6055 59 32 𝑛 𝑛 PRON cana-6055 59 33 then then ADV cana-6055 59 34 the the DET cana-6055 59 35 induced induced ADJ cana-6055 59 36 vertex vertex NOUN cana-6055 59 37 weight weight NOUN cana-6055 59 38 labeling labeling NOUN cana-6055 59 39 are be AUX cana-6055 59 40 as as SCONJ cana-6055 59 41 follows follow VERB cana-6055 59 42 . . PUNCT cana-6055 60 1 𝑀(𝑣𝑖 𝑀(𝑣𝑖 VERB cana-6055 60 2 ) ) PUNCT cana-6055 60 3 = = PRON cana-6055 60 4 { { PUNCT cana-6055 60 5 𝑛 𝑛 PROPN cana-6055 60 6 2 2 NUM cana-6055 60 7 + + NUM cana-6055 60 8 4 4 NUM cana-6055 60 9 , , PUNCT cana-6055 60 10 𝑖𝑓 𝑖𝑓 ADP cana-6055 60 11 𝑖 𝑖 SYM cana-6055 60 12 = = SYM cana-6055 60 13 1 1 NUM cana-6055 60 14 ( ( PUNCT cana-6055 60 15 βˆ’1)𝑖+1 βˆ’1)𝑖+1 PROPN cana-6055 60 16 ( ( PUNCT cana-6055 60 17 𝑛 𝑛 PROPN cana-6055 60 18 2 2 NUM cana-6055 60 19 + + NUM cana-6055 60 20 3𝑖 3𝑖 NUM cana-6055 60 21 ) ) PUNCT cana-6055 60 22 , , PUNCT cana-6055 60 23 𝑖𝑓 𝑖𝑓 ADP cana-6055 60 24 1 1 NUM cana-6055 60 25 < < X cana-6055 60 26 𝑖 𝑖 X cana-6055 60 27 < < X cana-6055 60 28 𝑛 𝑛 PRON cana-6055 60 29 2 2 NUM cana-6055 60 30 ( ( PUNCT cana-6055 60 31 βˆ’1)𝑖+12𝑛 βˆ’1)𝑖+12𝑛 NOUN cana-6055 60 32 βˆ’ βˆ’ PROPN cana-6055 60 33 1 1 NUM cana-6055 60 34 , , PUNCT cana-6055 60 35 𝑖𝑓 𝑖𝑓 ADP cana-6055 60 36 𝑖 𝑖 NOUN cana-6055 60 37 = = SYM cana-6055 60 38 𝑛 𝑛 DET cana-6055 60 39 2 2 NUM cana-6055 60 40 , , PUNCT cana-6055 60 41 𝑛 𝑛 DET cana-6055 60 42 2 2 NUM cana-6055 60 43 + + NUM cana-6055 60 44 1 1 NUM cana-6055 60 45 ( ( PUNCT cana-6055 60 46 βˆ’1)𝑖+12𝑛 βˆ’1)𝑖+12𝑛 PROPN cana-6055 61 1 + + CCONJ cana-6055 61 2 3 3 NUM cana-6055 61 3 ( ( PUNCT cana-6055 61 4 𝑛 𝑛 PROPN cana-6055 61 5 2 2 NUM cana-6055 61 6 + + CCONJ cana-6055 61 7 1 1 NUM cana-6055 61 8 βˆ’ βˆ’ NUM cana-6055 61 9 𝑖 𝑖 NUM cana-6055 61 10 ) ) PUNCT cana-6055 61 11 , , PUNCT cana-6055 61 12 𝑖𝑓 𝑖𝑓 ADP cana-6055 61 13 𝑖 𝑖 X cana-6055 61 14 = = SYM cana-6055 61 15 𝑛 𝑛 PRON cana-6055 61 16 2 2 NUM cana-6055 61 17 + + CCONJ cana-6055 61 18 2 2 NUM cana-6055 61 19 , , PUNCT cana-6055 61 20 𝑛 𝑛 DET cana-6055 61 21 2 2 NUM cana-6055 61 22 + + SYM cana-6055 61 23 3 3 NUM cana-6055 61 24 , , PUNCT cana-6055 61 25 . . PUNCT cana-6055 61 26 . . PUNCT cana-6055 61 27 . . PUNCT cana-6055 62 1 , , PUNCT cana-6055 62 2 𝑛 𝑛 DET cana-6055 62 3 βˆ’ βˆ’ PROPN cana-6055 62 4 1 1 NUM cana-6055 62 5 βˆ’ βˆ’ PROPN cana-6055 62 6 ( ( PUNCT cana-6055 62 7 𝑛 𝑛 PROPN cana-6055 62 8 2 2 NUM cana-6055 62 9 + + NUM cana-6055 62 10 4 4 NUM cana-6055 62 11 ) ) PUNCT cana-6055 62 12 , , PUNCT cana-6055 62 13 𝑖𝑓 𝑖𝑓 ADP cana-6055 62 14 𝑖 𝑖 X cana-6055 62 15 = = SYM cana-6055 62 16 𝑛 𝑛 PRON cana-6055 62 17 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 62 18 ) ) PUNCT cana-6055 62 19 = = PRON cana-6055 62 20 { { PUNCT cana-6055 62 21 ( ( PUNCT cana-6055 62 22 βˆ’1)𝑖(𝑖 βˆ’1)𝑖(𝑖 NOUN cana-6055 62 23 ) ) PUNCT cana-6055 62 24 , , PUNCT cana-6055 62 25 𝑖𝑓 𝑖𝑓 ADP cana-6055 62 26 1 1 NUM cana-6055 62 27 ≀ ≀ NUM cana-6055 62 28 𝑖 𝑖 SYM cana-6055 62 29 ≀ ≀ NUM cana-6055 62 30 𝑛 𝑛 DET cana-6055 62 31 2 2 NUM cana-6055 62 32 ( ( PUNCT cana-6055 62 33 βˆ’1)𝑖(𝑛 βˆ’1)𝑖(𝑛 PRON cana-6055 62 34 + + NOUN cana-6055 62 35 1 1 NUM cana-6055 62 36 βˆ’ βˆ’ NOUN cana-6055 62 37 𝑖 𝑖 NOUN cana-6055 62 38 ) ) PUNCT cana-6055 62 39 , , PUNCT cana-6055 62 40 𝑖𝑓 𝑖𝑓 ADP cana-6055 62 41 𝑛 𝑛 PRON cana-6055 62 42 2 2 NUM cana-6055 62 43 < < X cana-6055 62 44 𝑖 𝑖 X cana-6055 62 45 ≀ ≀ NUM cana-6055 62 46 𝑛 𝑛 PRON cana-6055 62 47 hence hence ADV cana-6055 62 48 𝑆𝑛 𝑆𝑛 ADJ cana-6055 62 49 is be AUX cana-6055 62 50 a a DET cana-6055 62 51 distance distance NOUN cana-6055 62 52 pair pair NOUN cana-6055 62 53 antimagic antimagic ADJ cana-6055 62 54 graph graph NOUN cana-6055 62 55 if if SCONJ cana-6055 62 56 𝑛 𝑛 PROPN cana-6055 62 57 = = SYM cana-6055 62 58 4,6,8 4,6,8 NUM cana-6055 62 59 , , PUNCT cana-6055 62 60 … … PUNCT cana-6055 62 61 corollary corollary ADJ cana-6055 62 62 3.2 3.2 NUM cana-6055 62 63 the the DET cana-6055 62 64 helm helm NOUN cana-6055 62 65 graph graph NOUN cana-6055 62 66 𝐻𝑛 𝐻𝑛 PROPN cana-6055 62 67 is be AUX cana-6055 62 68 a a DET cana-6055 62 69 distance distance NOUN cana-6055 62 70 pair pair NOUN cana-6055 62 71 antimagic antimagic ADJ cana-6055 62 72 graph graph NOUN cana-6055 62 73 if if SCONJ cana-6055 62 74 𝑛 𝑛 PROPN cana-6055 62 75 = = SYM cana-6055 62 76 4,6,8 4,6,8 NUM cana-6055 62 77 , , PUNCT cana-6055 62 78 … … PUNCT cana-6055 62 79 by by ADP cana-6055 62 80 using use VERB cana-6055 62 81 theorem theorem ADJ cana-6055 62 82 2.10 2.10 NUM cana-6055 62 83 [ [ NOUN cana-6055 62 84 4 4 NUM cana-6055 62 85 ] ] PUNCT cana-6055 62 86 , , PUNCT cana-6055 62 87 if if SCONJ cana-6055 62 88 𝐺 𝐺 PROPN cana-6055 62 89 is be AUX cana-6055 62 90 a a DET cana-6055 62 91 distance distance NOUN cana-6055 62 92 pair pair NOUN cana-6055 62 93 antimagic antimagic ADJ cana-6055 62 94 graph graph NOUN cana-6055 62 95 with with ADP cana-6055 62 96 even even ADV cana-6055 62 97 number number NOUN cana-6055 62 98 of of ADP cana-6055 62 99 vertices vertex NOUN cana-6055 62 100 , , PUNCT cana-6055 62 101 then then ADV cana-6055 62 102 the the DET cana-6055 62 103 join join NOUN cana-6055 62 104 graph graph NOUN cana-6055 62 105 𝐺 𝐺 PROPN cana-6055 62 106 + + CCONJ cana-6055 62 107 𝐾1 𝐾1 NOUN cana-6055 62 108 is be AUX cana-6055 62 109 a a DET cana-6055 62 110 distance distance NOUN cana-6055 62 111 pair pair NOUN cana-6055 62 112 antimagic antimagic ADJ cana-6055 62 113 graph graph NOUN cana-6055 62 114 . . PUNCT cana-6055 63 1 theorem theorem VERB cana-6055 63 2 3.3 3.3 NUM cana-6055 63 3 the the DET cana-6055 63 4 triangular triangular NOUN cana-6055 63 5 snake snake NOUN cana-6055 63 6 graph graph NOUN cana-6055 63 7 𝑇𝑛 𝑇𝑛 NOUN cana-6055 63 8 is be AUX cana-6055 63 9 a a DET cana-6055 63 10 distance distance NOUN cana-6055 63 11 pair pair NOUN cana-6055 63 12 antimagic antimagic ADJ cana-6055 63 13 graph graph NOUN cana-6055 63 14 for for ADP cana-6055 63 15 all all DET cana-6055 63 16 𝑛 𝑛 PRON cana-6055 63 17 β‰₯ β‰₯ NUM cana-6055 63 18 2 2 NUM cana-6055 63 19 . . PUNCT cana-6055 63 20 proof proof NOUN cana-6055 63 21 . . PUNCT cana-6055 64 1 let let VERB cana-6055 64 2 𝑉(𝑇𝑛 𝑉(𝑇𝑛 ADV cana-6055 64 3 ) ) PUNCT cana-6055 65 1 = = PRON cana-6055 65 2 { { PUNCT cana-6055 65 3 𝑣𝑖 𝑣𝑖 ADP cana-6055 65 4 : : PUNCT cana-6055 65 5 1 1 NUM cana-6055 65 6 ≀ ≀ NUM cana-6055 65 7 𝑖 𝑖 SYM cana-6055 65 8 ≀ ≀ NUM cana-6055 65 9 𝑛 𝑛 PRON cana-6055 65 10 } } PUNCT cana-6055 65 11 βˆͺ βˆͺ ADJ cana-6055 65 12 { { PUNCT cana-6055 65 13 𝑒𝑖 𝑒𝑖 NOUN cana-6055 65 14 : : PUNCT cana-6055 65 15 1 1 NUM cana-6055 65 16 ≀ ≀ NUM cana-6055 65 17 𝑖 𝑖 SYM cana-6055 65 18 ≀ ≀ NUM cana-6055 65 19 𝑛 𝑛 PRON cana-6055 65 20 βˆ’ βˆ’ PROPN cana-6055 65 21 1 1 NUM cana-6055 65 22 } } PUNCT cana-6055 65 23 be be AUX cana-6055 65 24 the the DET cana-6055 65 25 vertex vertex NOUN cana-6055 65 26 set set NOUN cana-6055 65 27 and and CCONJ cana-6055 65 28 𝐸(𝑇𝑛 𝐸(𝑇𝑛 NUM cana-6055 65 29 ) ) PUNCT cana-6055 66 1 = = PRON cana-6055 66 2 { { PUNCT cana-6055 66 3 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-6055 66 4 : : PUNCT cana-6055 66 5 1 1 NUM cana-6055 66 6 ≀ ≀ NUM cana-6055 66 7 𝑖 𝑖 SYM cana-6055 66 8 ≀ ≀ NUM cana-6055 66 9 𝑛 𝑛 PRON cana-6055 66 10 βˆ’ βˆ’ PROPN cana-6055 66 11 1 1 NUM cana-6055 66 12 } } PUNCT cana-6055 66 13 βˆͺ βˆͺ ADJ cana-6055 66 14 { { PUNCT cana-6055 66 15 𝑒𝑖𝑣𝑖 𝑒𝑖𝑣𝑖 NOUN cana-6055 66 16 , , PUNCT cana-6055 66 17 𝑒𝑖𝑣𝑖+1 𝑒𝑖𝑣𝑖+1 PROPN cana-6055 66 18 : : PUNCT cana-6055 66 19 1 1 NUM cana-6055 66 20 ≀ ≀ NUM cana-6055 66 21 𝑖 𝑖 SYM cana-6055 66 22 ≀ ≀ NUM cana-6055 66 23 𝑛 𝑛 PRON cana-6055 66 24 βˆ’ βˆ’ PROPN cana-6055 66 25 1 1 NUM cana-6055 66 26 } } PUNCT cana-6055 66 27 be be AUX cana-6055 66 28 the the DET cana-6055 66 29 edge edge NOUN cana-6055 66 30 set set NOUN cana-6055 66 31 of of ADP cana-6055 66 32 𝑇𝑛. 𝑇𝑛. NOUN cana-6055 66 33 consider consider VERB cana-6055 66 34 following follow VERB cana-6055 66 35 two two NUM cana-6055 66 36 cases case NOUN cana-6055 66 37 . . PUNCT cana-6055 67 1 case(i case(i NOUN cana-6055 67 2 ): ): PUNCT cana-6055 67 3 𝑛 𝑛 PRON cana-6055 67 4 = = PUNCT cana-6055 67 5 is be AUX cana-6055 67 6 odd odd ADJ cana-6055 67 7 figure figure NOUN cana-6055 67 8 3.1 3.1 NUM cana-6055 67 9 dpam dpam NOUN cana-6055 67 10 labeling labeling NOUN cana-6055 67 11 of of ADP cana-6055 67 12 𝑇3 𝑇3 PROPN cana-6055 67 13 . . PUNCT cana-6055 68 1 ( ( PUNCT cana-6055 68 2 the the DET cana-6055 68 3 vertex vertex NOUN cana-6055 68 4 labels label NOUN cana-6055 68 5 are be AUX cana-6055 68 6 in in ADP cana-6055 68 7 usual usual ADJ cana-6055 68 8 font font NOUN cana-6055 68 9 and and CCONJ cana-6055 68 10 weights weight NOUN cana-6055 68 11 are be AUX cana-6055 68 12 in in ADP cana-6055 68 13 bold bold ADJ cana-6055 68 14 font font NOUN cana-6055 68 15 ) ) PUNCT cana-6055 68 16 communications communication NOUN cana-6055 68 17 on on ADP cana-6055 68 18 applied apply VERB cana-6055 68 19 nonlinear nonlinear ADJ cana-6055 68 20 analysis analysis NOUN cana-6055 68 21 issn issn NOUN cana-6055 68 22 : : PUNCT cana-6055 68 23 1074 1074 NUM cana-6055 68 24 - - PUNCT cana-6055 68 25 133x 133x NUM cana-6055 68 26 vol vol NOUN cana-6055 68 27 31 31 NUM cana-6055 68 28 no no NOUN cana-6055 68 29 . . PUNCT cana-6055 69 1 8s 8s PROPN cana-6055 69 2 ( ( PUNCT cana-6055 69 3 2024 2024 NUM cana-6055 69 4 ) ) PUNCT cana-6055 69 5 1158 1158 NUM cana-6055 69 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 69 7 clearly clearly ADV cana-6055 69 8 from from ADP cana-6055 69 9 figure figure NOUN cana-6055 69 10 3.1 3.1 NUM cana-6055 69 11 𝑇3 𝑇3 PROPN cana-6055 69 12 is be AUX cana-6055 69 13 a a DET cana-6055 69 14 distance distance NOUN cana-6055 69 15 pair pair NOUN cana-6055 69 16 antimagic antimagic ADJ cana-6055 69 17 graph graph NOUN cana-6055 69 18 . . PUNCT cana-6055 70 1 for for ADP cana-6055 70 2 𝑛 𝑛 PROPN cana-6055 70 3 = = SYM cana-6055 70 4 5,7,9 5,7,9 PROPN cana-6055 70 5 , , PUNCT cana-6055 70 6 . . PUNCT cana-6055 71 1 .. .. PUNCT cana-6055 71 2 define define VERB cana-6055 71 3 𝑓 𝑓 PRON cana-6055 71 4 : : PUNCT cana-6055 71 5 𝑉(𝑇𝑛 𝑉(𝑇𝑛 NUM cana-6055 71 6 ) ) PUNCT cana-6055 71 7 ⟢ ⟢ NOUN cana-6055 71 8 { { PUNCT cana-6055 71 9 0 0 NUM cana-6055 71 10 , , PUNCT cana-6055 71 11 Β±1 Β±1 VERB cana-6055 71 12 , , PUNCT cana-6055 71 13 Β±2,β‹― Β±2,β‹― NUM cana-6055 71 14 , , PUNCT cana-6055 71 15 ±𝑛 ±𝑛 PROPN cana-6055 71 16 βˆ’ βˆ’ NOUN cana-6055 71 17 1 1 NUM cana-6055 71 18 } } PUNCT cana-6055 71 19 by by ADP cana-6055 71 20 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 71 21 ) ) PUNCT cana-6055 71 22 = = SYM cana-6055 72 1 ( ( PUNCT cana-6055 72 2 𝑛 𝑛 PROPN cana-6055 72 3 + + NOUN cana-6055 72 4 1 1 NUM cana-6055 72 5 ) ) PUNCT cana-6055 72 6 βˆ’ βˆ’ NOUN cana-6055 72 7 2𝑖 2𝑖 NOUN cana-6055 72 8 , , PUNCT cana-6055 72 9 if if SCONJ cana-6055 72 10 𝑖 𝑖 PRON cana-6055 72 11 = = NOUN cana-6055 72 12 1,2,3 1,2,3 NUM cana-6055 72 13 , , PUNCT cana-6055 72 14 . . PUNCT cana-6055 72 15 . . PUNCT cana-6055 73 1 . . PUNCT cana-6055 74 1 , , PUNCT cana-6055 74 2 𝑛 𝑛 DET cana-6055 74 3 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 74 4 ) ) PUNCT cana-6055 74 5 = = SYM cana-6055 74 6 𝑛 𝑛 DET cana-6055 74 7 βˆ’ βˆ’ NOUN cana-6055 74 8 2𝑖 2𝑖 NOUN cana-6055 74 9 , , PUNCT cana-6055 74 10 if if SCONJ cana-6055 74 11 𝑖 𝑖 PRON cana-6055 74 12 = = NOUN cana-6055 74 13 1,2,3 1,2,3 NUM cana-6055 74 14 , , PUNCT cana-6055 74 15 . . PUNCT cana-6055 74 16 . . PUNCT cana-6055 75 1 . . PUNCT cana-6055 76 1 , , PUNCT cana-6055 76 2 𝑛 𝑛 DET cana-6055 76 3 βˆ’ βˆ’ NOUN cana-6055 76 4 1 1 NUM cana-6055 76 5 then then ADV cana-6055 76 6 the the DET cana-6055 76 7 induced induced ADJ cana-6055 76 8 vertex vertex NOUN cana-6055 76 9 weight weight NOUN cana-6055 76 10 labeling labeling NOUN cana-6055 76 11 are be AUX cana-6055 76 12 as as SCONJ cana-6055 76 13 follows follow VERB cana-6055 76 14 . . PUNCT cana-6055 77 1 𝑀(𝑣𝑖 𝑀(𝑣𝑖 VERB cana-6055 77 2 ) ) PUNCT cana-6055 77 3 = = PRON cana-6055 77 4 { { PUNCT cana-6055 77 5 2𝑛 2𝑛 PROPN cana-6055 77 6 βˆ’ βˆ’ PROPN cana-6055 77 7 5 5 NUM cana-6055 77 8 , , PUNCT cana-6055 77 9 𝑖𝑓 𝑖𝑓 ADP cana-6055 77 10 𝑖 𝑖 SYM cana-6055 78 1 = = NOUN cana-6055 78 2 1 1 NUM cana-6055 78 3 4(𝑛 4(𝑛 NUM cana-6055 78 4 βˆ’ βˆ’ NUM cana-6055 78 5 3 3 NUM cana-6055 78 6 ) ) PUNCT cana-6055 78 7 βˆ’ βˆ’ NOUN cana-6055 78 8 8(𝑖 8(𝑖 NUM cana-6055 78 9 βˆ’ βˆ’ PROPN cana-6055 78 10 2 2 NUM cana-6055 78 11 ) ) PUNCT cana-6055 78 12 , , PUNCT cana-6055 78 13 𝑖𝑓 𝑖𝑓 ADP cana-6055 78 14 𝑖 𝑖 X cana-6055 78 15 = = PUNCT cana-6055 78 16 2,3,4 2,3,4 NUM cana-6055 78 17 , , PUNCT cana-6055 78 18 . . PUNCT cana-6055 78 19 . . PUNCT cana-6055 78 20 . . PUNCT cana-6055 79 1 , , PUNCT cana-6055 80 1 𝑛 𝑛 PRON cana-6055 80 2 βˆ’ βˆ’ NOUN cana-6055 80 3 1 1 NUM cana-6055 80 4 βˆ’(2𝑛 βˆ’(2𝑛 NUM cana-6055 80 5 βˆ’ βˆ’ NOUN cana-6055 80 6 5 5 NUM cana-6055 80 7 ) ) PUNCT cana-6055 80 8 , , PUNCT cana-6055 80 9 𝑖𝑓 𝑖𝑓 ADP cana-6055 80 10 𝑖 𝑖 X cana-6055 80 11 = = SYM cana-6055 80 12 𝑛 𝑛 PRON cana-6055 80 13 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 80 14 ) ) PUNCT cana-6055 80 15 = = SYM cana-6055 80 16 2(𝑛 2(𝑛 NUM cana-6055 80 17 βˆ’ βˆ’ PROPN cana-6055 80 18 2𝑖 2𝑖 NOUN cana-6055 80 19 ) ) PUNCT cana-6055 81 1 if if SCONJ cana-6055 81 2 𝑖 𝑖 PRON cana-6055 81 3 = = NOUN cana-6055 81 4 1,2,3 1,2,3 NUM cana-6055 81 5 , , PUNCT cana-6055 81 6 . . PUNCT cana-6055 81 7 . . PUNCT cana-6055 82 1 . . PUNCT cana-6055 83 1 , , PUNCT cana-6055 83 2 𝑛 𝑛 DET cana-6055 83 3 βˆ’ βˆ’ PROPN cana-6055 83 4 1 1 NUM cana-6055 83 5 case(ii case(ii ADJ cana-6055 83 6 ): ): PUNCT cana-6055 83 7 𝑛 𝑛 PRON cana-6055 83 8 is be AUX cana-6055 83 9 even even ADV cana-6055 83 10 figure figure VERB cana-6055 83 11 3.2 3.2 NUM cana-6055 83 12 dpam dpam NOUN cana-6055 83 13 labeling labeling NOUN cana-6055 83 14 of of ADP cana-6055 83 15 𝑇2 𝑇2 NOUN cana-6055 83 16 and and CCONJ cana-6055 83 17 𝑇4 𝑇4 PROPN cana-6055 83 18 . . PUNCT cana-6055 84 1 clearly clearly ADV cana-6055 84 2 from from ADP cana-6055 84 3 figure figure NOUN cana-6055 84 4 3.2 3.2 NUM cana-6055 84 5 𝑇2 𝑇2 NOUN cana-6055 84 6 and and CCONJ cana-6055 84 7 𝑇4 𝑇4 PROPN cana-6055 84 8 are be AUX cana-6055 84 9 distance distance NOUN cana-6055 84 10 pair pair NOUN cana-6055 84 11 antimagic antimagic ADJ cana-6055 84 12 graphs graph NOUN cana-6055 84 13 . . PUNCT cana-6055 85 1 for for ADP cana-6055 85 2 𝑛 𝑛 PRON cana-6055 85 3 = = SYM cana-6055 85 4 6,8,10 6,8,10 NUM cana-6055 85 5 , , PUNCT cana-6055 85 6 . . PUNCT cana-6055 86 1 .. .. PUNCT cana-6055 86 2 define define VERB cana-6055 86 3 𝑓 𝑓 PRON cana-6055 86 4 : : PUNCT cana-6055 86 5 𝑉(𝑇𝑛 𝑉(𝑇𝑛 NUM cana-6055 86 6 ) ) PUNCT cana-6055 86 7 ⟢ ⟢ NOUN cana-6055 86 8 { { PUNCT cana-6055 86 9 0,Β±1,Β±2,β‹― 0,Β±1,Β±2,β‹― NUM cana-6055 86 10 , , PUNCT cana-6055 86 11 ±𝑛 ±𝑛 PROPN cana-6055 86 12 βˆ’ βˆ’ NOUN cana-6055 86 13 1 1 NUM cana-6055 86 14 } } PUNCT cana-6055 86 15 by by ADP cana-6055 86 16 𝑓(𝑣1 𝑓(𝑣1 NOUN cana-6055 86 17 ) ) PUNCT cana-6055 86 18 = = SYM cana-6055 87 1 𝑛 𝑛 PRON cana-6055 87 2 βˆ’ βˆ’ NUM cana-6055 87 3 1 1 NUM cana-6055 87 4 = = SYM cana-6055 87 5 βˆ’π‘“(𝑣𝑛 βˆ’π‘“(𝑣𝑛 NOUN cana-6055 87 6 ) ) PUNCT cana-6055 87 7 and and CCONJ cana-6055 87 8 remaining remain VERB cana-6055 87 9 vertices vertex NOUN cana-6055 87 10 has have AUX cana-6055 87 11 following follow VERB cana-6055 87 12 labeling labeling NOUN cana-6055 87 13 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 87 14 ) ) PUNCT cana-6055 87 15 = = PRON cana-6055 87 16 { { PUNCT cana-6055 87 17 𝑛 𝑛 PRON cana-6055 87 18 βˆ’ βˆ’ NOUN cana-6055 87 19 2(𝑖 2(𝑖 NUM cana-6055 87 20 βˆ’ βˆ’ NOUN cana-6055 87 21 1 1 NUM cana-6055 87 22 ) ) PUNCT cana-6055 87 23 , , PUNCT cana-6055 87 24 𝑖𝑓 𝑖𝑓 ADP cana-6055 87 25 1 1 NUM cana-6055 87 26 < < X cana-6055 87 27 𝑖 𝑖 X cana-6055 87 28 < < X cana-6055 87 29 𝑛 𝑛 PROPN cana-6055 87 30 2 2 NUM cana-6055 87 31 𝑛 𝑛 DET cana-6055 87 32 βˆ’ βˆ’ PROPN cana-6055 87 33 2𝑖 2𝑖 NOUN cana-6055 87 34 , , PUNCT cana-6055 87 35 𝑖𝑓 𝑖𝑓 ADP cana-6055 87 36 𝑛 𝑛 DET cana-6055 87 37 2 2 NUM cana-6055 87 38 ≀ ≀ NOUN cana-6055 87 39 𝑖 𝑖 ADP cana-6055 87 40 < < X cana-6055 87 41 𝑛 𝑛 PRON cana-6055 87 42 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 87 43 ) ) PUNCT cana-6055 87 44 = = SYM cana-6055 87 45 { { PUNCT cana-6055 87 46 𝑛 𝑛 PRON cana-6055 87 47 βˆ’ βˆ’ NUM cana-6055 87 48 1 1 NUM cana-6055 87 49 βˆ’ βˆ’ NOUN cana-6055 87 50 2𝑖 2𝑖 NOUN cana-6055 87 51 , , PUNCT cana-6055 87 52 𝑖𝑓 𝑖𝑓 ADP cana-6055 87 53 𝑖 𝑖 X cana-6055 87 54 = = NOUN cana-6055 87 55 1,2,3 1,2,3 NUM cana-6055 87 56 , , PUNCT cana-6055 87 57 . . PUNCT cana-6055 87 58 . . PUNCT cana-6055 88 1 . . PUNCT cana-6055 89 1 , , PUNCT cana-6055 89 2 𝑛 𝑛 PRON cana-6055 89 3 2 2 NUM cana-6055 89 4 βˆ’ βˆ’ NOUN cana-6055 89 5 1 1 NUM cana-6055 89 6 0 0 NUM cana-6055 89 7 , , PUNCT cana-6055 89 8 𝑖𝑓 𝑖𝑓 ADP cana-6055 89 9 𝑖 𝑖 NOUN cana-6055 89 10 = = SYM cana-6055 89 11 𝑛 𝑛 PRON cana-6055 89 12 2 2 NUM cana-6055 89 13 𝑛 𝑛 PRON cana-6055 89 14 βˆ’ βˆ’ PROPN cana-6055 89 15 1 1 NUM cana-6055 89 16 βˆ’ βˆ’ NOUN cana-6055 89 17 2(𝑖 2(𝑖 NUM cana-6055 89 18 βˆ’ βˆ’ NOUN cana-6055 89 19 1 1 NUM cana-6055 89 20 ) ) PUNCT cana-6055 89 21 𝑖𝑓 𝑖𝑓 ADP cana-6055 89 22 𝑖 𝑖 NOUN cana-6055 90 1 = = SYM cana-6055 90 2 𝑛 𝑛 PRON cana-6055 90 3 2 2 NUM cana-6055 90 4 + + NUM cana-6055 90 5 1 1 NUM cana-6055 90 6 , , PUNCT cana-6055 90 7 𝑛 𝑛 DET cana-6055 90 8 2 2 NUM cana-6055 90 9 + + SYM cana-6055 90 10 2 2 NUM cana-6055 90 11 , , PUNCT cana-6055 90 12 . . PUNCT cana-6055 90 13 . . PUNCT cana-6055 90 14 . . PUNCT cana-6055 91 1 , , PUNCT cana-6055 91 2 𝑛 𝑛 DET cana-6055 91 3 βˆ’ βˆ’ NOUN cana-6055 91 4 1 1 NUM cana-6055 91 5 then then ADV cana-6055 91 6 the the DET cana-6055 91 7 induced induced ADJ cana-6055 91 8 vertex vertex NOUN cana-6055 91 9 weight weight NOUN cana-6055 91 10 labeling labeling NOUN cana-6055 91 11 are be AUX cana-6055 91 12 as as SCONJ cana-6055 91 13 follows follow VERB cana-6055 91 14 . . PUNCT cana-6055 92 1 𝑣𝑖 𝑣𝑖 ADP cana-6055 92 2 𝑀(𝑣𝑖 𝑀(𝑣𝑖 ADV cana-6055 92 3 ) ) PUNCT cana-6055 92 4 𝑖 𝑖 SYM cana-6055 93 1 = = SYM cana-6055 93 2 1 1 NUM cana-6055 93 3 2𝑛 2𝑛 NOUN cana-6055 93 4 βˆ’ βˆ’ PROPN cana-6055 93 5 5 5 NUM cana-6055 93 6 𝑖 𝑖 NOUN cana-6055 93 7 = = NOUN cana-6055 93 8 2 2 NUM cana-6055 93 9 3 3 NUM cana-6055 93 10 + + SYM cana-6055 93 11 4(𝑛 4(𝑛 NUM cana-6055 93 12 βˆ’ βˆ’ NOUN cana-6055 93 13 4 4 NUM cana-6055 93 14 ) ) PUNCT cana-6055 93 15 𝑖 𝑖 NOUN cana-6055 93 16 = = PUNCT cana-6055 93 17 3,4,5 3,4,5 NUM cana-6055 93 18 , , PUNCT cana-6055 93 19 . . PUNCT cana-6055 93 20 . . PUNCT cana-6055 93 21 . . PUNCT cana-6055 94 1 , , PUNCT cana-6055 94 2 𝑛 𝑛 PRON cana-6055 94 3 2 2 NUM cana-6055 94 4 βˆ’ βˆ’ NOUN cana-6055 94 5 1 1 NUM cana-6055 94 6 4(𝑛 4(𝑛 NUM cana-6055 94 7 + + CCONJ cana-6055 94 8 1 1 NUM cana-6055 94 9 βˆ’ βˆ’ NOUN cana-6055 94 10 2𝑖 2𝑖 NUM cana-6055 94 11 ) ) PUNCT cana-6055 94 12 communications communication NOUN cana-6055 94 13 on on ADP cana-6055 94 14 applied apply VERB cana-6055 94 15 nonlinear nonlinear ADJ cana-6055 94 16 analysis analysis NOUN cana-6055 94 17 issn issn NOUN cana-6055 94 18 : : PUNCT cana-6055 94 19 1074 1074 NUM cana-6055 94 20 - - PUNCT cana-6055 94 21 133x 133x NUM cana-6055 94 22 vol vol NOUN cana-6055 94 23 31 31 NUM cana-6055 94 24 no no NOUN cana-6055 94 25 . . PUNCT cana-6055 95 1 8s 8s PROPN cana-6055 95 2 ( ( PUNCT cana-6055 95 3 2024 2024 NUM cana-6055 95 4 ) ) PUNCT cana-6055 95 5 1159 1159 NUM cana-6055 95 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 95 7 𝑖 𝑖 SYM cana-6055 95 8 = = SYM cana-6055 95 9 𝑛 𝑛 PRON cana-6055 95 10 2 2 NUM cana-6055 95 11 3 3 NUM cana-6055 95 12 𝑖 𝑖 NOUN cana-6055 95 13 = = SYM cana-6055 95 14 𝑛 𝑛 PRON cana-6055 95 15 2 2 NUM cana-6055 95 16 + + CCONJ cana-6055 95 17 1 1 NUM cana-6055 95 18 βˆ’3 βˆ’3 NOUN cana-6055 95 19 𝑖 𝑖 NOUN cana-6055 95 20 = = SYM cana-6055 95 21 𝑛 𝑛 PRON cana-6055 95 22 2 2 NUM cana-6055 95 23 + + CCONJ cana-6055 95 24 2 2 NUM cana-6055 95 25 , , PUNCT cana-6055 95 26 𝑛 𝑛 DET cana-6055 95 27 2 2 NUM cana-6055 95 28 + + SYM cana-6055 95 29 3 3 NUM cana-6055 95 30 , , PUNCT cana-6055 95 31 . . PUNCT cana-6055 95 32 . . PUNCT cana-6055 95 33 . . PUNCT cana-6055 96 1 , , PUNCT cana-6055 96 2 𝑛 𝑛 DET cana-6055 96 3 βˆ’ βˆ’ PROPN cana-6055 96 4 2 2 NUM cana-6055 96 5 4(𝑛 4(𝑛 NUM cana-6055 96 6 + + CCONJ cana-6055 96 7 1 1 NUM cana-6055 96 8 βˆ’ βˆ’ NOUN cana-6055 96 9 2𝑖 2𝑖 NOUN cana-6055 96 10 ) ) PUNCT cana-6055 97 1 𝑖 𝑖 NOUN cana-6055 98 1 = = NOUN cana-6055 98 2 𝑛 𝑛 PRON cana-6055 98 3 βˆ’ βˆ’ NUM cana-6055 98 4 1 1 NUM cana-6055 98 5 βˆ’(3 βˆ’(3 NOUN cana-6055 98 6 + + NUM cana-6055 98 7 4(𝑛 4(𝑛 NUM cana-6055 98 8 βˆ’ βˆ’ NOUN cana-6055 98 9 4 4 NUM cana-6055 98 10 ) ) PUNCT cana-6055 98 11 ) ) PUNCT cana-6055 99 1 𝑖 𝑖 X cana-6055 99 2 = = PUNCT cana-6055 99 3 𝑛 𝑛 PRON cana-6055 99 4 βˆ’(2𝑛 βˆ’(2𝑛 NUM cana-6055 99 5 βˆ’ βˆ’ NOUN cana-6055 99 6 5 5 NUM cana-6055 99 7 ) ) PUNCT cana-6055 99 8 𝑒𝑖 𝑒𝑖 PRON cana-6055 99 9 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 99 10 ) ) PUNCT cana-6055 99 11 𝑖 𝑖 SYM cana-6055 100 1 = = SYM cana-6055 100 2 1 1 NUM cana-6055 100 3 2𝑛 2𝑛 NOUN cana-6055 100 4 βˆ’ βˆ’ PROPN cana-6055 100 5 3 3 NUM cana-6055 100 6 𝑖 𝑖 NOUN cana-6055 100 7 = = NOUN cana-6055 100 8 2,3,4 2,3,4 NUM cana-6055 100 9 , , PUNCT cana-6055 100 10 . . PUNCT cana-6055 100 11 . . PUNCT cana-6055 100 12 . . PUNCT cana-6055 101 1 , , PUNCT cana-6055 101 2 𝑛 𝑛 DET cana-6055 101 3 2 2 NUM cana-6055 101 4 βˆ’ βˆ’ NOUN cana-6055 101 5 1 1 NUM cana-6055 101 6 2(𝑛 2(𝑛 NUM cana-6055 101 7 + + CCONJ cana-6055 101 8 1 1 NUM cana-6055 101 9 ) ) PUNCT cana-6055 101 10 βˆ’ βˆ’ NOUN cana-6055 101 11 4𝑖 4𝑖 NOUN cana-6055 101 12 𝑖 𝑖 NOUN cana-6055 102 1 = = SYM cana-6055 102 2 𝑛 𝑛 PRON cana-6055 102 3 2 2 NUM cana-6055 102 4 0 0 NUM cana-6055 102 5 𝑛 𝑛 DET cana-6055 102 6 2 2 NUM cana-6055 102 7 + + NUM cana-6055 102 8 1 1 NUM cana-6055 102 9 , , PUNCT cana-6055 102 10 𝑛 𝑛 DET cana-6055 102 11 2 2 NUM cana-6055 102 12 + + SYM cana-6055 102 13 2 2 NUM cana-6055 102 14 , , PUNCT cana-6055 102 15 . . PUNCT cana-6055 102 16 . . PUNCT cana-6055 102 17 . . PUNCT cana-6055 103 1 , , PUNCT cana-6055 103 2 𝑛 𝑛 DET cana-6055 103 3 βˆ’ βˆ’ NUM cana-6055 103 4 2 2 NUM cana-6055 103 5 2(𝑛 2(𝑛 NUM cana-6055 103 6 βˆ’ βˆ’ NUM cana-6055 103 7 1 1 NUM cana-6055 103 8 ) ) PUNCT cana-6055 103 9 βˆ’ βˆ’ PROPN cana-6055 104 1 4𝑖 4𝑖 NOUN cana-6055 104 2 𝑖 𝑖 PUNCT cana-6055 104 3 = = VERB cana-6055 104 4 𝑛 𝑛 PRON cana-6055 104 5 βˆ’ βˆ’ NUM cana-6055 104 6 1 1 NUM cana-6055 104 7 βˆ’(2𝑛 βˆ’(2𝑛 NUM cana-6055 104 8 βˆ’ βˆ’ NOUN cana-6055 104 9 3 3 NUM cana-6055 104 10 ) ) PUNCT cana-6055 104 11 hence hence ADV cana-6055 104 12 𝑇𝑛 𝑇𝑛 ADV cana-6055 104 13 is be AUX cana-6055 104 14 a a DET cana-6055 104 15 distance distance NOUN cana-6055 104 16 pair pair NOUN cana-6055 104 17 antimagic antimagic ADJ cana-6055 104 18 graph graph NOUN cana-6055 104 19 for for ADP cana-6055 104 20 all all DET cana-6055 104 21 𝑛 𝑛 PRON cana-6055 104 22 β‰₯ β‰₯ NUM cana-6055 104 23 2 2 NUM cana-6055 104 24 . . PUNCT cana-6055 104 25 theorem theorem VERB cana-6055 104 26 3.4 3.4 NUM cana-6055 104 27 the the DET cana-6055 104 28 n n CCONJ cana-6055 104 29 - - PUNCT cana-6055 104 30 gear gear NOUN cana-6055 104 31 graph graph NOUN cana-6055 104 32 𝐺𝑛 𝐺𝑛 PROPN cana-6055 104 33 is be AUX cana-6055 104 34 a a DET cana-6055 104 35 distance distance NOUN cana-6055 104 36 pair pair NOUN cana-6055 104 37 antimagic antimagic ADJ cana-6055 104 38 graph graph NOUN cana-6055 104 39 , , PUNCT cana-6055 104 40 if if SCONJ cana-6055 104 41 𝑛 𝑛 PRON cana-6055 104 42 = = SYM cana-6055 104 43 4,6,8 4,6,8 NUM cana-6055 104 44 , , PUNCT cana-6055 104 45 … … PUNCT cana-6055 104 46 proof proof NOUN cana-6055 104 47 . . PUNCT cana-6055 105 1 let let VERB cana-6055 105 2 𝑉(𝐺𝑛 𝑉(𝐺𝑛 NUM cana-6055 105 3 ) ) PUNCT cana-6055 105 4 = = PRON cana-6055 105 5 { { PUNCT cana-6055 105 6 𝑒0 𝑒0 PROPN cana-6055 105 7 , , PUNCT cana-6055 105 8 𝑒𝑖 𝑒𝑖 INTJ cana-6055 105 9 , , PUNCT cana-6055 105 10 𝑣𝑖 𝑣𝑖 ADP cana-6055 105 11 : : PUNCT cana-6055 105 12 𝑖 𝑖 SYM cana-6055 105 13 = = NOUN cana-6055 105 14 1,2,3 1,2,3 NUM cana-6055 105 15 , , PUNCT cana-6055 105 16 . . PUNCT cana-6055 105 17 . . PUNCT cana-6055 106 1 . . PUNCT cana-6055 107 1 , , PUNCT cana-6055 107 2 𝑛 𝑛 X cana-6055 107 3 } } PUNCT cana-6055 107 4 and and CCONJ cana-6055 107 5 𝐸(𝐺𝑛 𝐸(𝐺𝑛 NOUN cana-6055 107 6 ) ) PUNCT cana-6055 107 7 = = SYM cana-6055 107 8 { { PUNCT cana-6055 107 9 𝑒𝑖𝑣𝑖 𝑒𝑖𝑣𝑖 PROPN cana-6055 107 10 , , PUNCT cana-6055 107 11 𝑣𝑖𝑒𝑖+1 𝑣𝑖𝑒𝑖+1 PROPN cana-6055 107 12 : : PUNCT cana-6055 107 13 𝑖 𝑖 SYM cana-6055 107 14 = = NOUN cana-6055 107 15 1,2,3 1,2,3 NUM cana-6055 107 16 , , PUNCT cana-6055 107 17 . . PUNCT cana-6055 107 18 . . PUNCT cana-6055 108 1 . . PUNCT cana-6055 109 1 , , PUNCT cana-6055 109 2 𝑛 𝑛 DET cana-6055 109 3 βˆ’ βˆ’ NOUN cana-6055 109 4 1 1 NUM cana-6055 109 5 } } PUNCT cana-6055 109 6 βˆͺ βˆͺ ADJ cana-6055 109 7 { { PUNCT cana-6055 109 8 𝑣𝑛𝑒1 𝑣𝑛𝑒1 PROPN cana-6055 109 9 } } PUNCT cana-6055 109 10 βˆͺ βˆͺ X cana-6055 109 11 { { PUNCT cana-6055 109 12 𝑒0𝑒𝑖 𝑒0𝑒𝑖 X cana-6055 109 13 : : PUNCT cana-6055 109 14 𝑖 𝑖 X cana-6055 109 15 = = NOUN cana-6055 109 16 1,2,3 1,2,3 NUM cana-6055 109 17 , , PUNCT cana-6055 109 18 . . PUNCT cana-6055 109 19 . . PUNCT cana-6055 110 1 . . PUNCT cana-6055 111 1 , , PUNCT cana-6055 111 2 𝑛 𝑛 X cana-6055 111 3 } } PUNCT cana-6055 111 4 be be VERB cana-6055 111 5 vertex vertex NOUN cana-6055 111 6 set set NOUN cana-6055 111 7 and and CCONJ cana-6055 111 8 edge edge NOUN cana-6055 111 9 set set NOUN cana-6055 111 10 of of ADP cana-6055 111 11 𝐺𝑛. 𝐺𝑛. NOUN cana-6055 111 12 define define VERB cana-6055 111 13 𝑓 𝑓 PRON cana-6055 111 14 : : PUNCT cana-6055 111 15 𝑉(𝐺𝑛 𝑉(𝐺𝑛 NUM cana-6055 111 16 ) ) PUNCT cana-6055 111 17 ⟢ ⟢ NOUN cana-6055 111 18 { { PUNCT cana-6055 111 19 0 0 NUM cana-6055 111 20 , , PUNCT cana-6055 111 21 Β±1 Β±1 VERB cana-6055 111 22 , , PUNCT cana-6055 111 23 Β±2,β‹― Β±2,β‹― NUM cana-6055 111 24 , , PUNCT cana-6055 111 25 ±𝑛 ±𝑛 PROPN cana-6055 111 26 } } PUNCT cana-6055 111 27 by by ADP cana-6055 111 28 the the DET cana-6055 111 29 following follow VERB cana-6055 111 30 two two NUM cana-6055 111 31 cases case NOUN cana-6055 111 32 case case NOUN cana-6055 111 33 : : PUNCT cana-6055 111 34 ( ( PUNCT cana-6055 111 35 i i NOUN cana-6055 111 36 ) ) PUNCT cana-6055 111 37 𝑛 𝑛 PRON cana-6055 111 38 ≑ ≑ PROPN cana-6055 111 39 2 2 NUM cana-6055 111 40 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ NOUN cana-6055 111 41 4 4 NUM cana-6055 111 42 𝑒𝑖 𝑒𝑖 NOUN cana-6055 111 43 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 111 44 ) ) PUNCT cana-6055 111 45 𝑖 𝑖 X cana-6055 112 1 = = NOUN cana-6055 112 2 0 0 PUNCT cana-6055 112 3 0 0 NUM cana-6055 112 4 𝑖 𝑖 NOUN cana-6055 112 5 = = NOUN cana-6055 112 6 1,2,3,4 1,2,3,4 NUM cana-6055 112 7 , , PUNCT cana-6055 112 8 . . PUNCT cana-6055 112 9 . . PUNCT cana-6055 112 10 . . PUNCT cana-6055 113 1 , , PUNCT cana-6055 113 2 𝑛 𝑛 PRON cana-6055 113 3 2 2 NUM cana-6055 113 4 βˆ’ βˆ’ NOUN cana-6055 113 5 1 1 NUM cana-6055 113 6 𝑖 𝑖 SYM cana-6055 113 7 𝑖 𝑖 NOUN cana-6055 113 8 = = VERB cana-6055 113 9 𝑛 𝑛 PRON cana-6055 113 10 2 2 NUM cana-6055 113 11 𝑛 𝑛 NOUN cana-6055 113 12 𝑖 𝑖 NOUN cana-6055 113 13 = = SYM cana-6055 113 14 𝑛 𝑛 PRON cana-6055 113 15 2 2 NUM cana-6055 113 16 + + NUM cana-6055 113 17 1 1 NUM cana-6055 113 18 , , PUNCT cana-6055 113 19 𝑛 𝑛 DET cana-6055 113 20 2 2 NUM cana-6055 113 21 + + SYM cana-6055 113 22 2 2 NUM cana-6055 113 23 , , PUNCT cana-6055 113 24 . . PUNCT cana-6055 113 25 . . PUNCT cana-6055 113 26 . . PUNCT cana-6055 114 1 , , PUNCT cana-6055 114 2 𝑛 𝑛 DET cana-6055 114 3 βˆ’ βˆ’ NUM cana-6055 114 4 1 1 NUM cana-6055 114 5 𝑛 𝑛 DET cana-6055 114 6 2 2 NUM cana-6055 114 7 βˆ’ βˆ’ NOUN cana-6055 114 8 𝑖 𝑖 SYM cana-6055 114 9 𝑖 𝑖 NOUN cana-6055 114 10 = = PUNCT cana-6055 114 11 𝑛 𝑛 PRON cana-6055 114 12 βˆ’π‘› βˆ’π‘› NOUN cana-6055 114 13 𝑣𝑖 𝑣𝑖 ADP cana-6055 114 14 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 114 15 ) ) PUNCT cana-6055 114 16 𝑖 𝑖 SYM cana-6055 114 17 = = NOUN cana-6055 114 18 1,2,3,4 1,2,3,4 NUM cana-6055 114 19 , , PUNCT cana-6055 114 20 . . PUNCT cana-6055 114 21 . . PUNCT cana-6055 115 1 . . PUNCT cana-6055 116 1 , , PUNCT cana-6055 116 2 𝑛 𝑛 PROPN cana-6055 116 3 2 2 NUM cana-6055 116 4 𝑛 𝑛 DET cana-6055 116 5 2 2 NUM cana-6055 116 6 + + CCONJ cana-6055 116 7 ( ( PUNCT cana-6055 116 8 𝑖 𝑖 NOUN cana-6055 116 9 βˆ’ βˆ’ PROPN cana-6055 116 10 1 1 NUM cana-6055 116 11 ) ) PUNCT cana-6055 116 12 communications communication NOUN cana-6055 116 13 on on ADP cana-6055 116 14 applied apply VERB cana-6055 116 15 nonlinear nonlinear ADJ cana-6055 116 16 analysis analysis NOUN cana-6055 116 17 issn issn NOUN cana-6055 116 18 : : PUNCT cana-6055 116 19 1074 1074 NUM cana-6055 116 20 - - PUNCT cana-6055 116 21 133x 133x NUM cana-6055 116 22 vol vol NOUN cana-6055 116 23 31 31 NUM cana-6055 116 24 no no NOUN cana-6055 116 25 . . PUNCT cana-6055 117 1 8s 8s PROPN cana-6055 117 2 ( ( PUNCT cana-6055 117 3 2024 2024 NUM cana-6055 117 4 ) ) PUNCT cana-6055 117 5 1160 1160 NUM cana-6055 117 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 117 7 𝑖 𝑖 SYM cana-6055 117 8 = = SYM cana-6055 117 9 𝑛 𝑛 PRON cana-6055 117 10 2 2 NUM cana-6055 117 11 + + NUM cana-6055 117 12 1 1 NUM cana-6055 117 13 , , PUNCT cana-6055 117 14 𝑛 𝑛 DET cana-6055 117 15 2 2 NUM cana-6055 117 16 + + SYM cana-6055 117 17 2 2 NUM cana-6055 117 18 , , PUNCT cana-6055 117 19 . . PUNCT cana-6055 117 20 . . PUNCT cana-6055 117 21 . . PUNCT cana-6055 118 1 , , PUNCT cana-6055 118 2 𝑛 𝑛 DET cana-6055 118 3 βˆ’(𝑖 βˆ’(𝑖 NOUN cana-6055 118 4 βˆ’ βˆ’ PROPN cana-6055 118 5 1 1 NUM cana-6055 118 6 ) ) PUNCT cana-6055 118 7 then then ADV cana-6055 118 8 the the DET cana-6055 118 9 induced induced ADJ cana-6055 118 10 vertex vertex NOUN cana-6055 118 11 weight weight NOUN cana-6055 118 12 labeling labeling NOUN cana-6055 118 13 are be AUX cana-6055 118 14 as as SCONJ cana-6055 118 15 follows follow VERB cana-6055 118 16 . . PUNCT cana-6055 119 1 𝑒𝑖 𝑒𝑖 PRON cana-6055 119 2 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 119 3 ) ) PUNCT cana-6055 119 4 𝑖 𝑖 SYM cana-6055 119 5 = = NOUN cana-6055 120 1 1 1 NUM cana-6055 120 2 βˆ’ βˆ’ NOUN cana-6055 120 3 ( ( PUNCT cana-6055 120 4 𝑛 𝑛 PROPN cana-6055 120 5 2 2 NUM cana-6055 120 6 βˆ’ βˆ’ NOUN cana-6055 120 7 1 1 NUM cana-6055 120 8 ) ) PUNCT cana-6055 120 9 𝑖 𝑖 NOUN cana-6055 120 10 = = PUNCT cana-6055 120 11 2,3,4 2,3,4 NUM cana-6055 120 12 , , PUNCT cana-6055 120 13 . . PUNCT cana-6055 120 14 . . PUNCT cana-6055 120 15 . . PUNCT cana-6055 121 1 , , PUNCT cana-6055 121 2 𝑛 𝑛 PROPN cana-6055 121 3 2 2 NUM cana-6055 121 4 ( ( PUNCT cana-6055 121 5 𝑛 𝑛 PROPN cana-6055 121 6 βˆ’ βˆ’ PROPN cana-6055 121 7 1 1 NUM cana-6055 121 8 ) ) PUNCT cana-6055 121 9 + + CCONJ cana-6055 121 10 2(𝑖 2(𝑖 NUM cana-6055 121 11 βˆ’ βˆ’ NOUN cana-6055 121 12 1 1 NUM cana-6055 121 13 ) ) PUNCT cana-6055 121 14 𝑖 𝑖 NOUN cana-6055 121 15 = = SYM cana-6055 121 16 𝑛 𝑛 PRON cana-6055 121 17 2 2 NUM cana-6055 121 18 + + SYM cana-6055 121 19 1 1 NUM cana-6055 121 20 𝑛 𝑛 DET cana-6055 121 21 2 2 NUM cana-6055 121 22 βˆ’ βˆ’ NOUN cana-6055 121 23 1 1 NUM cana-6055 121 24 𝑖 𝑖 NOUN cana-6055 121 25 = = SYM cana-6055 121 26 𝑛 𝑛 PRON cana-6055 121 27 2 2 NUM cana-6055 121 28 + + CCONJ cana-6055 121 29 2 2 NUM cana-6055 121 30 , , PUNCT cana-6055 121 31 𝑛 𝑛 DET cana-6055 121 32 2 2 NUM cana-6055 121 33 + + SYM cana-6055 121 34 3 3 NUM cana-6055 121 35 , , PUNCT cana-6055 121 36 . . PUNCT cana-6055 121 37 . . PUNCT cana-6055 121 38 . . PUNCT cana-6055 122 1 , , PUNCT cana-6055 122 2 𝑛 𝑛 DET cana-6055 122 3 3 3 NUM cana-6055 122 4 βˆ’ βˆ’ NOUN cana-6055 122 5 2𝑖 2𝑖 NOUN cana-6055 122 6 𝑣𝑖 𝑣𝑖 ADP cana-6055 122 7 𝑀(𝑣𝑖 𝑀(𝑣𝑖 VERB cana-6055 122 8 ) ) PUNCT cana-6055 122 9 𝑖 𝑖 X cana-6055 123 1 = = NOUN cana-6055 123 2 1,2,3 1,2,3 NUM cana-6055 123 3 , , PUNCT cana-6055 123 4 . . PUNCT cana-6055 123 5 . . PUNCT cana-6055 124 1 . . PUNCT cana-6055 125 1 , , PUNCT cana-6055 125 2 𝑛 𝑛 PRON cana-6055 125 3 2 2 NUM cana-6055 125 4 βˆ’ βˆ’ NUM cana-6055 125 5 2 2 NUM cana-6055 125 6 2𝑖 2𝑖 NOUN cana-6055 125 7 + + CCONJ cana-6055 125 8 1 1 NUM cana-6055 125 9 𝑖 𝑖 NOUN cana-6055 125 10 = = SYM cana-6055 125 11 𝑛 𝑛 PRON cana-6055 125 12 2 2 NUM cana-6055 125 13 βˆ’ βˆ’ NOUN cana-6055 125 14 1 1 NUM cana-6055 125 15 3𝑛 3𝑛 NUM cana-6055 125 16 2 2 NUM cana-6055 125 17 βˆ’ βˆ’ PROPN cana-6055 125 18 1 1 NUM cana-6055 125 19 𝑖 𝑖 NOUN cana-6055 125 20 = = SYM cana-6055 125 21 𝑛 𝑛 DET cana-6055 125 22 2 2 NUM cana-6055 125 23 𝑛 𝑛 DET cana-6055 125 24 βˆ’ βˆ’ PROPN cana-6055 125 25 1 1 NUM cana-6055 125 26 𝑖 𝑖 NOUN cana-6055 125 27 = = NOUN cana-6055 125 28 𝑛 𝑛 PRON cana-6055 125 29 2 2 NUM cana-6055 125 30 + + NUM cana-6055 125 31 1 1 NUM cana-6055 125 32 , , PUNCT cana-6055 125 33 𝑛 𝑛 DET cana-6055 125 34 2 2 NUM cana-6055 125 35 + + SYM cana-6055 125 36 2 2 NUM cana-6055 125 37 , , PUNCT cana-6055 125 38 . . PUNCT cana-6055 125 39 . . PUNCT cana-6055 125 40 . . PUNCT cana-6055 126 1 , , PUNCT cana-6055 127 1 𝑛 𝑛 DET cana-6055 127 2 βˆ’ βˆ’ PROPN cana-6055 127 3 2 2 NUM cana-6055 127 4 𝑛 𝑛 PRON cana-6055 127 5 βˆ’ βˆ’ NUM cana-6055 127 6 1 1 NUM cana-6055 127 7 βˆ’ βˆ’ NOUN cana-6055 127 8 2𝑖 2𝑖 NOUN cana-6055 127 9 𝑖 𝑖 X cana-6055 127 10 = = SYM cana-6055 127 11 𝑛 𝑛 PRON cana-6055 127 12 βˆ’ βˆ’ NUM cana-6055 127 13 1 1 NUM cana-6055 127 14 βˆ’ βˆ’ PROPN cana-6055 127 15 ( ( PUNCT cana-6055 127 16 3𝑛 3𝑛 NUM cana-6055 127 17 2 2 NUM cana-6055 127 18 βˆ’ βˆ’ NOUN cana-6055 127 19 1 1 NUM cana-6055 127 20 ) ) PUNCT cana-6055 127 21 𝑖 𝑖 NOUN cana-6055 127 22 = = SYM cana-6055 127 23 𝑛 𝑛 DET cana-6055 127 24 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-6055 127 25 βˆ’ βˆ’ NOUN cana-6055 127 26 1 1 NUM cana-6055 127 27 ) ) PUNCT cana-6055 127 28 case case NOUN cana-6055 127 29 : : PUNCT cana-6055 127 30 ( ( PUNCT cana-6055 127 31 ii ii NOUN cana-6055 127 32 ) ) PUNCT cana-6055 127 33 𝑛 𝑛 PRON cana-6055 127 34 ≑ ≑ PROPN cana-6055 127 35 0 0 NUM cana-6055 127 36 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ PROPN cana-6055 127 37 4 4 NUM cana-6055 127 38 𝑒𝑖 𝑒𝑖 NOUN cana-6055 127 39 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 127 40 ) ) PUNCT cana-6055 127 41 𝑖 𝑖 X cana-6055 128 1 = = NOUN cana-6055 128 2 0 0 PUNCT cana-6055 128 3 0 0 NUM cana-6055 128 4 𝑖 𝑖 NOUN cana-6055 128 5 = = NOUN cana-6055 128 6 1,2,3,4 1,2,3,4 NUM cana-6055 128 7 , , PUNCT cana-6055 128 8 . . PUNCT cana-6055 128 9 . . PUNCT cana-6055 128 10 . . PUNCT cana-6055 129 1 , , PUNCT cana-6055 129 2 𝑛 𝑛 DET cana-6055 129 3 4 4 NUM cana-6055 129 4 2𝑖 2𝑖 NOUN cana-6055 129 5 βˆ’ βˆ’ NOUN cana-6055 129 6 1 1 NUM cana-6055 129 7 𝑖 𝑖 NOUN cana-6055 129 8 = = NOUN cana-6055 129 9 𝑛 𝑛 PRON cana-6055 129 10 4 4 NUM cana-6055 129 11 + + SYM cana-6055 129 12 1 1 NUM cana-6055 129 13 , , PUNCT cana-6055 129 14 𝑛 𝑛 DET cana-6055 129 15 4 4 NUM cana-6055 129 16 + + SYM cana-6055 129 17 2 2 NUM cana-6055 129 18 , , PUNCT cana-6055 129 19 . . PUNCT cana-6055 129 20 . . PUNCT cana-6055 130 1 . . PUNCT cana-6055 131 1 , , PUNCT cana-6055 131 2 𝑛 𝑛 PRON cana-6055 131 3 2 2 NUM cana-6055 131 4 2(𝑖 2(𝑖 NUM cana-6055 131 5 βˆ’ βˆ’ NOUN cana-6055 131 6 𝑛 𝑛 PROPN cana-6055 131 7 4 4 NUM cana-6055 131 8 ) ) PUNCT cana-6055 131 9 𝑖 𝑖 NOUN cana-6055 131 10 = = SYM cana-6055 131 11 𝑛 𝑛 PRON cana-6055 131 12 2 2 NUM cana-6055 131 13 + + NUM cana-6055 131 14 1 1 NUM cana-6055 131 15 , , PUNCT cana-6055 131 16 𝑛 𝑛 DET cana-6055 131 17 2 2 NUM cana-6055 131 18 + + SYM cana-6055 131 19 2 2 NUM cana-6055 131 20 , , PUNCT cana-6055 131 21 . . PUNCT cana-6055 131 22 . . PUNCT cana-6055 132 1 . . PUNCT cana-6055 133 1 , , PUNCT cana-6055 133 2 3𝑛 3𝑛 NUM cana-6055 133 3 4 4 NUM cana-6055 133 4 𝑛 𝑛 NOUN cana-6055 133 5 + + NOUN cana-6055 133 6 1 1 NUM cana-6055 133 7 βˆ’ βˆ’ NOUN cana-6055 133 8 2𝑖 2𝑖 NOUN cana-6055 133 9 𝑖 𝑖 SYM cana-6055 133 10 = = SYM cana-6055 133 11 3𝑛 3𝑛 NUM cana-6055 133 12 4 4 NUM cana-6055 133 13 + + SYM cana-6055 133 14 1 1 NUM cana-6055 133 15 , , PUNCT cana-6055 133 16 3𝑛 3𝑛 NUM cana-6055 133 17 4 4 NUM cana-6055 133 18 + + SYM cana-6055 133 19 2 2 NUM cana-6055 133 20 , , PUNCT cana-6055 133 21 . . PUNCT cana-6055 133 22 . . PUNCT cana-6055 133 23 . . PUNCT cana-6055 134 1 , , PUNCT cana-6055 135 1 𝑛 𝑛 DET cana-6055 135 2 3𝑛 3𝑛 NUM cana-6055 135 3 2 2 NUM cana-6055 135 4 βˆ’ βˆ’ NOUN cana-6055 135 5 2𝑖 2𝑖 NOUN cana-6055 135 6 𝑣𝑖 𝑣𝑖 ADP cana-6055 135 7 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 135 8 ) ) PUNCT cana-6055 135 9 𝑖 𝑖 X cana-6055 135 10 = = NOUN cana-6055 135 11 1,2,3 1,2,3 NUM cana-6055 135 12 , , PUNCT cana-6055 135 13 . . PUNCT cana-6055 135 14 . . PUNCT cana-6055 136 1 . . PUNCT cana-6055 137 1 , , PUNCT cana-6055 137 2 𝑛 𝑛 PROPN cana-6055 137 3 4 4 NUM cana-6055 137 4 𝑛 𝑛 DET cana-6055 137 5 βˆ’ βˆ’ NOUN cana-6055 137 6 2(𝑖 2(𝑖 NUM cana-6055 137 7 βˆ’ βˆ’ NOUN cana-6055 137 8 1 1 NUM cana-6055 137 9 ) ) PUNCT cana-6055 137 10 𝑖 𝑖 NOUN cana-6055 138 1 = = SYM cana-6055 138 2 𝑛 𝑛 PRON cana-6055 138 3 4 4 NUM cana-6055 138 4 + + SYM cana-6055 138 5 1 1 NUM cana-6055 138 6 , , PUNCT cana-6055 138 7 𝑛 𝑛 DET cana-6055 138 8 4 4 NUM cana-6055 138 9 + + SYM cana-6055 138 10 2 2 NUM cana-6055 138 11 , , PUNCT cana-6055 138 12 . . PUNCT cana-6055 138 13 . . PUNCT cana-6055 139 1 . . PUNCT cana-6055 140 1 , , PUNCT cana-6055 140 2 𝑛 𝑛 PRON cana-6055 140 3 2 2 NUM cana-6055 140 4 2(𝑖 2(𝑖 NUM cana-6055 140 5 βˆ’ βˆ’ NOUN cana-6055 140 6 1 1 NUM cana-6055 140 7 ) ) PUNCT cana-6055 140 8 + + CCONJ cana-6055 140 9 1 1 NUM cana-6055 140 10 communications communication NOUN cana-6055 140 11 on on ADP cana-6055 140 12 applied apply VERB cana-6055 140 13 nonlinear nonlinear ADJ cana-6055 140 14 analysis analysis NOUN cana-6055 140 15 issn issn NOUN cana-6055 140 16 : : PUNCT cana-6055 140 17 1074 1074 NUM cana-6055 140 18 - - PUNCT cana-6055 140 19 133x 133x NUM cana-6055 140 20 vol vol NOUN cana-6055 140 21 31 31 NUM cana-6055 140 22 no no NOUN cana-6055 140 23 . . PUNCT cana-6055 141 1 8s 8s PROPN cana-6055 141 2 ( ( PUNCT cana-6055 141 3 2024 2024 NUM cana-6055 141 4 ) ) PUNCT cana-6055 141 5 1161 1161 NUM cana-6055 141 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 141 7 𝑖 𝑖 SYM cana-6055 141 8 = = SYM cana-6055 141 9 𝑛 𝑛 PRON cana-6055 141 10 2 2 NUM cana-6055 141 11 + + NUM cana-6055 141 12 1 1 NUM cana-6055 141 13 , , PUNCT cana-6055 141 14 𝑛 𝑛 DET cana-6055 141 15 2 2 NUM cana-6055 141 16 + + SYM cana-6055 141 17 2 2 NUM cana-6055 141 18 , , PUNCT cana-6055 141 19 . . PUNCT cana-6055 141 20 . . PUNCT cana-6055 141 21 . . PUNCT cana-6055 142 1 , , PUNCT cana-6055 142 2 3𝑛 3𝑛 NUM cana-6055 142 3 4 4 NUM cana-6055 142 4 βˆ’2(𝑛 βˆ’2(𝑛 NOUN cana-6055 142 5 + + CCONJ cana-6055 142 6 1 1 NUM cana-6055 142 7 βˆ’ βˆ’ NUM cana-6055 142 8 𝑖 𝑖 SYM cana-6055 142 9 ) ) PUNCT cana-6055 142 10 𝑖 𝑖 NOUN cana-6055 142 11 = = SYM cana-6055 142 12 3𝑛 3𝑛 NUM cana-6055 142 13 4 4 NUM cana-6055 142 14 + + SYM cana-6055 142 15 1 1 NUM cana-6055 142 16 , , PUNCT cana-6055 142 17 3𝑛 3𝑛 NUM cana-6055 142 18 4 4 NUM cana-6055 142 19 + + SYM cana-6055 142 20 2 2 NUM cana-6055 142 21 , , PUNCT cana-6055 142 22 . . PUNCT cana-6055 142 23 . . PUNCT cana-6055 143 1 . . PUNCT cana-6055 144 1 , , PUNCT cana-6055 144 2 𝑛 𝑛 PRON cana-6055 144 3 𝑛 𝑛 NOUN cana-6055 144 4 + + NOUN cana-6055 144 5 1 1 NUM cana-6055 144 6 βˆ’ βˆ’ NOUN cana-6055 144 7 2𝑖 2𝑖 NOUN cana-6055 144 8 then then ADV cana-6055 144 9 the the DET cana-6055 144 10 induced induced ADJ cana-6055 144 11 vertex vertex NOUN cana-6055 144 12 weight weight NOUN cana-6055 144 13 labeling labeling NOUN cana-6055 144 14 are be AUX cana-6055 144 15 as as SCONJ cana-6055 144 16 follows follow VERB cana-6055 144 17 . . PUNCT cana-6055 145 1 𝑒𝑖 𝑒𝑖 PRON cana-6055 145 2 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 145 3 ) ) PUNCT cana-6055 145 4 𝑖 𝑖 SYM cana-6055 146 1 = = NOUN cana-6055 146 2 0 0 PUNCT cana-6055 146 3 0 0 NUM cana-6055 146 4 𝑖 𝑖 SYM cana-6055 146 5 = = NOUN cana-6055 147 1 1 1 NUM cana-6055 147 2 1 1 NUM cana-6055 147 3 𝑖 𝑖 NOUN cana-6055 147 4 = = NOUN cana-6055 147 5 2,3 2,3 NUM cana-6055 147 6 , , PUNCT cana-6055 147 7 . . PUNCT cana-6055 147 8 . . PUNCT cana-6055 147 9 . . PUNCT cana-6055 148 1 , , PUNCT cana-6055 148 2 𝑛 𝑛 DET cana-6055 148 3 4 4 NUM cana-6055 148 4 2(𝑛 2(𝑛 NUM cana-6055 148 5 + + CCONJ cana-6055 148 6 1 1 NUM cana-6055 148 7 ) ) PUNCT cana-6055 148 8 βˆ’ βˆ’ PROPN cana-6055 148 9 4(𝑖 4(𝑖 NUM cana-6055 149 1 βˆ’ βˆ’ NOUN cana-6055 149 2 1 1 NUM cana-6055 149 3 ) ) PUNCT cana-6055 149 4 𝑖 𝑖 NOUN cana-6055 149 5 = = SYM cana-6055 149 6 𝑛 𝑛 PRON cana-6055 149 7 4 4 NUM cana-6055 149 8 + + SYM cana-6055 149 9 1 1 NUM cana-6055 149 10 𝑛 𝑛 NOUN cana-6055 149 11 + + NUM cana-6055 149 12 3 3 NUM cana-6055 149 13 𝑖 𝑖 NOUN cana-6055 149 14 = = NOUN cana-6055 149 15 𝑛 𝑛 PRON cana-6055 149 16 4 4 NUM cana-6055 149 17 + + SYM cana-6055 149 18 2 2 NUM cana-6055 149 19 , , PUNCT cana-6055 149 20 𝑛 𝑛 DET cana-6055 149 21 4 4 NUM cana-6055 149 22 + + SYM cana-6055 149 23 3 3 NUM cana-6055 149 24 , , PUNCT cana-6055 149 25 . . PUNCT cana-6055 149 26 . . PUNCT cana-6055 149 27 . . PUNCT cana-6055 150 1 , , PUNCT cana-6055 150 2 𝑛 𝑛 PRON cana-6055 150 3 2 2 NUM cana-6055 150 4 4(𝑖 4(𝑖 NUM cana-6055 150 5 βˆ’ βˆ’ NOUN cana-6055 150 6 1 1 NUM cana-6055 150 7 ) ) PUNCT cana-6055 150 8 𝑖 𝑖 NOUN cana-6055 150 9 = = SYM cana-6055 150 10 𝑛 𝑛 PRON cana-6055 150 11 2 2 NUM cana-6055 150 12 + + SYM cana-6055 150 13 1 1 NUM cana-6055 150 14 βˆ’1 βˆ’1 NOUN cana-6055 150 15 𝑖 𝑖 NOUN cana-6055 150 16 = = SYM cana-6055 150 17 𝑛 𝑛 PRON cana-6055 150 18 2 2 NUM cana-6055 150 19 + + CCONJ cana-6055 150 20 2 2 NUM cana-6055 150 21 , , PUNCT cana-6055 150 22 𝑛 𝑛 DET cana-6055 150 23 2 2 NUM cana-6055 150 24 + + SYM cana-6055 150 25 3 3 NUM cana-6055 150 26 , , PUNCT cana-6055 150 27 . . PUNCT cana-6055 150 28 . . PUNCT cana-6055 150 29 . . PUNCT cana-6055 151 1 , , PUNCT cana-6055 151 2 3𝑛 3𝑛 NUM cana-6055 151 3 4 4 NUM cana-6055 151 4 βˆ’(4(𝑛 βˆ’(4(𝑛 VERB cana-6055 151 5 βˆ’ βˆ’ NOUN cana-6055 151 6 𝑖 𝑖 SYM cana-6055 151 7 ) ) PUNCT cana-6055 151 8 + + CCONJ cana-6055 151 9 6 6 X cana-6055 151 10 ) ) PUNCT cana-6055 151 11 𝑖 𝑖 NOUN cana-6055 151 12 = = SYM cana-6055 151 13 3𝑛 3𝑛 NUM cana-6055 151 14 4 4 NUM cana-6055 151 15 + + SYM cana-6055 151 16 1 1 NUM cana-6055 151 17 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-6055 151 18 + + NOUN cana-6055 151 19 3 3 X cana-6055 151 20 ) ) PUNCT cana-6055 151 21 𝑖 𝑖 NOUN cana-6055 151 22 = = SYM cana-6055 151 23 3𝑛 3𝑛 NUM cana-6055 151 24 4 4 NUM cana-6055 151 25 + + SYM cana-6055 151 26 2 2 NUM cana-6055 151 27 , , PUNCT cana-6055 151 28 3𝑛 3𝑛 NUM cana-6055 151 29 4 4 NUM cana-6055 151 30 + + CCONJ cana-6055 151 31 3 3 NUM cana-6055 151 32 , , PUNCT cana-6055 151 33 . . PUNCT cana-6055 151 34 . . PUNCT cana-6055 152 1 . . PUNCT cana-6055 153 1 , , PUNCT cana-6055 153 2 𝑛 𝑛 DET cana-6055 153 3 βˆ’(4 βˆ’(4 PROPN cana-6055 153 4 ( ( PUNCT cana-6055 153 5 𝑖 𝑖 NOUN cana-6055 153 6 βˆ’ βˆ’ NOUN cana-6055 153 7 𝑛 𝑛 PRON cana-6055 153 8 2 2 NUM cana-6055 153 9 βˆ’ βˆ’ NUM cana-6055 153 10 1 1 NUM cana-6055 153 11 ) ) PUNCT cana-6055 153 12 ) ) PUNCT cana-6055 153 13 𝑣𝑖 𝑣𝑖 ADP cana-6055 153 14 𝑀(𝑣𝑖 𝑀(𝑣𝑖 VERB cana-6055 153 15 ) ) PUNCT cana-6055 153 16 𝑖 𝑖 X cana-6055 154 1 = = NOUN cana-6055 154 2 1,2,3 1,2,3 NUM cana-6055 154 3 , , PUNCT cana-6055 154 4 . . PUNCT cana-6055 154 5 . . PUNCT cana-6055 155 1 . . PUNCT cana-6055 156 1 , , PUNCT cana-6055 156 2 𝑛 𝑛 DET cana-6055 156 3 4 4 NUM cana-6055 156 4 βˆ’ βˆ’ NOUN cana-6055 156 5 1 1 NUM cana-6055 156 6 4𝑖 4𝑖 NOUN cana-6055 156 7 𝑖 𝑖 NOUN cana-6055 156 8 = = SYM cana-6055 156 9 𝑛 𝑛 DET cana-6055 156 10 4 4 NUM cana-6055 156 11 𝑛 𝑛 ADP cana-6055 156 12 2 2 NUM cana-6055 156 13 + + CCONJ cana-6055 156 14 1 1 NUM cana-6055 156 15 𝑖 𝑖 NOUN cana-6055 156 16 = = NOUN cana-6055 156 17 𝑛 𝑛 PRON cana-6055 156 18 4 4 NUM cana-6055 156 19 + + SYM cana-6055 156 20 1 1 NUM cana-6055 156 21 , , PUNCT cana-6055 156 22 𝑛 𝑛 PRON cana-6055 156 23 4 4 NUM cana-6055 156 24 + + SYM cana-6055 156 25 2 2 NUM cana-6055 156 26 , , PUNCT cana-6055 156 27 . . PUNCT cana-6055 156 28 . . PUNCT cana-6055 157 1 . . PUNCT cana-6055 158 1 , , PUNCT cana-6055 158 2 𝑛 𝑛 DET cana-6055 158 3 2 2 NUM cana-6055 158 4 βˆ’ βˆ’ PROPN cana-6055 158 5 1 1 NUM cana-6055 158 6 4(𝑖 4(𝑖 NUM cana-6055 158 7 βˆ’ βˆ’ NOUN cana-6055 158 8 𝑛 𝑛 PRON cana-6055 158 9 4 4 NUM cana-6055 158 10 ) ) PUNCT cana-6055 159 1 + + CCONJ cana-6055 159 2 2 2 NUM cana-6055 159 3 𝑖 𝑖 NOUN cana-6055 159 4 = = NOUN cana-6055 159 5 𝑛 𝑛 PRON cana-6055 159 6 2 2 NUM cana-6055 159 7 𝑛 𝑛 DET cana-6055 159 8 2 2 NUM cana-6055 159 9 βˆ’ βˆ’ NOUN cana-6055 159 10 1 1 NUM cana-6055 159 11 𝑖 𝑖 NOUN cana-6055 159 12 = = SYM cana-6055 159 13 𝑛 𝑛 PRON cana-6055 159 14 2 2 NUM cana-6055 159 15 + + NUM cana-6055 159 16 1 1 NUM cana-6055 159 17 , , PUNCT cana-6055 159 18 𝑛 𝑛 DET cana-6055 159 19 2 2 NUM cana-6055 159 20 + + SYM cana-6055 159 21 2 2 NUM cana-6055 159 22 , , PUNCT cana-6055 159 23 . . PUNCT cana-6055 159 24 . . PUNCT cana-6055 159 25 . . PUNCT cana-6055 160 1 , , PUNCT cana-6055 160 2 3𝑛 3𝑛 NUM cana-6055 160 3 4 4 NUM cana-6055 160 4 βˆ’ βˆ’ NOUN cana-6055 160 5 1 1 NUM cana-6055 160 6 4 4 NUM cana-6055 160 7 ( ( PUNCT cana-6055 160 8 𝑛 𝑛 PROPN cana-6055 160 9 2 2 NUM cana-6055 160 10 βˆ’ βˆ’ NOUN cana-6055 160 11 𝑖 𝑖 SYM cana-6055 160 12 ) ) PUNCT cana-6055 160 13 𝑖 𝑖 NOUN cana-6055 160 14 = = SYM cana-6055 160 15 3𝑛 3𝑛 NUM cana-6055 160 16 4 4 NUM cana-6055 160 17 βˆ’ βˆ’ NOUN cana-6055 160 18 ( ( PUNCT cana-6055 160 19 𝑛 𝑛 PROPN cana-6055 160 20 2 2 NUM cana-6055 160 21 + + NUM cana-6055 160 22 1 1 NUM cana-6055 160 23 ) ) PUNCT cana-6055 160 24 𝑖 𝑖 NOUN cana-6055 160 25 = = SYM cana-6055 160 26 3𝑛 3𝑛 NUM cana-6055 160 27 4 4 NUM cana-6055 160 28 + + SYM cana-6055 160 29 1 1 NUM cana-6055 160 30 , , PUNCT cana-6055 160 31 3𝑛 3𝑛 NUM cana-6055 160 32 4 4 NUM cana-6055 160 33 + + SYM cana-6055 160 34 2 2 NUM cana-6055 160 35 , , PUNCT cana-6055 160 36 . . PUNCT cana-6055 160 37 . . PUNCT cana-6055 160 38 . . PUNCT cana-6055 161 1 , , PUNCT cana-6055 161 2 𝑛 𝑛 DET cana-6055 161 3 βˆ’ βˆ’ PROPN cana-6055 161 4 1 1 NUM cana-6055 161 5 βˆ’(4 βˆ’(4 X cana-6055 161 6 ( ( PUNCT cana-6055 161 7 𝑖 𝑖 SYM cana-6055 161 8 βˆ’ βˆ’ X cana-6055 161 9 3𝑛 3𝑛 NUM cana-6055 161 10 4 4 NUM cana-6055 161 11 ) ) PUNCT cana-6055 161 12 + + CCONJ cana-6055 161 13 2 2 X cana-6055 161 14 ) ) PUNCT cana-6055 161 15 𝑖 𝑖 NOUN cana-6055 161 16 = = SYM cana-6055 161 17 𝑛 𝑛 DET cana-6055 161 18 βˆ’ βˆ’ PROPN cana-6055 161 19 ( ( PUNCT cana-6055 161 20 𝑛 𝑛 PROPN cana-6055 161 21 2 2 NUM cana-6055 161 22 βˆ’ βˆ’ NOUN cana-6055 161 23 1 1 NUM cana-6055 161 24 ) ) PUNCT cana-6055 161 25 hence hence ADV cana-6055 161 26 the the DET cana-6055 161 27 n n CCONJ cana-6055 161 28 - - PUNCT cana-6055 161 29 gear gear NOUN cana-6055 161 30 graph graph NOUN cana-6055 161 31 𝐺𝑛 𝐺𝑛 PROPN cana-6055 161 32 is be AUX cana-6055 161 33 a a DET cana-6055 161 34 distance distance NOUN cana-6055 161 35 pair pair NOUN cana-6055 161 36 antimagic antimagic ADJ cana-6055 161 37 graph graph NOUN cana-6055 161 38 , , PUNCT cana-6055 161 39 if if SCONJ cana-6055 161 40 𝑛 𝑛 PRON cana-6055 161 41 = = SYM cana-6055 161 42 4,6,8 4,6,8 NUM cana-6055 161 43 , , PUNCT cana-6055 161 44 … … PUNCT cana-6055 161 45 communications communication NOUN cana-6055 161 46 on on ADP cana-6055 161 47 applied apply VERB cana-6055 161 48 nonlinear nonlinear ADJ cana-6055 161 49 analysis analysis NOUN cana-6055 161 50 issn issn NOUN cana-6055 161 51 : : PUNCT cana-6055 161 52 1074 1074 NUM cana-6055 161 53 - - PUNCT cana-6055 161 54 133x 133x NUM cana-6055 161 55 vol vol NOUN cana-6055 161 56 31 31 NUM cana-6055 161 57 no no NOUN cana-6055 161 58 . . PUNCT cana-6055 162 1 8s 8s PROPN cana-6055 162 2 ( ( PUNCT cana-6055 162 3 2024 2024 NUM cana-6055 162 4 ) ) PUNCT cana-6055 162 5 1162 1162 NUM cana-6055 162 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 162 7 theorem theorem VERB cana-6055 162 8 3.5 3.5 NUM cana-6055 162 9 the the DET cana-6055 162 10 book book NOUN cana-6055 162 11 graph graph NOUN cana-6055 162 12 𝐡𝑛 𝐡𝑛 PROPN cana-6055 162 13 = = PUNCT cana-6055 162 14 𝑆𝑛+1 𝑆𝑛+1 PROPN cana-6055 162 15 Γ— Γ— NOUN cana-6055 162 16 𝑃2 𝑃2 NOUN cana-6055 162 17 is be AUX cana-6055 162 18 a a DET cana-6055 162 19 distance distance NOUN cana-6055 162 20 pair pair NOUN cana-6055 162 21 antimagic antimagic ADJ cana-6055 162 22 graph graph NOUN cana-6055 162 23 if if SCONJ cana-6055 162 24 𝑛 𝑛 PRON cana-6055 162 25 β‰₯ β‰₯ NOUN cana-6055 162 26 2 2 NUM cana-6055 162 27 . . PUNCT cana-6055 163 1 proof proof NOUN cana-6055 163 2 . . PUNCT cana-6055 164 1 let let VERB cana-6055 164 2 the the DET cana-6055 164 3 vertex vertex NOUN cana-6055 164 4 set set NOUN cana-6055 164 5 of of ADP cana-6055 164 6 book book NOUN cana-6055 164 7 graph graph NOUN cana-6055 164 8 𝑉(𝐡𝑛 𝑉(𝐡𝑛 PROPN cana-6055 164 9 ) ) PUNCT cana-6055 164 10 be be AUX cana-6055 164 11 { { PUNCT cana-6055 164 12 𝑒𝑖 𝑒𝑖 ADP cana-6055 164 13 , , PUNCT cana-6055 164 14 𝑣𝑖 𝑣𝑖 ADP cana-6055 164 15 : : PUNCT cana-6055 164 16 𝑖 𝑖 X cana-6055 164 17 = = PUNCT cana-6055 164 18 0,1,2,3 0,1,2,3 NUM cana-6055 164 19 , , PUNCT cana-6055 164 20 . . PUNCT cana-6055 164 21 . . PUNCT cana-6055 165 1 . . PUNCT cana-6055 166 1 , , PUNCT cana-6055 166 2 𝑛 𝑛 X cana-6055 166 3 } } PUNCT cana-6055 166 4 and and CCONJ cana-6055 166 5 the the DET cana-6055 166 6 edge edge NOUN cana-6055 166 7 set set VERB cana-6055 166 8 be be AUX cana-6055 166 9 { { PUNCT cana-6055 166 10 𝑒0𝑒𝑖 𝑒0𝑒𝑖 X cana-6055 166 11 , , PUNCT cana-6055 166 12 𝑣0𝑣𝑖 𝑣0𝑣𝑖 NOUN cana-6055 166 13 : : PUNCT cana-6055 166 14 𝑖 𝑖 X cana-6055 166 15 = = NOUN cana-6055 166 16 1,2,3 1,2,3 NUM cana-6055 166 17 , , PUNCT cana-6055 166 18 . . PUNCT cana-6055 166 19 . . PUNCT cana-6055 167 1 . . PUNCT cana-6055 168 1 , , PUNCT cana-6055 168 2 𝑛 𝑛 X cana-6055 168 3 } } PUNCT cana-6055 168 4 βˆͺ βˆͺ ADJ cana-6055 168 5 { { PUNCT cana-6055 168 6 𝑒𝑖𝑣𝑖 𝑒𝑖𝑣𝑖 NOUN cana-6055 168 7 : : PUNCT cana-6055 169 1 𝑖 𝑖 SYM cana-6055 169 2 = = PUNCT cana-6055 169 3 0,1,2,3 0,1,2,3 NUM cana-6055 169 4 , , PUNCT cana-6055 169 5 . . PUNCT cana-6055 169 6 . . PUNCT cana-6055 170 1 . . PUNCT cana-6055 171 1 , , PUNCT cana-6055 171 2 𝑛 𝑛 X cana-6055 171 3 } } PUNCT cana-6055 171 4 . . PUNCT cana-6055 172 1 define define VERB cana-6055 172 2 𝑓 𝑓 DET cana-6055 172 3 : : PUNCT cana-6055 172 4 𝑉(𝑆𝑛+1 𝑉(𝑆𝑛+1 ADJ cana-6055 172 5 Γ— Γ— PROPN cana-6055 172 6 𝑃2 𝑃2 PROPN cana-6055 172 7 ) ) PUNCT cana-6055 172 8 ⟢ ⟢ NOUN cana-6055 172 9 { { PUNCT cana-6055 172 10 Β±1 Β±1 VERB cana-6055 172 11 , , PUNCT cana-6055 172 12 Β±2,β‹― Β±2,β‹― NUM cana-6055 172 13 , , PUNCT cana-6055 172 14 ±𝑛 ±𝑛 PROPN cana-6055 172 15 + + ADP cana-6055 172 16 1 1 NUM cana-6055 172 17 } } PUNCT cana-6055 172 18 by by ADP cana-6055 172 19 𝑓(𝑒0 𝑓(𝑒0 ADJ cana-6055 172 20 ) ) PUNCT cana-6055 172 21 = = SYM cana-6055 172 22 1 1 NUM cana-6055 172 23 ; ; PUNCT cana-6055 172 24 𝑓(𝑣0 𝑓(𝑣0 X cana-6055 172 25 ) ) PUNCT cana-6055 172 26 = = SYM cana-6055 172 27 βˆ’1 βˆ’1 NOUN cana-6055 172 28 ; ; PUNCT cana-6055 172 29 𝑓(𝑒𝑖 𝑓(𝑒𝑖 X cana-6055 172 30 ) ) PUNCT cana-6055 172 31 = = SYM cana-6055 172 32 𝑖 𝑖 PROPN cana-6055 173 1 + + NOUN cana-6055 173 2 1 1 NUM cana-6055 173 3 , , PUNCT cana-6055 173 4 𝑖𝑓 𝑖𝑓 ADP cana-6055 173 5 𝑖 𝑖 SYM cana-6055 173 6 = = NOUN cana-6055 173 7 1,2,3 1,2,3 NUM cana-6055 173 8 , , PUNCT cana-6055 173 9 . . PUNCT cana-6055 173 10 . . PUNCT cana-6055 174 1 . . PUNCT cana-6055 175 1 , , PUNCT cana-6055 175 2 𝑛 𝑛 NOUN cana-6055 175 3 ; ; PUNCT cana-6055 175 4 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NUM cana-6055 175 5 ) ) PUNCT cana-6055 176 1 = = SYM cana-6055 176 2 βˆ’(𝑖 βˆ’(𝑖 NOUN cana-6055 176 3 + + NOUN cana-6055 176 4 1 1 NUM cana-6055 176 5 ) ) PUNCT cana-6055 176 6 , , PUNCT cana-6055 176 7 𝑖𝑓 𝑖𝑓 ADP cana-6055 176 8 𝑖 𝑖 X cana-6055 176 9 = = NOUN cana-6055 176 10 1,2,3 1,2,3 NUM cana-6055 176 11 , , PUNCT cana-6055 176 12 . . PUNCT cana-6055 176 13 . . PUNCT cana-6055 177 1 . . PUNCT cana-6055 178 1 , , PUNCT cana-6055 178 2 𝑛 𝑛 DET cana-6055 178 3 then then ADV cana-6055 178 4 the the DET cana-6055 178 5 induced induced ADJ cana-6055 178 6 vertex vertex NOUN cana-6055 178 7 weight weight NOUN cana-6055 178 8 labeling labeling NOUN cana-6055 178 9 are be AUX cana-6055 178 10 as as SCONJ cana-6055 178 11 follows follow VERB cana-6055 178 12 . . PUNCT cana-6055 179 1 𝑀(𝑒0 𝑀(𝑒0 NOUN cana-6055 179 2 ) ) PUNCT cana-6055 180 1 = = PRON cana-6055 180 2 ( ( PUNCT cana-6055 180 3 𝑛+1)(𝑛+2 𝑛+1)(𝑛+2 X cana-6055 180 4 ) ) PUNCT cana-6055 180 5 2 2 NUM cana-6055 180 6 βˆ’ βˆ’ NOUN cana-6055 180 7 1 1 NUM cana-6055 180 8 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 180 9 ) ) PUNCT cana-6055 180 10 = = SYM cana-6055 181 1 βˆ’π‘– βˆ’π‘– PROPN cana-6055 181 2 , , PUNCT cana-6055 181 3 𝑖𝑓 𝑖𝑓 ADP cana-6055 181 4 𝑖 𝑖 X cana-6055 181 5 = = NOUN cana-6055 181 6 1,2,3 1,2,3 NUM cana-6055 181 7 , , PUNCT cana-6055 181 8 . . PUNCT cana-6055 181 9 . . PUNCT cana-6055 182 1 . . PUNCT cana-6055 183 1 , , PUNCT cana-6055 184 1 𝑛 𝑛 PRON cana-6055 184 2 𝑀(𝑣0 𝑀(𝑣0 ADJ cana-6055 184 3 ) ) PUNCT cana-6055 184 4 = = SYM cana-6055 184 5 1 1 NUM cana-6055 184 6 βˆ’ βˆ’ PROPN cana-6055 184 7 ( ( PUNCT cana-6055 184 8 𝑛+1)(𝑛+2 𝑛+1)(𝑛+2 X cana-6055 184 9 ) ) PUNCT cana-6055 184 10 2 2 NUM cana-6055 184 11 𝑀(𝑣𝑖 𝑀(𝑣𝑖 ADJ cana-6055 184 12 ) ) PUNCT cana-6055 184 13 = = SYM cana-6055 184 14 𝑖 𝑖 PROPN cana-6055 184 15 , , PUNCT cana-6055 184 16 𝑖𝑓 𝑖𝑓 ADP cana-6055 184 17 𝑖 𝑖 SYM cana-6055 184 18 = = NOUN cana-6055 184 19 1,2,3 1,2,3 NUM cana-6055 184 20 , , PUNCT cana-6055 184 21 . . PUNCT cana-6055 184 22 . . PUNCT cana-6055 185 1 . . PUNCT cana-6055 186 1 , , PUNCT cana-6055 186 2 𝑛 𝑛 PRON cana-6055 186 3 hence hence ADV cana-6055 186 4 𝐡𝑛 𝐡𝑛 PROPN cana-6055 186 5 is be AUX cana-6055 186 6 a a DET cana-6055 186 7 distance distance NOUN cana-6055 186 8 pair pair NOUN cana-6055 186 9 antimagic antimagic ADJ cana-6055 186 10 graph graph NOUN cana-6055 186 11 if if SCONJ cana-6055 186 12 𝑛 𝑛 PRON cana-6055 186 13 β‰₯ β‰₯ NOUN cana-6055 186 14 2 2 NUM cana-6055 186 15 . . PUNCT cana-6055 186 16 theorem theorem VERB cana-6055 186 17 3.6 3.6 NUM cana-6055 186 18 the the DET cana-6055 186 19 friendship friendship NOUN cana-6055 186 20 graph graph NOUN cana-6055 186 21 𝐹𝑛 𝐹𝑛 PROPN cana-6055 186 22 is be AUX cana-6055 186 23 a a DET cana-6055 186 24 distance distance NOUN cana-6055 186 25 pair pair NOUN cana-6055 186 26 antimagic antimagic ADJ cana-6055 186 27 graph graph NOUN cana-6055 186 28 . . PUNCT cana-6055 187 1 proof proof NOUN cana-6055 187 2 . . PUNCT cana-6055 188 1 let let VERB cana-6055 188 2 the the DET cana-6055 188 3 vertex vertex NOUN cana-6055 188 4 set set NOUN cana-6055 188 5 of of ADP cana-6055 188 6 friendship friendship NOUN cana-6055 188 7 graph graph NOUN cana-6055 188 8 𝑉(𝐹𝑛 𝑉(𝐹𝑛 PROPN cana-6055 188 9 ) ) PUNCT cana-6055 188 10 be be AUX cana-6055 188 11 { { PUNCT cana-6055 188 12 𝑣0 𝑣0 PROPN cana-6055 188 13 , , PUNCT cana-6055 188 14 𝑣1 𝑣1 PROPN cana-6055 188 15 , , PUNCT cana-6055 188 16 𝑣2 𝑣2 PROPN cana-6055 188 17 , , PUNCT cana-6055 188 18 . . PUNCT cana-6055 188 19 . . PUNCT cana-6055 189 1 . . PUNCT cana-6055 190 1 , , PUNCT cana-6055 190 2 𝑣2𝑛 𝑣2𝑛 PROPN cana-6055 190 3 } } PUNCT cana-6055 190 4 and and CCONJ cana-6055 190 5 the the DET cana-6055 190 6 edge edge NOUN cana-6055 190 7 set set VERB cana-6055 190 8 be be AUX cana-6055 190 9 { { PUNCT cana-6055 190 10 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 NOUN cana-6055 190 11 : : PUNCT cana-6055 190 12 𝑖 𝑖 SYM cana-6055 190 13 = = SYM cana-6055 190 14 1,3 1,3 NUM cana-6055 190 15 , , PUNCT cana-6055 190 16 . . PUNCT cana-6055 190 17 . . PUNCT cana-6055 191 1 . . PUNCT cana-6055 192 1 , , PUNCT cana-6055 192 2 2𝑛 2𝑛 PROPN cana-6055 192 3 βˆ’ βˆ’ PROPN cana-6055 192 4 1 1 NUM cana-6055 192 5 } } PUNCT cana-6055 192 6 βˆͺ βˆͺ X cana-6055 192 7 { { PUNCT cana-6055 192 8 𝑣0𝑣𝑖 𝑣0𝑣𝑖 NOUN cana-6055 192 9 : : PUNCT cana-6055 192 10 𝑖 𝑖 SYM cana-6055 192 11 = = NOUN cana-6055 192 12 1,2,3 1,2,3 NUM cana-6055 192 13 , , PUNCT cana-6055 192 14 . . PUNCT cana-6055 192 15 . . PUNCT cana-6055 192 16 . . PUNCT cana-6055 193 1 , , PUNCT cana-6055 193 2 2𝑛 2𝑛 NUM cana-6055 193 3 } } PUNCT cana-6055 193 4 . . PUNCT cana-6055 194 1 define define VERB cana-6055 194 2 𝑓 𝑓 DET cana-6055 194 3 : : PUNCT cana-6055 194 4 𝑉(𝐹𝑛 𝑉(𝐹𝑛 PROPN cana-6055 194 5 ) ) PUNCT cana-6055 194 6 ⟢ ⟢ NOUN cana-6055 194 7 { { PUNCT cana-6055 194 8 0,Β±1,Β±2,β‹― 0,Β±1,Β±2,β‹― NUM cana-6055 194 9 , , PUNCT cana-6055 194 10 ±𝑛 ±𝑛 PROPN cana-6055 194 11 } } PUNCT cana-6055 194 12 by by ADP cana-6055 194 13 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 194 14 ) ) PUNCT cana-6055 194 15 = = SYM cana-6055 194 16 { { PUNCT cana-6055 194 17 0 0 NUM cana-6055 194 18 , , PUNCT cana-6055 194 19 𝑖𝑓 𝑖𝑓 ADP cana-6055 194 20 𝑖 𝑖 PUNCT cana-6055 195 1 = = NOUN cana-6055 195 2 0 0 PUNCT cana-6055 195 3 𝑖+1 𝑖+1 NUM cana-6055 195 4 2 2 NUM cana-6055 195 5 , , PUNCT cana-6055 195 6 𝑖𝑓 𝑖𝑓 ADP cana-6055 195 7 𝑖 𝑖 PUNCT cana-6055 195 8 𝑖𝑠 𝑖𝑠 NOUN cana-6055 195 9 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-6055 196 1 βˆ’ βˆ’ NOUN cana-6055 196 2 ( ( PUNCT cana-6055 196 3 𝑖 𝑖 SYM cana-6055 196 4 2 2 NUM cana-6055 196 5 ) ) PUNCT cana-6055 196 6 , , PUNCT cana-6055 196 7 𝑖𝑓 𝑖𝑓 X cana-6055 196 8 𝑖 𝑖 PRON cana-6055 196 9 𝑖𝑠 𝑖𝑠 INTJ cana-6055 197 1 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-6055 197 2 then then ADV cana-6055 197 3 the the DET cana-6055 197 4 induced induced ADJ cana-6055 197 5 vertex vertex NOUN cana-6055 197 6 weight weight NOUN cana-6055 197 7 labeling labeling NOUN cana-6055 197 8 are be AUX cana-6055 197 9 as as SCONJ cana-6055 197 10 follows follow VERB cana-6055 197 11 . . PUNCT cana-6055 198 1 𝑀(𝑣𝑖 𝑀(𝑣𝑖 VERB cana-6055 198 2 ) ) PUNCT cana-6055 199 1 = = PRON cana-6055 199 2 { { PUNCT cana-6055 199 3 0 0 NUM cana-6055 199 4 , , PUNCT cana-6055 199 5 𝑖𝑓 𝑖𝑓 ADP cana-6055 199 6 𝑖 𝑖 SYM cana-6055 200 1 = = NOUN cana-6055 200 2 0 0 NUM cana-6055 200 3 βˆ’ βˆ’ PROPN cana-6055 200 4 ( ( PUNCT cana-6055 200 5 𝑖+1 𝑖+1 NUM cana-6055 200 6 2 2 NUM cana-6055 200 7 ) ) PUNCT cana-6055 200 8 , , PUNCT cana-6055 200 9 𝑖𝑓 𝑖𝑓 X cana-6055 200 10 𝑖 𝑖 PRON cana-6055 200 11 𝑖𝑠 𝑖𝑠 NOUN cana-6055 200 12 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ PROPN cana-6055 200 13 𝑖 𝑖 PRON cana-6055 200 14 2 2 NUM cana-6055 200 15 , , PUNCT cana-6055 200 16 𝑖𝑓 𝑖𝑓 ADP cana-6055 200 17 𝑖 𝑖 PRON cana-6055 200 18 𝑖𝑠 𝑖𝑠 INTJ cana-6055 200 19 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-6055 201 1 hence hence ADV cana-6055 201 2 𝐹𝑛 𝐹𝑛 PROPN cana-6055 201 3 is be AUX cana-6055 201 4 a a DET cana-6055 201 5 distance distance NOUN cana-6055 201 6 pair pair NOUN cana-6055 201 7 antimagic antimagic ADJ cana-6055 201 8 graph graph NOUN cana-6055 201 9 . . PUNCT cana-6055 202 1 theorem theorem VERB cana-6055 202 2 3.7 3.7 NUM cana-6055 202 3 the the DET cana-6055 202 4 prism prism NOUN cana-6055 202 5 graph graph NOUN cana-6055 203 1 𝐢𝑛 𝐢𝑛 PROPN cana-6055 203 2 Γ— Γ— NOUN cana-6055 203 3 𝐾2 𝐾2 NOUN cana-6055 203 4 is be AUX cana-6055 203 5 a a DET cana-6055 203 6 distance distance NOUN cana-6055 203 7 pair pair NOUN cana-6055 203 8 antimagic antimagic ADJ cana-6055 203 9 graph graph NOUN cana-6055 203 10 . . PUNCT cana-6055 204 1 proof proof NOUN cana-6055 204 2 . . PUNCT cana-6055 205 1 let let VERB cana-6055 205 2 𝑉(𝐢𝑛 𝑉(𝐢𝑛 NOUN cana-6055 205 3 Γ— Γ— NOUN cana-6055 205 4 𝐾2 𝐾2 NOUN cana-6055 205 5 ) ) PUNCT cana-6055 206 1 = = PRON cana-6055 206 2 { { PUNCT cana-6055 206 3 𝑣𝑖 𝑣𝑖 NOUN cana-6055 206 4 , , PUNCT cana-6055 206 5 𝑒𝑖 𝑒𝑖 ADP cana-6055 206 6 : : PUNCT cana-6055 206 7 1 1 NUM cana-6055 206 8 ≀ ≀ NUM cana-6055 206 9 𝑖 𝑖 SYM cana-6055 206 10 ≀ ≀ NUM cana-6055 206 11 𝑛 𝑛 PRON cana-6055 206 12 } } PUNCT cana-6055 206 13 be be VERB cana-6055 206 14 the the DET cana-6055 206 15 vertex vertex NOUN cana-6055 206 16 set set NOUN cana-6055 206 17 and and CCONJ cana-6055 206 18 𝐸(𝐢𝑛 𝐸(𝐢𝑛 NUM cana-6055 206 19 Γ— Γ— NOUN cana-6055 206 20 𝐾2 𝐾2 NOUN cana-6055 206 21 ) ) PUNCT cana-6055 207 1 = = PRON cana-6055 207 2 { { PUNCT cana-6055 207 3 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-6055 207 4 , , PUNCT cana-6055 207 5 𝑒𝑖𝑒𝑖+1 𝑒𝑖𝑒𝑖+1 PROPN cana-6055 207 6 , , PUNCT cana-6055 207 7 𝑣𝑖𝑒𝑖 𝑣𝑖𝑒𝑖 VERB cana-6055 207 8 : : PUNCT cana-6055 207 9 1 1 NUM cana-6055 207 10 ≀ ≀ NUM cana-6055 207 11 𝑖 𝑖 SYM cana-6055 207 12 ≀ ≀ NUM cana-6055 207 13 𝑛 𝑛 PRON cana-6055 207 14 βˆ’ βˆ’ PROPN cana-6055 207 15 1 1 NUM cana-6055 207 16 } } PUNCT cana-6055 207 17 βˆͺ βˆͺ ADJ cana-6055 207 18 { { PUNCT cana-6055 207 19 𝑒𝑛𝑒1 𝑒𝑛𝑒1 PROPN cana-6055 207 20 , , PUNCT cana-6055 207 21 𝑣𝑛𝑣1 𝑣𝑛𝑣1 PROPN cana-6055 207 22 , , PUNCT cana-6055 207 23 𝑣𝑛𝑒𝑛 𝑣𝑛𝑒𝑛 PROPN cana-6055 207 24 } } PUNCT cana-6055 207 25 be be AUX cana-6055 207 26 the the DET cana-6055 207 27 edge edge NOUN cana-6055 207 28 set set NOUN cana-6055 207 29 of of ADP cana-6055 207 30 𝐢𝑛 𝐢𝑛 PROPN cana-6055 207 31 Γ— Γ— NOUN cana-6055 207 32 𝐾2 𝐾2 NOUN cana-6055 207 33 . . PUNCT cana-6055 208 1 communications communication NOUN cana-6055 208 2 on on ADP cana-6055 208 3 applied apply VERB cana-6055 208 4 nonlinear nonlinear ADJ cana-6055 208 5 analysis analysis NOUN cana-6055 208 6 issn issn NOUN cana-6055 208 7 : : PUNCT cana-6055 208 8 1074 1074 NUM cana-6055 208 9 - - PUNCT cana-6055 208 10 133x 133x NUM cana-6055 208 11 vol vol NOUN cana-6055 208 12 31 31 NUM cana-6055 208 13 no no NOUN cana-6055 208 14 . . PUNCT cana-6055 209 1 8s 8s PROPN cana-6055 209 2 ( ( PUNCT cana-6055 209 3 2024 2024 NUM cana-6055 209 4 ) ) PUNCT cana-6055 209 5 1163 1163 NUM cana-6055 209 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 209 7 define define VERB cana-6055 209 8 𝑓 𝑓 PRON cana-6055 209 9 : : PUNCT cana-6055 209 10 𝑉(𝐢𝑛 𝑉(𝐢𝑛 ADJ cana-6055 209 11 Γ— Γ— NOUN cana-6055 209 12 𝐾2 𝐾2 NOUN cana-6055 209 13 ) ) PUNCT cana-6055 209 14 ⟢ ⟢ NOUN cana-6055 209 15 { { PUNCT cana-6055 209 16 Β±1,Β±2,β‹― Β±1,Β±2,β‹― NOUN cana-6055 209 17 , , PUNCT cana-6055 209 18 ±𝑛 ±𝑛 PROPN cana-6055 209 19 } } PUNCT cana-6055 209 20 by by ADP cana-6055 209 21 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 209 22 ) ) PUNCT cana-6055 209 23 = = SYM cana-6055 210 1 𝑖 𝑖 NOUN cana-6055 210 2 , , PUNCT cana-6055 210 3 if if SCONJ cana-6055 210 4 𝑖 𝑖 PRON cana-6055 210 5 = = NOUN cana-6055 210 6 1,2,3 1,2,3 NUM cana-6055 210 7 , , PUNCT cana-6055 210 8 . . PUNCT cana-6055 210 9 . . PUNCT cana-6055 210 10 . . PUNCT cana-6055 211 1 , , PUNCT cana-6055 211 2 𝑛 𝑛 DET cana-6055 211 3 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 211 4 ) ) PUNCT cana-6055 211 5 = = SYM cana-6055 211 6 { { PUNCT cana-6055 211 7 βˆ’π‘› βˆ’π‘› NOUN cana-6055 211 8 , , PUNCT cana-6055 211 9 βˆ’(𝑖 βˆ’(𝑖 PROPN cana-6055 212 1 βˆ’ βˆ’ PROPN cana-6055 212 2 1 1 NUM cana-6055 212 3 ) ) PUNCT cana-6055 212 4 , , PUNCT cana-6055 212 5 𝑖𝑓 𝑖𝑓 ADP cana-6055 212 6 𝑖 𝑖 X cana-6055 212 7 = = PUNCT cana-6055 212 8 1 1 NUM cana-6055 212 9 𝑖𝑓 𝑖𝑓 ADP cana-6055 212 10 𝑖 𝑖 SYM cana-6055 212 11 = = PUNCT cana-6055 213 1 2,3,4 2,3,4 NUM cana-6055 213 2 , , PUNCT cana-6055 213 3 . . PUNCT cana-6055 213 4 . . PUNCT cana-6055 214 1 . . PUNCT cana-6055 215 1 , , PUNCT cana-6055 215 2 𝑛 𝑛 DET cana-6055 215 3 then then ADV cana-6055 215 4 the the DET cana-6055 215 5 induced induced ADJ cana-6055 215 6 vertex vertex NOUN cana-6055 215 7 weight weight NOUN cana-6055 215 8 labeling labeling NOUN cana-6055 215 9 are be AUX cana-6055 215 10 as as SCONJ cana-6055 215 11 follows follow VERB cana-6055 215 12 . . PUNCT cana-6055 216 1 𝑀(𝑣𝑖 𝑀(𝑣𝑖 VERB cana-6055 216 2 ) ) PUNCT cana-6055 216 3 = = PUNCT cana-6055 217 1 𝑖 𝑖 PROPN cana-6055 218 1 + + NOUN cana-6055 218 2 1 1 NUM cana-6055 218 3 , , PUNCT cana-6055 218 4 𝑖𝑓 𝑖𝑓 ADP cana-6055 218 5 𝑖 𝑖 SYM cana-6055 218 6 = = NOUN cana-6055 218 7 1,2,3 1,2,3 NUM cana-6055 218 8 , , PUNCT cana-6055 218 9 . . PUNCT cana-6055 218 10 . . PUNCT cana-6055 219 1 . . PUNCT cana-6055 220 1 , , PUNCT cana-6055 221 1 𝑛 𝑛 DET cana-6055 221 2 βˆ’ βˆ’ PROPN cana-6055 221 3 1 1 NUM cana-6055 221 4 ; ; PUNCT cana-6055 221 5 𝑀(𝑣𝑛 𝑀(𝑣𝑛 X cana-6055 221 6 ) ) PUNCT cana-6055 221 7 = = SYM cana-6055 221 8 1 1 NUM cana-6055 221 9 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 221 10 ) ) PUNCT cana-6055 221 11 = = PRON cana-6055 221 12 { { PUNCT cana-6055 221 13 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-6055 221 14 βˆ’ βˆ’ NOUN cana-6055 221 15 1 1 NUM cana-6055 221 16 ) ) PUNCT cana-6055 221 17 , , PUNCT cana-6055 221 18 𝑖𝑓 𝑖𝑓 ADP cana-6055 221 19 𝑖 𝑖 X cana-6055 221 20 = = SYM cana-6055 221 21 1 1 NUM cana-6055 221 22 βˆ’π‘› βˆ’π‘› NOUN cana-6055 221 23 , , PUNCT cana-6055 221 24 𝑖𝑓 𝑖𝑓 ADP cana-6055 221 25 𝑖 𝑖 SYM cana-6055 222 1 = = SYM cana-6055 222 2 2 2 NUM cana-6055 222 3 βˆ’(𝑖 βˆ’(𝑖 NOUN cana-6055 222 4 βˆ’ βˆ’ PROPN cana-6055 222 5 2 2 NUM cana-6055 222 6 ) ) PUNCT cana-6055 222 7 , , PUNCT cana-6055 222 8 𝑖𝑓 𝑖𝑓 ADP cana-6055 222 9 𝑖 𝑖 X cana-6055 222 10 = = NOUN cana-6055 222 11 3,4,5 3,4,5 NUM cana-6055 222 12 , , PUNCT cana-6055 222 13 . . PUNCT cana-6055 222 14 . . PUNCT cana-6055 222 15 . . PUNCT cana-6055 223 1 , , PUNCT cana-6055 223 2 𝑛 𝑛 PRON cana-6055 223 3 hence hence ADV cana-6055 223 4 𝐢𝑛 𝐢𝑛 INTJ cana-6055 223 5 Γ— Γ— NOUN cana-6055 223 6 𝐾2 𝐾2 NOUN cana-6055 223 7 is be AUX cana-6055 223 8 a a DET cana-6055 223 9 distance distance NOUN cana-6055 223 10 pair pair NOUN cana-6055 223 11 antimagic antimagic ADJ cana-6055 223 12 graph graph NOUN cana-6055 223 13 . . PUNCT cana-6055 224 1 theorem theorem VERB cana-6055 224 2 3.8 3.8 NUM cana-6055 224 3 the the DET cana-6055 224 4 crossed cross VERB cana-6055 224 5 prism prism NOUN cana-6055 224 6 𝐢𝑃𝑛 𝐢𝑃𝑛 NOUN cana-6055 224 7 is be AUX cana-6055 224 8 a a DET cana-6055 224 9 distance distance NOUN cana-6055 224 10 pair pair NOUN cana-6055 224 11 antimagic antimagic ADJ cana-6055 224 12 graph graph NOUN cana-6055 224 13 if if SCONJ cana-6055 224 14 𝑛 𝑛 PRON cana-6055 224 15 ≑ ≑ PROPN cana-6055 224 16 0 0 PUNCT cana-6055 225 1 ( ( PUNCT cana-6055 225 2 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ NOUN cana-6055 225 3 4 4 NUM cana-6055 225 4 ) ) PUNCT cana-6055 225 5 . . PUNCT cana-6055 226 1 proof proof NOUN cana-6055 226 2 . . PUNCT cana-6055 227 1 let let VERB cana-6055 227 2 𝑉(𝐢𝑃𝑛 𝑉(𝐢𝑃𝑛 PROPN cana-6055 227 3 ) ) PUNCT cana-6055 228 1 = = PRON cana-6055 228 2 { { PUNCT cana-6055 228 3 𝑣𝑖 𝑣𝑖 NOUN cana-6055 228 4 , , PUNCT cana-6055 228 5 𝑒𝑖 𝑒𝑖 ADP cana-6055 228 6 : : PUNCT cana-6055 228 7 1 1 NUM cana-6055 228 8 ≀ ≀ NUM cana-6055 228 9 𝑖 𝑖 SYM cana-6055 228 10 ≀ ≀ NUM cana-6055 228 11 𝑛 𝑛 PRON cana-6055 228 12 } } PUNCT cana-6055 228 13 be be VERB cana-6055 228 14 the the DET cana-6055 228 15 vertex vertex NOUN cana-6055 228 16 set set NOUN cana-6055 228 17 of of ADP cana-6055 228 18 𝐢𝑃𝑛 𝐢𝑃𝑛 NOUN cana-6055 228 19 and and CCONJ cana-6055 228 20 edge edge NOUN cana-6055 228 21 sets set NOUN cana-6055 228 22 as as SCONJ cana-6055 228 23 follows follow VERB cana-6055 228 24 : : PUNCT cana-6055 228 25 𝐸(𝐢𝑃𝑛 𝐸(𝐢𝑃𝑛 PROPN cana-6055 228 26 ) ) PUNCT cana-6055 229 1 = = PRON cana-6055 229 2 { { PUNCT cana-6055 229 3 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-6055 229 4 , , PUNCT cana-6055 229 5 𝑣1𝑣𝑛 𝑣1𝑣𝑛 PROPN cana-6055 229 6 : : PUNCT cana-6055 229 7 1 1 NUM cana-6055 229 8 ≀ ≀ NUM cana-6055 229 9 𝑖 𝑖 SYM cana-6055 229 10 ≀ ≀ NUM cana-6055 229 11 𝑛 𝑛 PRON cana-6055 229 12 βˆ’ βˆ’ PROPN cana-6055 229 13 1 1 NUM cana-6055 229 14 } } PUNCT cana-6055 229 15 βˆͺ βˆͺ ADJ cana-6055 229 16 { { PUNCT cana-6055 229 17 𝑒𝑖𝑒𝑖+1 𝑒𝑖𝑒𝑖+1 NOUN cana-6055 229 18 , , PUNCT cana-6055 229 19 𝑒1𝑒𝑛 𝑒1𝑒𝑛 NOUN cana-6055 229 20 : : PUNCT cana-6055 229 21 1 1 NUM cana-6055 229 22 ≀ ≀ NUM cana-6055 229 23 𝑖 𝑖 SYM cana-6055 229 24 ≀ ≀ NUM cana-6055 229 25 𝑛 𝑛 PRON cana-6055 229 26 βˆ’ βˆ’ PROPN cana-6055 229 27 1 1 NUM cana-6055 229 28 } } PUNCT cana-6055 229 29 and and CCONJ cana-6055 229 30 adding add VERB cana-6055 229 31 edges edge NOUN cana-6055 229 32 𝑒𝑠𝑣𝑠+1 𝑒𝑠𝑣𝑠+1 NOUN cana-6055 229 33 for for ADP cana-6055 229 34 𝑠 𝑠 PROPN cana-6055 229 35 ∈ ∈ PROPN cana-6055 229 36 { { PUNCT cana-6055 229 37 1,3 1,3 NUM cana-6055 229 38 , , PUNCT cana-6055 229 39 . . PUNCT cana-6055 229 40 . . PUNCT cana-6055 229 41 . . PUNCT cana-6055 230 1 , , PUNCT cana-6055 230 2 𝑛 𝑛 DET cana-6055 230 3 βˆ’ βˆ’ NOUN cana-6055 230 4 1 1 NUM cana-6055 230 5 } } PUNCT cana-6055 230 6 and and CCONJ cana-6055 230 7 π‘’π‘‘π‘£π‘‘βˆ’1 π‘’π‘‘π‘£π‘‘βˆ’1 PROPN cana-6055 230 8 for for ADP cana-6055 230 9 𝑑 𝑑 PROPN cana-6055 230 10 ∈ ∈ PROPN cana-6055 230 11 { { PUNCT cana-6055 230 12 2,4 2,4 NUM cana-6055 230 13 , , PUNCT cana-6055 230 14 . . PUNCT cana-6055 230 15 . . PUNCT cana-6055 230 16 . . PUNCT cana-6055 231 1 , , PUNCT cana-6055 231 2 𝑛 𝑛 PROPN cana-6055 231 3 } } PUNCT cana-6055 231 4 , , PUNCT cana-6055 231 5 where where SCONJ cana-6055 231 6 𝑣𝑖 𝑣𝑖 ADV cana-6055 231 7 , , PUNCT cana-6055 231 8 𝑒𝑖 𝑒𝑖 NOUN cana-6055 231 9 are be AUX cana-6055 231 10 inner inner ADJ cana-6055 231 11 and and CCONJ cana-6055 231 12 outer outer ADJ cana-6055 231 13 vertices vertex NOUN cana-6055 231 14 of of ADP cana-6055 231 15 𝐢𝑃𝑛. 𝐢𝑃𝑛. PRON cana-6055 231 16 define define VERB cana-6055 231 17 𝑓 𝑓 PRON cana-6055 231 18 : : PUNCT cana-6055 231 19 𝑉(𝐢𝑃𝑛 𝑉(𝐢𝑃𝑛 PROPN cana-6055 231 20 ) ) PUNCT cana-6055 231 21 ⟢ ⟢ NOUN cana-6055 231 22 { { PUNCT cana-6055 231 23 Β±1,Β±2,β‹― Β±1,Β±2,β‹― NOUN cana-6055 231 24 , , PUNCT cana-6055 231 25 ±𝑛 ±𝑛 PROPN cana-6055 231 26 } } PUNCT cana-6055 231 27 by by ADP cana-6055 231 28 𝑣𝑖 𝑣𝑖 ADP cana-6055 231 29 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-6055 231 30 ) ) PUNCT cana-6055 231 31 𝑒𝑖 𝑒𝑖 NOUN cana-6055 231 32 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 231 33 ) ) PUNCT cana-6055 231 34 𝑖 𝑖 SYM cana-6055 232 1 = = NOUN cana-6055 232 2 1 1 NUM cana-6055 232 3 2 2 NUM cana-6055 232 4 𝑖 𝑖 NOUN cana-6055 232 5 = = NOUN cana-6055 232 6 1 1 NUM cana-6055 232 7 1 1 NUM cana-6055 232 8 𝑖 𝑖 NOUN cana-6055 232 9 = = SYM cana-6055 232 10 2 2 NUM cana-6055 232 11 -2 -2 NOUN cana-6055 232 12 𝑖 𝑖 SYM cana-6055 232 13 = = SYM cana-6055 232 14 2 2 NUM cana-6055 232 15 -1 -1 NOUN cana-6055 232 16 𝑖 𝑖 SYM cana-6055 232 17 = = PUNCT cana-6055 232 18 3,4,5 3,4,5 NUM cana-6055 232 19 , , PUNCT cana-6055 232 20 . . PUNCT cana-6055 232 21 . . PUNCT cana-6055 232 22 . . PUNCT cana-6055 233 1 , , PUNCT cana-6055 233 2 𝑛 𝑛 PROPN cana-6055 233 3 2 2 NUM cana-6055 233 4 ( ( PUNCT cana-6055 233 5 βˆ’1)𝑖+1(2𝑖 βˆ’1)𝑖+1(2𝑖 NOUN cana-6055 233 6 βˆ’ βˆ’ PROPN cana-6055 233 7 3 3 NUM cana-6055 233 8 ) ) PUNCT cana-6055 233 9 + + CCONJ cana-6055 233 10 1 1 NUM cana-6055 233 11 𝑖 𝑖 SYM cana-6055 233 12 = = NOUN cana-6055 233 13 3,4,5 3,4,5 NUM cana-6055 233 14 , , PUNCT cana-6055 233 15 . . PUNCT cana-6055 233 16 . . PUNCT cana-6055 233 17 . . PUNCT cana-6055 234 1 , , PUNCT cana-6055 234 2 𝑛 𝑛 PROPN cana-6055 234 3 2 2 NUM cana-6055 234 4 ( ( PUNCT cana-6055 234 5 βˆ’1)𝑖+1 βˆ’1)𝑖+1 PROPN cana-6055 234 6 ( ( PUNCT cana-6055 234 7 2𝑖 2𝑖 NOUN cana-6055 234 8 βˆ’ βˆ’ PROPN cana-6055 234 9 3 3 X cana-6055 234 10 ) ) PUNCT cana-6055 234 11 𝑖 𝑖 NOUN cana-6055 234 12 = = SYM cana-6055 234 13 𝑛 𝑛 DET cana-6055 234 14 2 2 NUM cana-6055 234 15 + + SYM cana-6055 234 16 1 1 NUM cana-6055 234 17 𝑛 𝑛 NOUN cana-6055 234 18 𝑖 𝑖 SYM cana-6055 234 19 = = SYM cana-6055 234 20 𝑛 𝑛 PRON cana-6055 234 21 2 2 NUM cana-6055 234 22 + + SYM cana-6055 234 23 1 1 NUM cana-6055 234 24 𝑛 𝑛 PRON cana-6055 234 25 βˆ’ βˆ’ NUM cana-6055 234 26 1 1 NUM cana-6055 234 27 𝑖 𝑖 NOUN cana-6055 234 28 = = NOUN cana-6055 234 29 𝑛 𝑛 PRON cana-6055 234 30 2 2 NUM cana-6055 234 31 + + SYM cana-6055 234 32 2 2 NUM cana-6055 234 33 βˆ’π‘› βˆ’π‘› NOUN cana-6055 234 34 𝑖 𝑖 NOUN cana-6055 234 35 = = SYM cana-6055 234 36 𝑛 𝑛 PRON cana-6055 234 37 2 2 NUM cana-6055 234 38 + + SYM cana-6055 234 39 2 2 NUM cana-6055 234 40 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-6055 234 41 βˆ’ βˆ’ NOUN cana-6055 234 42 1 1 NUM cana-6055 234 43 ) ) PUNCT cana-6055 234 44 𝑖 𝑖 NOUN cana-6055 234 45 = = SYM cana-6055 234 46 𝑛 𝑛 PRON cana-6055 234 47 2 2 NUM cana-6055 234 48 + + CCONJ cana-6055 234 49 3 3 NUM cana-6055 234 50 , , PUNCT cana-6055 234 51 𝑛 𝑛 DET cana-6055 234 52 2 2 NUM cana-6055 234 53 + + NUM cana-6055 234 54 4 4 NUM cana-6055 234 55 , , PUNCT cana-6055 234 56 . . PUNCT cana-6055 234 57 . . PUNCT cana-6055 235 1 . . PUNCT cana-6055 236 1 , , PUNCT cana-6055 236 2 𝑛 𝑛 PROPN cana-6055 236 3 ( ( PUNCT cana-6055 236 4 βˆ’1)𝑖+1[2(𝑛 βˆ’1)𝑖+1[2(𝑛 NOUN cana-6055 236 5 βˆ’ βˆ’ PROPN cana-6055 236 6 𝑖 𝑖 SYM cana-6055 236 7 ) ) PUNCT cana-6055 236 8 + + CCONJ cana-6055 236 9 4 4 X cana-6055 236 10 ] ] SYM cana-6055 236 11 𝑖 𝑖 NOUN cana-6055 236 12 = = SYM cana-6055 236 13 𝑛 𝑛 PRON cana-6055 236 14 2 2 NUM cana-6055 236 15 + + CCONJ cana-6055 236 16 3 3 NUM cana-6055 236 17 , , PUNCT cana-6055 236 18 𝑛 𝑛 DET cana-6055 236 19 2 2 NUM cana-6055 236 20 + + NUM cana-6055 236 21 4 4 NUM cana-6055 236 22 , , PUNCT cana-6055 236 23 . . PUNCT cana-6055 236 24 . . PUNCT cana-6055 237 1 . . PUNCT cana-6055 238 1 , , PUNCT cana-6055 238 2 𝑛 𝑛 PROPN cana-6055 238 3 ( ( PUNCT cana-6055 238 4 βˆ’1)𝑖+1 βˆ’1)𝑖+1 PROPN cana-6055 238 5 2(𝑛 2(𝑛 NUM cana-6055 238 6 βˆ’ βˆ’ NOUN cana-6055 238 7 𝑖 𝑖 SYM cana-6055 238 8 ) ) PUNCT cana-6055 239 1 + + CCONJ cana-6055 239 2 3 3 NUM cana-6055 239 3 then then ADV cana-6055 239 4 the the DET cana-6055 239 5 induced induced ADJ cana-6055 239 6 vertex vertex NOUN cana-6055 239 7 weight weight NOUN cana-6055 239 8 labeling labeling NOUN cana-6055 239 9 are be AUX cana-6055 239 10 as as SCONJ cana-6055 239 11 follows follow VERB cana-6055 239 12 . . PUNCT cana-6055 240 1 𝑣𝑖 𝑣𝑖 ADP cana-6055 240 2 𝑀(𝑣𝑖 𝑀(𝑣𝑖 VERB cana-6055 240 3 ) ) PUNCT cana-6055 240 4 𝑒𝑖 𝑒𝑖 PRON cana-6055 240 5 𝑀(𝑒𝑖 𝑀(𝑒𝑖 NOUN cana-6055 240 6 ) ) PUNCT cana-6055 240 7 𝑖 𝑖 SYM cana-6055 241 1 = = NOUN cana-6055 241 2 1 1 NUM cana-6055 241 3 7 7 NUM cana-6055 241 4 𝑖 𝑖 NOUN cana-6055 241 5 = = NOUN cana-6055 241 6 1 1 NUM cana-6055 241 7 6 6 NUM cana-6055 241 8 𝑖 𝑖 NOUN cana-6055 241 9 = = SYM cana-6055 241 10 2 2 NUM cana-6055 241 11 -7 -7 NOUN cana-6055 241 12 𝑖 𝑖 SYM cana-6055 241 13 = = SYM cana-6055 241 14 2 2 NUM cana-6055 241 15 -6 -6 NOUN cana-6055 241 16 𝑖 𝑖 NOUN cana-6055 241 17 = = NOUN cana-6055 241 18 3,5 3,5 NUM cana-6055 241 19 , , PUNCT cana-6055 241 20 . . PUNCT cana-6055 241 21 . . PUNCT cana-6055 241 22 . . PUNCT cana-6055 242 1 , , PUNCT cana-6055 242 2 𝑛 𝑛 PRON cana-6055 242 3 2 2 NUM cana-6055 242 4 βˆ’ βˆ’ NUM cana-6055 242 5 1 1 NUM cana-6055 242 6 βˆ’[6(𝑖 βˆ’[6(𝑖 SYM cana-6055 242 7 βˆ’ βˆ’ NUM cana-6055 242 8 1 1 NUM cana-6055 242 9 ) ) PUNCT cana-6055 242 10 + + CCONJ cana-6055 242 11 1 1 X cana-6055 242 12 ] ] PUNCT cana-6055 242 13 𝑖 𝑖 PUNCT cana-6055 242 14 = = NOUN cana-6055 242 15 3,5 3,5 NUM cana-6055 242 16 , , PUNCT cana-6055 242 17 . . PUNCT cana-6055 242 18 . . PUNCT cana-6055 243 1 . . PUNCT cana-6055 244 1 , , PUNCT cana-6055 244 2 𝑛 𝑛 PRON cana-6055 244 3 2 2 NUM cana-6055 244 4 βˆ’ βˆ’ NUM cana-6055 244 5 1 1 NUM cana-6055 244 6 βˆ’[6(𝑖 βˆ’[6(𝑖 SYM cana-6055 244 7 βˆ’ βˆ’ NUM cana-6055 244 8 1 1 NUM cana-6055 244 9 ) ) PUNCT cana-6055 244 10 ] ] PUNCT cana-6055 245 1 𝑖 𝑖 X cana-6055 245 2 = = SYM cana-6055 245 3 4,6 4,6 NUM cana-6055 245 4 , , PUNCT cana-6055 245 5 . . PUNCT cana-6055 245 6 . . PUNCT cana-6055 246 1 . . PUNCT cana-6055 247 1 , , PUNCT cana-6055 247 2 𝑛 𝑛 DET cana-6055 247 3 2 2 NUM cana-6055 247 4 6𝑖 6𝑖 NOUN cana-6055 247 5 βˆ’ βˆ’ NOUN cana-6055 247 6 9 9 NUM cana-6055 247 7 𝑖 𝑖 SYM cana-6055 247 8 = = NOUN cana-6055 247 9 4,6 4,6 NUM cana-6055 247 10 , , PUNCT cana-6055 247 11 . . PUNCT cana-6055 247 12 . . PUNCT cana-6055 248 1 . . PUNCT cana-6055 249 1 , , PUNCT cana-6055 249 2 𝑛 𝑛 DET cana-6055 249 3 2 2 NUM cana-6055 249 4 6𝑖 6𝑖 NOUN cana-6055 249 5 βˆ’ βˆ’ PROPN cana-6055 249 6 10 10 NUM cana-6055 249 7 𝑖 𝑖 NOUN cana-6055 249 8 = = SYM cana-6055 249 9 𝑛 𝑛 PRON cana-6055 249 10 2 2 NUM cana-6055 249 11 + + CCONJ cana-6055 249 12 1 1 NUM cana-6055 249 13 3 3 NUM cana-6055 249 14 βˆ’ βˆ’ NOUN cana-6055 249 15 6(𝑖 6(𝑖 NUM cana-6055 249 16 βˆ’ βˆ’ NOUN cana-6055 249 17 1 1 NUM cana-6055 249 18 ) ) PUNCT cana-6055 249 19 𝑖 𝑖 NOUN cana-6055 250 1 = = SYM cana-6055 250 2 𝑛 𝑛 PRON cana-6055 250 3 2 2 NUM cana-6055 250 4 + + CCONJ cana-6055 250 5 1 1 NUM cana-6055 250 6 4 4 NUM cana-6055 250 7 βˆ’ βˆ’ NOUN cana-6055 250 8 6(𝑖 6(𝑖 NUM cana-6055 250 9 βˆ’ βˆ’ NOUN cana-6055 250 10 1 1 NUM cana-6055 250 11 ) ) PUNCT cana-6055 250 12 communications communication NOUN cana-6055 250 13 on on ADP cana-6055 250 14 applied apply VERB cana-6055 250 15 nonlinear nonlinear ADJ cana-6055 250 16 analysis analysis NOUN cana-6055 250 17 issn issn NOUN cana-6055 250 18 : : PUNCT cana-6055 250 19 1074 1074 NUM cana-6055 250 20 - - PUNCT cana-6055 250 21 133x 133x NUM cana-6055 250 22 vol vol NOUN cana-6055 250 23 31 31 NUM cana-6055 250 24 no no NOUN cana-6055 250 25 . . PUNCT cana-6055 251 1 8s 8s PROPN cana-6055 251 2 ( ( PUNCT cana-6055 251 3 2024 2024 NUM cana-6055 251 4 ) ) PUNCT cana-6055 251 5 1164 1164 NUM cana-6055 251 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 251 7 𝑖 𝑖 SYM cana-6055 251 8 = = SYM cana-6055 251 9 𝑛 𝑛 PRON cana-6055 251 10 2 2 NUM cana-6055 251 11 + + CCONJ cana-6055 251 12 2 2 NUM cana-6055 251 13 6(𝑖 6(𝑖 NUM cana-6055 251 14 βˆ’ βˆ’ NOUN cana-6055 251 15 2 2 NUM cana-6055 251 16 ) ) PUNCT cana-6055 251 17 βˆ’ βˆ’ NOUN cana-6055 252 1 3 3 NUM cana-6055 252 2 𝑖 𝑖 SYM cana-6055 252 3 = = SYM cana-6055 252 4 𝑛 𝑛 PRON cana-6055 252 5 2 2 NUM cana-6055 252 6 + + CCONJ cana-6055 252 7 2 2 NUM cana-6055 252 8 6(𝑖 6(𝑖 NUM cana-6055 252 9 βˆ’ βˆ’ NOUN cana-6055 252 10 2 2 NUM cana-6055 252 11 ) ) PUNCT cana-6055 252 12 βˆ’ βˆ’ PROPN cana-6055 252 13 4 4 NUM cana-6055 252 14 𝑖 𝑖 NOUN cana-6055 252 15 = = SYM cana-6055 252 16 𝑛 𝑛 PRON cana-6055 252 17 2 2 NUM cana-6055 252 18 + + CCONJ cana-6055 252 19 3 3 NUM cana-6055 252 20 , , PUNCT cana-6055 252 21 𝑛 𝑛 DET cana-6055 252 22 2 2 NUM cana-6055 252 23 + + SYM cana-6055 252 24 5 5 NUM cana-6055 252 25 , , PUNCT cana-6055 252 26 . . PUNCT cana-6055 252 27 . . PUNCT cana-6055 252 28 . . PUNCT cana-6055 253 1 , , PUNCT cana-6055 254 1 𝑛 𝑛 DET cana-6055 254 2 βˆ’ βˆ’ PROPN cana-6055 254 3 1 1 NUM cana-6055 254 4 βˆ’[6(𝑛 βˆ’[6(𝑛 SYM cana-6055 254 5 βˆ’ βˆ’ PROPN cana-6055 254 6 𝑖 𝑖 SYM cana-6055 254 7 + + NOUN cana-6055 254 8 1 1 NUM cana-6055 254 9 ) ) PUNCT cana-6055 254 10 + + CCONJ cana-6055 254 11 3 3 X cana-6055 254 12 ] ] SYM cana-6055 254 13 𝑖 𝑖 NOUN cana-6055 254 14 = = SYM cana-6055 254 15 𝑛 𝑛 PRON cana-6055 254 16 2 2 NUM cana-6055 254 17 + + CCONJ cana-6055 254 18 3 3 NUM cana-6055 254 19 , , PUNCT cana-6055 254 20 𝑛 𝑛 DET cana-6055 254 21 2 2 NUM cana-6055 254 22 + + SYM cana-6055 254 23 5 5 NUM cana-6055 254 24 , , PUNCT cana-6055 254 25 . . PUNCT cana-6055 254 26 . . PUNCT cana-6055 255 1 . . PUNCT cana-6055 256 1 , , PUNCT cana-6055 257 1 𝑛 𝑛 DET cana-6055 257 2 βˆ’ βˆ’ PROPN cana-6055 257 3 1 1 NUM cana-6055 257 4 βˆ’[6(𝑛 βˆ’[6(𝑛 SYM cana-6055 257 5 βˆ’ βˆ’ PROPN cana-6055 257 6 𝑖 𝑖 SYM cana-6055 257 7 + + NOUN cana-6055 257 8 1 1 NUM cana-6055 257 9 ) ) PUNCT cana-6055 257 10 + + CCONJ cana-6055 257 11 2 2 X cana-6055 257 12 ] ] PUNCT cana-6055 257 13 𝑖 𝑖 NOUN cana-6055 257 14 = = SYM cana-6055 257 15 𝑛 𝑛 PRON cana-6055 257 16 2 2 NUM cana-6055 257 17 + + CCONJ cana-6055 257 18 4 4 NUM cana-6055 257 19 , , PUNCT cana-6055 257 20 𝑛 𝑛 DET cana-6055 257 21 2 2 NUM cana-6055 257 22 + + NUM cana-6055 257 23 6 6 NUM cana-6055 257 24 , , PUNCT cana-6055 257 25 . . PUNCT cana-6055 257 26 . . PUNCT cana-6055 258 1 . . PUNCT cana-6055 259 1 , , PUNCT cana-6055 259 2 𝑛 𝑛 PRON cana-6055 259 3 6(𝑛 6(𝑛 NUM cana-6055 259 4 βˆ’ βˆ’ PROPN cana-6055 259 5 𝑖 𝑖 SYM cana-6055 260 1 + + NOUN cana-6055 260 2 2 2 NUM cana-6055 260 3 ) ) PUNCT cana-6055 260 4 + + CCONJ cana-6055 260 5 1 1 NUM cana-6055 260 6 𝑖 𝑖 NOUN cana-6055 260 7 = = SYM cana-6055 260 8 𝑛 𝑛 PRON cana-6055 260 9 2 2 NUM cana-6055 260 10 + + CCONJ cana-6055 260 11 4 4 NUM cana-6055 260 12 , , PUNCT cana-6055 260 13 𝑛 𝑛 DET cana-6055 260 14 2 2 NUM cana-6055 260 15 + + NUM cana-6055 260 16 6 6 NUM cana-6055 260 17 , , PUNCT cana-6055 260 18 . . PUNCT cana-6055 260 19 . . PUNCT cana-6055 260 20 . . PUNCT cana-6055 261 1 , , PUNCT cana-6055 261 2 𝑛 𝑛 PRON cana-6055 261 3 6(𝑛 6(𝑛 NUM cana-6055 261 4 βˆ’ βˆ’ PROPN cana-6055 261 5 𝑖 𝑖 SYM cana-6055 262 1 + + NOUN cana-6055 262 2 2 2 NUM cana-6055 262 3 ) ) PUNCT cana-6055 262 4 hence hence ADV cana-6055 262 5 the the DET cana-6055 262 6 crossed cross VERB cana-6055 262 7 prism prism NOUN cana-6055 262 8 𝐢𝑃𝑛 𝐢𝑃𝑛 NOUN cana-6055 262 9 is be AUX cana-6055 262 10 a a DET cana-6055 262 11 distance distance NOUN cana-6055 262 12 pair pair NOUN cana-6055 262 13 antimagic antimagic ADJ cana-6055 262 14 graph graph NOUN cana-6055 262 15 if if SCONJ cana-6055 262 16 𝑛 𝑛 PRON cana-6055 262 17 ≑ ≑ PROPN cana-6055 262 18 0 0 PUNCT cana-6055 263 1 ( ( PUNCT cana-6055 263 2 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ NOUN cana-6055 263 3 4 4 NUM cana-6055 263 4 ) ) PUNCT cana-6055 263 5 . . PUNCT cana-6055 264 1 theorem theorem VERB cana-6055 264 2 3.9 3.9 NUM cana-6055 264 3 the the DET cana-6055 264 4 circulant circulant ADJ cana-6055 264 5 graph graph NOUN cana-6055 264 6 𝐢(2𝑛 𝐢(2𝑛 NOUN cana-6055 264 7 ; ; PUNCT cana-6055 264 8 { { PUNCT cana-6055 264 9 𝑑1 𝑑1 NOUN cana-6055 264 10 , , PUNCT cana-6055 264 11 𝑑2 𝑑2 NOUN cana-6055 264 12 , , PUNCT cana-6055 264 13 𝑛 𝑛 ADJ cana-6055 264 14 } } PUNCT cana-6055 264 15 ) ) PUNCT cana-6055 264 16 is be AUX cana-6055 264 17 a a DET cana-6055 264 18 distance distance NOUN cana-6055 264 19 pair pair NOUN cana-6055 264 20 antimagic antimagic NOUN cana-6055 264 21 if if SCONJ cana-6055 264 22 𝑛 𝑛 PRON cana-6055 264 23 = = SYM cana-6055 264 24 𝑑1 𝑑1 NOUN cana-6055 264 25 + + CCONJ cana-6055 264 26 𝑑2 𝑑2 NOUN cana-6055 264 27 , , PUNCT cana-6055 264 28 𝑑1 𝑑1 NOUN cana-6055 264 29 < < X cana-6055 264 30 𝑑2 𝑑2 NOUN cana-6055 264 31 and and CCONJ cana-6055 264 32 𝑛 𝑛 ADJ cana-6055 264 33 > > X cana-6055 265 1 3 3 X cana-6055 265 2 . . PUNCT cana-6055 265 3 proof proof NOUN cana-6055 265 4 . . PUNCT cana-6055 266 1 let let VERB cana-6055 266 2 𝐺 𝐺 NOUN cana-6055 266 3 = = SYM cana-6055 266 4 𝐢(2𝑛 𝐢(2𝑛 PROPN cana-6055 266 5 ; ; PUNCT cana-6055 266 6 { { PUNCT cana-6055 266 7 𝑑1 𝑑1 NOUN cana-6055 266 8 , , PUNCT cana-6055 266 9 𝑑2 𝑑2 NOUN cana-6055 266 10 , , PUNCT cana-6055 266 11 𝑛 𝑛 NOUN cana-6055 266 12 } } PUNCT cana-6055 266 13 ) ) PUNCT cana-6055 266 14 and and CCONJ cana-6055 266 15 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 266 16 ) ) PUNCT cana-6055 266 17 = = SYM cana-6055 266 18 { { PUNCT cana-6055 266 19 𝑒1 𝑒1 PROPN cana-6055 266 20 , , PUNCT cana-6055 266 21 𝑒2 𝑒2 PROPN cana-6055 266 22 , , PUNCT cana-6055 266 23 𝑒3 𝑒3 PROPN cana-6055 266 24 , , PUNCT cana-6055 266 25 . . PUNCT cana-6055 266 26 . . PUNCT cana-6055 267 1 . . PUNCT cana-6055 268 1 , , PUNCT cana-6055 268 2 𝑒2𝑛 𝑒2𝑛 NOUN cana-6055 268 3 } } PUNCT cana-6055 268 4 define define VERB cana-6055 268 5 𝑓 𝑓 DET cana-6055 268 6 : : PUNCT cana-6055 268 7 𝑉(𝐺 𝑉(𝐺 NOUN cana-6055 268 8 ) ) PUNCT cana-6055 268 9 ⟢ ⟢ NOUN cana-6055 268 10 { { PUNCT cana-6055 268 11 Β±1,Β±2,β‹― Β±1,Β±2,β‹― NOUN cana-6055 268 12 , , PUNCT cana-6055 268 13 ±𝑛 ±𝑛 PROPN cana-6055 268 14 } } PUNCT cana-6055 268 15 by by ADP cana-6055 268 16 following follow VERB cana-6055 268 17 two two NUM cana-6055 268 18 cases case NOUN cana-6055 268 19 . . PUNCT cana-6055 269 1 case case NOUN cana-6055 269 2 ( ( PUNCT cana-6055 269 3 i i NOUN cana-6055 269 4 ): ): PUNCT cana-6055 269 5 n n X cana-6055 269 6 is be AUX cana-6055 269 7 odd odd ADJ cana-6055 269 8 𝑓(𝑒𝑖 𝑓(𝑒𝑖 NOUN cana-6055 269 9 ) ) PUNCT cana-6055 269 10 = = SYM cana-6055 269 11 { { PUNCT cana-6055 269 12 βˆ’π‘– βˆ’π‘– PROPN cana-6055 269 13 , , PUNCT cana-6055 269 14 𝑖𝑓 𝑖𝑓 ADP cana-6055 269 15 𝑖 𝑖 X cana-6055 269 16 = = NOUN cana-6055 269 17 1,3,5 1,3,5 NUM cana-6055 269 18 , , PUNCT cana-6055 269 19 . . PUNCT cana-6055 269 20 . . PUNCT cana-6055 270 1 . . PUNCT cana-6055 271 1 , , PUNCT cana-6055 271 2 𝑛 𝑛 PROPN cana-6055 271 3 𝑖 𝑖 SYM cana-6055 271 4 , , PUNCT cana-6055 271 5 𝑖𝑓 𝑖𝑓 ADP cana-6055 271 6 𝑖 𝑖 SYM cana-6055 271 7 = = SYM cana-6055 271 8 2,4,6 2,4,6 NUM cana-6055 271 9 , , PUNCT cana-6055 271 10 . . PUNCT cana-6055 271 11 . . PUNCT cana-6055 272 1 . . PUNCT cana-6055 273 1 , , PUNCT cana-6055 273 2 𝑛 𝑛 DET cana-6055 273 3 βˆ’ βˆ’ PROPN cana-6055 273 4 1 1 NUM cana-6055 273 5 𝑛 𝑛 PRON cana-6055 273 6 βˆ’ βˆ’ PROPN cana-6055 273 7 𝑖 𝑖 SYM cana-6055 273 8 , , PUNCT cana-6055 273 9 𝑖𝑓 𝑖𝑓 ADP cana-6055 273 10 𝑖 𝑖 NOUN cana-6055 273 11 = = SYM cana-6055 273 12 𝑛 𝑛 PROPN cana-6055 273 13 + + ADJ cana-6055 273 14 2 2 NUM cana-6055 273 15 , , PUNCT cana-6055 273 16 𝑛 𝑛 PRON cana-6055 273 17 + + NOUN cana-6055 273 18 4 4 NUM cana-6055 273 19 , , PUNCT cana-6055 273 20 𝑛 𝑛 PRON cana-6055 273 21 + + NOUN cana-6055 273 22 6 6 NUM cana-6055 273 23 , , PUNCT cana-6055 273 24 . . PUNCT cana-6055 273 25 . . PUNCT cana-6055 274 1 . . PUNCT cana-6055 275 1 , , PUNCT cana-6055 275 2 2𝑛 2𝑛 PROPN cana-6055 275 3 βˆ’ βˆ’ PROPN cana-6055 276 1 1 1 NUM cana-6055 276 2 𝑖 𝑖 SYM cana-6055 276 3 βˆ’ βˆ’ PROPN cana-6055 276 4 𝑛 𝑛 PROPN cana-6055 276 5 , , PUNCT cana-6055 276 6 𝑖𝑓 𝑖𝑓 ADP cana-6055 276 7 𝑖 𝑖 NOUN cana-6055 276 8 = = SYM cana-6055 276 9 𝑛 𝑛 PROPN cana-6055 277 1 + + NOUN cana-6055 277 2 1 1 NUM cana-6055 277 3 , , PUNCT cana-6055 277 4 𝑛 𝑛 PRON cana-6055 277 5 + + NOUN cana-6055 277 6 3 3 NUM cana-6055 277 7 , , PUNCT cana-6055 277 8 𝑛 𝑛 PRON cana-6055 277 9 + + NOUN cana-6055 277 10 4 4 NUM cana-6055 277 11 , , PUNCT cana-6055 277 12 . . PUNCT cana-6055 277 13 . . PUNCT cana-6055 278 1 . . PUNCT cana-6055 279 1 , , PUNCT cana-6055 279 2 2𝑛 2𝑛 PROPN cana-6055 279 3 then then ADV cana-6055 279 4 the the DET cana-6055 279 5 induced induced ADJ cana-6055 279 6 vertex vertex NOUN cana-6055 279 7 weight weight NOUN cana-6055 279 8 labeling labeling NOUN cana-6055 279 9 are be AUX cana-6055 279 10 as as SCONJ cana-6055 279 11 follows follow NOUN cana-6055 279 12 . . PUNCT cana-6055 280 1 𝑀(𝑒𝑖 𝑀(𝑒𝑖 X cana-6055 280 2 ) ) PUNCT cana-6055 280 3 = = PRON cana-6055 280 4 { { PUNCT cana-6055 280 5 𝑖 𝑖 ADP cana-6055 280 6 , , PUNCT cana-6055 280 7 𝑖𝑓 𝑖𝑓 ADP cana-6055 280 8 𝑖 𝑖 SYM cana-6055 280 9 = = NOUN cana-6055 280 10 1,3,5 1,3,5 NUM cana-6055 280 11 , , PUNCT cana-6055 280 12 . . PUNCT cana-6055 280 13 . . PUNCT cana-6055 280 14 . . PUNCT cana-6055 281 1 , , PUNCT cana-6055 281 2 𝑛 𝑛 DET cana-6055 281 3 βˆ’π‘– βˆ’π‘– PROPN cana-6055 281 4 , , PUNCT cana-6055 281 5 𝑖𝑓 𝑖𝑓 ADP cana-6055 281 6 𝑖 𝑖 X cana-6055 281 7 = = SYM cana-6055 281 8 2,4,6 2,4,6 NUM cana-6055 281 9 , , PUNCT cana-6055 281 10 . . PUNCT cana-6055 281 11 . . PUNCT cana-6055 282 1 . . PUNCT cana-6055 283 1 , , PUNCT cana-6055 284 1 𝑛 𝑛 PRON cana-6055 284 2 βˆ’ βˆ’ NUM cana-6055 284 3 1 1 NUM cana-6055 284 4 𝑖 𝑖 NOUN cana-6055 284 5 βˆ’ βˆ’ PROPN cana-6055 284 6 𝑛 𝑛 PROPN cana-6055 284 7 , , PUNCT cana-6055 284 8 𝑖𝑓 𝑖𝑓 ADP cana-6055 284 9 𝑖 𝑖 NOUN cana-6055 284 10 = = SYM cana-6055 284 11 𝑛 𝑛 PROPN cana-6055 284 12 + + ADJ cana-6055 284 13 2 2 NUM cana-6055 284 14 , , PUNCT cana-6055 284 15 𝑛 𝑛 PRON cana-6055 284 16 + + NOUN cana-6055 284 17 4 4 NUM cana-6055 284 18 , , PUNCT cana-6055 284 19 𝑛 𝑛 PRON cana-6055 284 20 + + NOUN cana-6055 284 21 6 6 NUM cana-6055 284 22 , , PUNCT cana-6055 284 23 . . PUNCT cana-6055 284 24 . . PUNCT cana-6055 284 25 . . PUNCT cana-6055 285 1 , , PUNCT cana-6055 285 2 2𝑛 2𝑛 PROPN cana-6055 285 3 βˆ’ βˆ’ PROPN cana-6055 285 4 1 1 NUM cana-6055 285 5 𝑛 𝑛 PRON cana-6055 285 6 βˆ’ βˆ’ PROPN cana-6055 285 7 𝑖 𝑖 SYM cana-6055 285 8 , , PUNCT cana-6055 285 9 𝑖𝑓 𝑖𝑓 ADP cana-6055 285 10 𝑖 𝑖 NOUN cana-6055 285 11 = = SYM cana-6055 285 12 𝑛 𝑛 PROPN cana-6055 285 13 + + NOUN cana-6055 285 14 1 1 NUM cana-6055 285 15 , , PUNCT cana-6055 285 16 𝑛 𝑛 PRON cana-6055 285 17 + + NOUN cana-6055 285 18 3 3 NUM cana-6055 285 19 , , PUNCT cana-6055 285 20 𝑛 𝑛 PRON cana-6055 285 21 + + NOUN cana-6055 285 22 4 4 NUM cana-6055 285 23 , , PUNCT cana-6055 285 24 . . PUNCT cana-6055 285 25 . . PUNCT cana-6055 285 26 . . PUNCT cana-6055 286 1 , , PUNCT cana-6055 286 2 2𝑛 2𝑛 PROPN cana-6055 286 3 case case NOUN cana-6055 286 4 ( ( PUNCT cana-6055 286 5 ii ii NOUN cana-6055 286 6 ): ): PUNCT cana-6055 286 7 n n X cana-6055 286 8 is be AUX cana-6055 286 9 even even ADV cana-6055 286 10 𝑓(𝑒𝑖 𝑓(𝑒𝑖 X cana-6055 286 11 ) ) PUNCT cana-6055 286 12 = = SYM cana-6055 286 13 { { PUNCT cana-6055 286 14 βˆ’π‘– βˆ’π‘– PROPN cana-6055 286 15 , , PUNCT cana-6055 286 16 𝑖𝑓 𝑖𝑓 ADP cana-6055 286 17 𝑖 𝑖 X cana-6055 286 18 = = NOUN cana-6055 286 19 1,3,5 1,3,5 NUM cana-6055 286 20 , , PUNCT cana-6055 286 21 . . PUNCT cana-6055 286 22 . . PUNCT cana-6055 287 1 . . PUNCT cana-6055 288 1 , , PUNCT cana-6055 288 2 𝑛 𝑛 DET cana-6055 288 3 βˆ’ βˆ’ NUM cana-6055 288 4 1 1 NUM cana-6055 288 5 𝑖 𝑖 NOUN cana-6055 288 6 , , PUNCT cana-6055 288 7 𝑖𝑓 𝑖𝑓 ADP cana-6055 288 8 𝑖 𝑖 SYM cana-6055 288 9 = = SYM cana-6055 288 10 2,4,6 2,4,6 NUM cana-6055 288 11 , , PUNCT cana-6055 288 12 . . PUNCT cana-6055 288 13 . . PUNCT cana-6055 289 1 . . PUNCT cana-6055 290 1 , , PUNCT cana-6055 290 2 𝑛 𝑛 VERB cana-6055 290 3 𝑖 𝑖 SYM cana-6055 290 4 βˆ’ βˆ’ PROPN cana-6055 290 5 𝑛 𝑛 PROPN cana-6055 290 6 , , PUNCT cana-6055 290 7 𝑖𝑓 𝑖𝑓 ADP cana-6055 290 8 𝑖 𝑖 NOUN cana-6055 290 9 = = SYM cana-6055 290 10 𝑛 𝑛 PROPN cana-6055 291 1 + + NOUN cana-6055 291 2 1 1 NUM cana-6055 291 3 , , PUNCT cana-6055 291 4 𝑛 𝑛 PRON cana-6055 291 5 + + NOUN cana-6055 291 6 3 3 NUM cana-6055 291 7 , , PUNCT cana-6055 291 8 𝑛 𝑛 PRON cana-6055 291 9 + + NOUN cana-6055 291 10 5 5 NUM cana-6055 291 11 , , PUNCT cana-6055 291 12 . . PUNCT cana-6055 291 13 . . PUNCT cana-6055 292 1 . . PUNCT cana-6055 293 1 , , PUNCT cana-6055 293 2 2𝑛 2𝑛 PROPN cana-6055 293 3 βˆ’ βˆ’ PROPN cana-6055 293 4 1 1 NUM cana-6055 293 5 𝑛 𝑛 PRON cana-6055 293 6 βˆ’ βˆ’ PROPN cana-6055 293 7 𝑖 𝑖 SYM cana-6055 293 8 , , PUNCT cana-6055 293 9 𝑖𝑓 𝑖𝑓 ADP cana-6055 293 10 𝑖 𝑖 NOUN cana-6055 293 11 = = SYM cana-6055 293 12 𝑛 𝑛 PROPN cana-6055 293 13 + + ADJ cana-6055 293 14 2 2 NUM cana-6055 293 15 , , PUNCT cana-6055 293 16 𝑛 𝑛 PRON cana-6055 293 17 + + NOUN cana-6055 293 18 4 4 NUM cana-6055 293 19 , , PUNCT cana-6055 293 20 𝑛 𝑛 PRON cana-6055 293 21 + + NOUN cana-6055 293 22 6 6 NUM cana-6055 293 23 , , PUNCT cana-6055 293 24 . . PUNCT cana-6055 293 25 . . PUNCT cana-6055 293 26 . . PUNCT cana-6055 294 1 , , PUNCT cana-6055 294 2 2𝑛 2𝑛 PROPN cana-6055 294 3 then then ADV cana-6055 294 4 the the DET cana-6055 294 5 induced induced ADJ cana-6055 294 6 vertex vertex NOUN cana-6055 294 7 weight weight NOUN cana-6055 294 8 labeling labeling NOUN cana-6055 294 9 are be AUX cana-6055 294 10 as as SCONJ cana-6055 294 11 follows follow NOUN cana-6055 294 12 . . PUNCT cana-6055 295 1 𝑀(𝑒𝑖 𝑀(𝑒𝑖 X cana-6055 295 2 ) ) PUNCT cana-6055 295 3 = = PRON cana-6055 295 4 { { PUNCT cana-6055 295 5 𝑖 𝑖 ADP cana-6055 295 6 , , PUNCT cana-6055 295 7 𝑖𝑓 𝑖𝑓 ADP cana-6055 295 8 𝑖 𝑖 SYM cana-6055 295 9 = = NOUN cana-6055 295 10 1,3,5 1,3,5 NUM cana-6055 295 11 , , PUNCT cana-6055 295 12 . . PUNCT cana-6055 295 13 . . PUNCT cana-6055 295 14 . . PUNCT cana-6055 296 1 , , PUNCT cana-6055 296 2 𝑛 𝑛 DET cana-6055 296 3 βˆ’ βˆ’ PROPN cana-6055 296 4 1 1 NUM cana-6055 296 5 βˆ’π‘– βˆ’π‘– PROPN cana-6055 296 6 , , PUNCT cana-6055 296 7 𝑖𝑓 𝑖𝑓 ADP cana-6055 296 8 𝑖 𝑖 X cana-6055 296 9 = = SYM cana-6055 296 10 2,4,6 2,4,6 NUM cana-6055 296 11 , , PUNCT cana-6055 296 12 . . PUNCT cana-6055 296 13 . . PUNCT cana-6055 297 1 . . PUNCT cana-6055 298 1 , , PUNCT cana-6055 298 2 𝑛 𝑛 DET cana-6055 298 3 𝑛 𝑛 PRON cana-6055 298 4 βˆ’ βˆ’ PROPN cana-6055 298 5 𝑖 𝑖 SYM cana-6055 298 6 , , PUNCT cana-6055 298 7 𝑖𝑓 𝑖𝑓 ADP cana-6055 298 8 𝑖 𝑖 NOUN cana-6055 298 9 = = SYM cana-6055 298 10 𝑛 𝑛 PROPN cana-6055 299 1 + + NOUN cana-6055 299 2 1 1 NUM cana-6055 299 3 , , PUNCT cana-6055 299 4 𝑛 𝑛 PRON cana-6055 299 5 + + NOUN cana-6055 299 6 3 3 NUM cana-6055 299 7 , , PUNCT cana-6055 299 8 𝑛 𝑛 PRON cana-6055 299 9 + + NOUN cana-6055 299 10 5 5 NUM cana-6055 299 11 , , PUNCT cana-6055 299 12 . . PUNCT cana-6055 299 13 . . PUNCT cana-6055 300 1 . . PUNCT cana-6055 301 1 , , PUNCT cana-6055 301 2 2𝑛 2𝑛 PROPN cana-6055 301 3 βˆ’ βˆ’ PROPN cana-6055 302 1 1 1 NUM cana-6055 302 2 𝑖 𝑖 SYM cana-6055 302 3 βˆ’ βˆ’ PROPN cana-6055 302 4 𝑛 𝑛 PROPN cana-6055 302 5 , , PUNCT cana-6055 302 6 𝑖𝑓 𝑖𝑓 ADP cana-6055 302 7 𝑖 𝑖 NOUN cana-6055 302 8 = = SYM cana-6055 302 9 𝑛 𝑛 PROPN cana-6055 302 10 + + ADJ cana-6055 302 11 2 2 NUM cana-6055 302 12 , , PUNCT cana-6055 302 13 𝑛 𝑛 PRON cana-6055 302 14 + + NOUN cana-6055 302 15 4 4 NUM cana-6055 302 16 , , PUNCT cana-6055 302 17 𝑛 𝑛 PRON cana-6055 302 18 + + NOUN cana-6055 302 19 6 6 NUM cana-6055 302 20 , , PUNCT cana-6055 302 21 . . PUNCT cana-6055 302 22 . . PUNCT cana-6055 303 1 . . PUNCT cana-6055 304 1 , , PUNCT cana-6055 304 2 2𝑛 2𝑛 PROPN cana-6055 304 3 hence hence ADV cana-6055 304 4 𝐢(2𝑛 𝐢(2𝑛 PROPN cana-6055 304 5 ; ; PUNCT cana-6055 304 6 { { PUNCT cana-6055 304 7 𝑑1 𝑑1 NOUN cana-6055 304 8 , , PUNCT cana-6055 304 9 𝑑2 𝑑2 NOUN cana-6055 304 10 , , PUNCT cana-6055 304 11 𝑛 𝑛 ADJ cana-6055 304 12 } } PUNCT cana-6055 304 13 ) ) PUNCT cana-6055 304 14 is be AUX cana-6055 304 15 a a DET cana-6055 304 16 distance distance NOUN cana-6055 304 17 pair pair NOUN cana-6055 304 18 antimagic antimagic NOUN cana-6055 304 19 if if SCONJ cana-6055 304 20 𝑛 𝑛 PRON cana-6055 304 21 = = SYM cana-6055 304 22 𝑑1 𝑑1 NOUN cana-6055 304 23 + + CCONJ cana-6055 304 24 𝑑2 𝑑2 NOUN cana-6055 304 25 , , PUNCT cana-6055 304 26 𝑑1 𝑑1 NOUN cana-6055 304 27 < < X cana-6055 304 28 𝑑2 𝑑2 NOUN cana-6055 304 29 and and CCONJ cana-6055 304 30 𝑛 𝑛 ADJ cana-6055 304 31 > > SYM cana-6055 305 1 3 3 X cana-6055 305 2 . . PUNCT cana-6055 305 3 corollary corollary ADJ cana-6055 305 4 3.10 3.10 NUM cana-6055 305 5 the the DET cana-6055 305 6 circulant circulant ADJ cana-6055 305 7 graph graph NOUN cana-6055 305 8 𝐢̅(2𝑛 𝐢̅(2𝑛 NOUN cana-6055 305 9 ; ; PUNCT cana-6055 305 10 { { PUNCT cana-6055 305 11 𝑑1 𝑑1 NOUN cana-6055 305 12 , , PUNCT cana-6055 305 13 𝑑2 𝑑2 NOUN cana-6055 305 14 , , PUNCT cana-6055 305 15 𝑛 𝑛 ADJ cana-6055 305 16 } } PUNCT cana-6055 305 17 ) ) PUNCT cana-6055 305 18 is be AUX cana-6055 305 19 a a DET cana-6055 305 20 closed closed ADJ cana-6055 305 21 distance distance NOUN cana-6055 305 22 pair pair NOUN cana-6055 305 23 antimagic antimagic NOUN cana-6055 305 24 , , PUNCT cana-6055 305 25 if if SCONJ cana-6055 305 26 𝑛 𝑛 PRON cana-6055 305 27 = = SYM cana-6055 305 28 𝑑1 𝑑1 NOUN cana-6055 305 29 + + CCONJ cana-6055 305 30 𝑑2 𝑑2 NOUN cana-6055 305 31 , , PUNCT cana-6055 305 32 𝑑1 𝑑1 NOUN cana-6055 305 33 < < X cana-6055 305 34 𝑑2 𝑑2 NOUN cana-6055 305 35 and and CCONJ cana-6055 305 36 𝑛 𝑛 ADJ cana-6055 305 37 > > SYM cana-6055 306 1 3 3 X cana-6055 306 2 . . PUNCT cana-6055 306 3 corollary corollary NOUN cana-6055 306 4 3.11 3.11 NUM cana-6055 306 5 the the DET cana-6055 306 6 circulant circulant ADJ cana-6055 306 7 graph graph NOUN cana-6055 306 8 𝐢̅(2𝑛 𝐢̅(2𝑛 PROPN cana-6055 306 9 ; ; PUNCT cana-6055 306 10 { { PUNCT cana-6055 306 11 𝑑1 𝑑1 NOUN cana-6055 306 12 , , PUNCT cana-6055 306 13 𝑑2 𝑑2 NOUN cana-6055 306 14 } } PUNCT cana-6055 306 15 ) ) PUNCT cana-6055 306 16 is be AUX cana-6055 306 17 a a DET cana-6055 306 18 distance distance NOUN cana-6055 306 19 pair pair NOUN cana-6055 306 20 antimagic antimagic NOUN cana-6055 306 21 , , PUNCT cana-6055 306 22 if if SCONJ cana-6055 306 23 𝑛 𝑛 PRON cana-6055 306 24 = = SYM cana-6055 306 25 𝑑1 𝑑1 NOUN cana-6055 306 26 + + CCONJ cana-6055 306 27 𝑑2 𝑑2 NOUN cana-6055 306 28 , , PUNCT cana-6055 306 29 𝑑1 𝑑1 NOUN cana-6055 306 30 < < X cana-6055 306 31 𝑑2 𝑑2 NOUN cana-6055 306 32 and and CCONJ cana-6055 306 33 𝑛 𝑛 ADJ cana-6055 306 34 > > SYM cana-6055 307 1 3 3 X cana-6055 307 2 . . PUNCT cana-6055 307 3 communications communication NOUN cana-6055 307 4 on on ADP cana-6055 307 5 applied apply VERB cana-6055 307 6 nonlinear nonlinear ADJ cana-6055 307 7 analysis analysis NOUN cana-6055 307 8 issn issn NOUN cana-6055 307 9 : : PUNCT cana-6055 307 10 1074 1074 NUM cana-6055 307 11 - - PUNCT cana-6055 307 12 133x 133x NUM cana-6055 307 13 vol vol NOUN cana-6055 307 14 31 31 NUM cana-6055 307 15 no no NOUN cana-6055 307 16 . . PUNCT cana-6055 308 1 8s 8s PROPN cana-6055 308 2 ( ( PUNCT cana-6055 308 3 2024 2024 NUM cana-6055 308 4 ) ) PUNCT cana-6055 308 5 1165 1165 NUM cana-6055 308 6 https://internationalpubls.com https://internationalpubls.com X cana-6055 308 7 corollary corollary NOUN cana-6055 308 8 3.12 3.12 NUM cana-6055 308 9 the the DET cana-6055 308 10 circulant circulant ADJ cana-6055 308 11 graph graph NOUN cana-6055 308 12 𝐢(2𝑛;𝐷 𝐢(2𝑛;𝐷 PROPN cana-6055 308 13 ) ) PUNCT cana-6055 308 14 is be AUX cana-6055 308 15 a a DET cana-6055 308 16 distance distance NOUN cana-6055 308 17 pair pair NOUN cana-6055 308 18 antimagic antimagic NOUN cana-6055 308 19 , , PUNCT cana-6055 308 20 if if SCONJ cana-6055 308 21 𝑛 𝑛 PROPN cana-6055 308 22 > > X cana-6055 308 23 3 3 NUM cana-6055 308 24 where where SCONJ cana-6055 308 25 𝐷 𝐷 PROPN cana-6055 308 26 βŠ† βŠ† NUM cana-6055 308 27 𝐷𝑛 𝐷𝑛 NOUN cana-6055 308 28 = = SYM cana-6055 308 29 { { PUNCT cana-6055 308 30 ( ( PUNCT cana-6055 308 31 𝑑𝑖 𝑑𝑖 PROPN cana-6055 308 32 , , PUNCT cana-6055 308 33 𝑑𝑗 𝑑𝑗 PROPN cana-6055 308 34 ): ): PUNCT cana-6055 308 35 𝑑𝑖 𝑑𝑖 X cana-6055 309 1 + + CCONJ cana-6055 309 2 𝑑𝑗 𝑑𝑗 NOUN cana-6055 309 3 = = SYM cana-6055 309 4 𝑛 𝑛 NOUN cana-6055 309 5 } } PUNCT cana-6055 309 6 and and CCONJ cana-6055 309 7 𝑑𝑖 𝑑𝑖 VERB cana-6055 309 8 < < X cana-6055 309 9 𝑑𝑗 𝑑𝑗 PROPN cana-6055 309 10 . . PUNCT cana-6055 310 1 4 4 X cana-6055 310 2 . . X cana-6055 310 3 conclusion conclusion VERB cana-6055 310 4 the the DET cana-6055 310 5 distance distance NOUN cana-6055 310 6 pair pair NOUN cana-6055 310 7 antimagic antimagic ADJ cana-6055 310 8 labeling labeling NOUN cana-6055 310 9 on on ADP cana-6055 310 10 cycle cycle NOUN cana-6055 310 11 related relate VERB cana-6055 310 12 graphs graph NOUN cana-6055 310 13 is be AUX cana-6055 310 14 discussed discuss VERB cana-6055 310 15 in in ADP cana-6055 310 16 this this DET cana-6055 310 17 paper paper NOUN cana-6055 310 18 . . PUNCT cana-6055 311 1 in in ADP cana-6055 311 2 particular particular ADJ cana-6055 311 3 the the DET cana-6055 311 4 distance distance NOUN cana-6055 311 5 pair pair NOUN cana-6055 311 6 antimagicness antimagicness ADV cana-6055 311 7 of of ADP cana-6055 311 8 prism prism NOUN cana-6055 311 9 and and CCONJ cana-6055 311 10 crossed cross VERB cana-6055 311 11 prism prism NOUN cana-6055 311 12 is be AUX cana-6055 311 13 investigated investigate VERB cana-6055 311 14 for for ADP cana-6055 311 15 particular particular ADJ cana-6055 311 16 n. n. NOUN cana-6055 311 17 also also ADV cana-6055 311 18 it it PRON cana-6055 311 19 is be AUX cana-6055 311 20 proved prove VERB cana-6055 311 21 that that SCONJ cana-6055 311 22 the the DET cana-6055 311 23 circulant circulant ADJ cana-6055 311 24 graph graph NOUN cana-6055 311 25 𝐢(2𝑛 𝐢(2𝑛 NOUN cana-6055 311 26 ; ; PUNCT cana-6055 311 27 { { PUNCT cana-6055 311 28 𝑑1 𝑑1 NOUN cana-6055 311 29 , , PUNCT cana-6055 311 30 𝑑2 𝑑2 NOUN cana-6055 311 31 , , PUNCT cana-6055 311 32 𝑛 𝑛 ADJ cana-6055 311 33 } } PUNCT cana-6055 311 34 ) ) PUNCT cana-6055 311 35 is be AUX cana-6055 311 36 a a DET cana-6055 311 37 distance distance NOUN cana-6055 311 38 pair pair NOUN cana-6055 311 39 antimagic antimagic NOUN cana-6055 311 40 if if SCONJ cana-6055 311 41 𝑛 𝑛 PRON cana-6055 311 42 = = SYM cana-6055 311 43 𝑑1 𝑑1 NOUN cana-6055 311 44 + + CCONJ cana-6055 311 45 𝑑2 𝑑2 NOUN cana-6055 311 46 , , PUNCT cana-6055 311 47 d1 d1 PROPN cana-6055 311 48 < < X cana-6055 311 49 d2 d2 PROPN cana-6055 311 50 and and CCONJ cana-6055 311 51 𝑛 𝑛 ADP cana-6055 311 52 > > SYM cana-6055 311 53 3 3 NUM cana-6055 311 54 while while SCONJ cana-6055 311 55 its its PRON cana-6055 311 56 complement complement NOUN cana-6055 311 57 is be AUX cana-6055 311 58 closed close VERB cana-6055 311 59 distance distance NOUN cana-6055 311 60 pair pair NOUN cana-6055 311 61 antimagic antimagic NOUN cana-6055 311 62 with with ADP cana-6055 311 63 the the DET cana-6055 311 64 same same ADJ cana-6055 311 65 n. n. NOUN cana-6055 311 66 references reference NOUN cana-6055 311 67 [ [ X cana-6055 311 68 1 1 NUM cana-6055 311 69 ] ] X cana-6055 311 70 b.d b.d PROPN cana-6055 311 71 . . PROPN cana-6055 311 72 acharya acharya PROPN cana-6055 311 73 , , PUNCT cana-6055 311 74 s.b s.b PROPN cana-6055 311 75 . . PROPN cana-6055 311 76 rao rao PROPN cana-6055 311 77 , , PUNCT cana-6055 311 78 t. t. PROPN cana-6055 311 79 singh singh PROPN cana-6055 311 80 and and CCONJ cana-6055 311 81 v. v. ADP cana-6055 311 82 parameswaran parameswaran NOUN cana-6055 311 83 , , PUNCT cana-6055 311 84 neighborhood neighborhood NOUN cana-6055 311 85 magic magic NOUN cana-6055 311 86 graphs graph NOUN cana-6055 311 87 , , PUNCT cana-6055 311 88 in in ADP cana-6055 311 89 national national ADJ cana-6055 311 90 conference conference NOUN cana-6055 311 91 on on ADP cana-6055 311 92 graph graph NOUN cana-6055 311 93 theory theory NOUN cana-6055 311 94 , , PUNCT cana-6055 311 95 combinatorics combinatoric NOUN cana-6055 311 96 and and CCONJ cana-6055 311 97 algorithm algorithm PROPN cana-6055 311 98 , , PUNCT cana-6055 311 99 ( ( PUNCT cana-6055 311 100 2004 2004 NUM cana-6055 311 101 ) ) PUNCT cana-6055 311 102 . . PUNCT cana-6055 312 1 [ [ X cana-6055 312 2 2 2 NUM cana-6055 312 3 ] ] PUNCT cana-6055 312 4 f. f. PROPN cana-6055 312 5 harary harary PROPN cana-6055 312 6 , , PUNCT cana-6055 312 7 graph graph NOUN cana-6055 312 8 therory therory NOUN cana-6055 312 9 , , PUNCT cana-6055 312 10 narosa narosa PROPN cana-6055 312 11 publishing publishing PROPN cana-6055 312 12 house house PROPN cana-6055 312 13 , , PUNCT cana-6055 312 14 new new PROPN cana-6055 312 15 delhi delhi PROPN cana-6055 312 16 , , PUNCT cana-6055 312 17 1998 1998 NUM cana-6055 312 18 . . PUNCT cana-6055 313 1 [ [ X cana-6055 313 2 3 3 X cana-6055 313 3 ] ] X cana-6055 313 4 n. n. PROPN cana-6055 313 5 kamatchi kamatchi PROPN cana-6055 313 6 , , PUNCT cana-6055 313 7 s. s. PROPN cana-6055 313 8 arumugam arumugam PROPN cana-6055 313 9 , , PUNCT cana-6055 313 10 distance distance NOUN cana-6055 313 11 antimagic antimagic ADJ cana-6055 313 12 graphs graph NOUN cana-6055 313 13 , , PUNCT cana-6055 313 14 jcmcc jcmcc ADJ cana-6055 313 15 , , PUNCT cana-6055 313 16 84 84 NUM cana-6055 313 17 ( ( PUNCT cana-6055 313 18 2013 2013 NUM cana-6055 313 19 ) ) PUNCT cana-6055 313 20 , , PUNCT cana-6055 313 21 61 61 NUM cana-6055 313 22 - - SYM cana-6055 313 23 67 67 NUM cana-6055 313 24 . . PUNCT cana-6055 314 1 [ [ X cana-6055 314 2 4 4 X cana-6055 314 3 ] ] PUNCT cana-6055 314 4 m. m. NOUN cana-6055 314 5 bala bala PROPN cana-6055 314 6 , , PUNCT cana-6055 314 7 t. t. PROPN cana-6055 314 8 saratha saratha PROPN cana-6055 314 9 devi devi PROPN cana-6055 314 10 , , PUNCT cana-6055 314 11 on on ADP cana-6055 314 12 distance distance NOUN cana-6055 314 13 pair pair NOUN cana-6055 314 14 antimagic antimagic ADJ cana-6055 314 15 labeling labeling NOUN cana-6055 314 16 of of ADP cana-6055 314 17 graphs graph NOUN cana-6055 314 18 , , PUNCT cana-6055 314 19 indian indian ADJ cana-6055 314 20 journal journal NOUN cana-6055 314 21 of of ADP cana-6055 314 22 natural natural ADJ cana-6055 314 23 sciences science NOUN cana-6055 314 24 , , PUNCT cana-6055 314 25 vol.15 vol.15 NOUN cana-6055 314 26 , , PUNCT cana-6055 314 27 issue issue NOUN cana-6055 314 28 86 86 NUM cana-6055 314 29 , , PUNCT cana-6055 314 30 oct oct NOUN cana-6055 314 31 2024 2024 NUM cana-6055 314 32 . . PUNCT cana-6055 315 1 [ [ X cana-6055 315 2 5 5 NUM cana-6055 315 3 ] ] PUNCT cana-6055 315 4 m. m. NOUN cana-6055 315 5 miller miller PROPN cana-6055 315 6 , , PUNCT cana-6055 315 7 c. c. PROPN cana-6055 315 8 rodger rodger PROPN cana-6055 315 9 and and CCONJ cana-6055 315 10 r. r. PROPN cana-6055 315 11 simanjuntak simanjuntak PROPN cana-6055 315 12 , , PUNCT cana-6055 315 13 distance distance NOUN cana-6055 315 14 magic magic ADJ cana-6055 315 15 labelings labeling NOUN cana-6055 315 16 of of ADP cana-6055 315 17 graphs graph NOUN cana-6055 315 18 , , PUNCT cana-6055 315 19 australas australa NOUN cana-6055 315 20 . . PUNCT cana-6055 316 1 j. j. PROPN cana-6055 316 2 combin combin PROPN cana-6055 316 3 . . PROPN cana-6055 316 4 , , PUNCT cana-6055 316 5 28 28 NUM cana-6055 316 6 ( ( PUNCT cana-6055 316 7 2003 2003 NUM cana-6055 316 8 ) ) PUNCT cana-6055 316 9 , , PUNCT cana-6055 316 10 305 305 NUM cana-6055 316 11 - - SYM cana-6055 316 12 315 315 NUM cana-6055 316 13 [ [ X cana-6055 316 14 6 6 NUM cana-6055 316 15 ] ] PUNCT cana-6055 316 16 r. r. PROPN cana-6055 316 17 ponraj ponraj PROPN cana-6055 316 18 , , PUNCT cana-6055 316 19 j.v.x j.v.x NOUN cana-6055 316 20 . . PUNCT cana-6055 317 1 parthipan parthipan PROPN cana-6055 317 2 , , PUNCT cana-6055 317 3 pair pair NOUN cana-6055 317 4 sum sum NOUN cana-6055 317 5 labeling labeling NOUN cana-6055 317 6 of of ADP cana-6055 317 7 graphs graph NOUN cana-6055 317 8 , , PUNCT cana-6055 317 9 the the DET cana-6055 317 10 journal journal NOUN cana-6055 317 11 of of ADP cana-6055 317 12 indian indian PROPN cana-6055 317 13 acad acad PROPN cana-6055 317 14 . . PUNCT cana-6055 318 1 math math NOUN cana-6055 318 2 , , PUNCT cana-6055 318 3 vol vol NOUN cana-6055 318 4 32 32 NUM cana-6055 318 5 , , PUNCT cana-6055 318 6 no no INTJ cana-6055 318 7 . . PUNCT cana-6055 319 1 2(2010 2(2010 NUM cana-6055 319 2 ) ) PUNCT cana-6055 319 3 , , PUNCT cana-6055 319 4 587 587 NUM cana-6055 319 5 - - SYM cana-6055 319 6 595 595 NUM cana-6055 319 7 . . PUNCT cana-6055 320 1 [ [ X cana-6055 320 2 7 7 NUM cana-6055 320 3 ] ] X cana-6055 320 4 k.a k.a PROPN cana-6055 320 5 . . PROPN cana-6055 320 6 sugeng sugeng PROPN cana-6055 320 7 , , PUNCT cana-6055 320 8 d. d. PROPN cana-6055 320 9 froncek froncek PROPN cana-6055 320 10 , , PUNCT cana-6055 320 11 m. m. PROPN cana-6055 320 12 miller miller PROPN cana-6055 320 13 , , PUNCT cana-6055 320 14 j. j. PROPN cana-6055 320 15 ryan ryan PROPN cana-6055 320 16 and and CCONJ cana-6055 320 17 j. j. PROPN cana-6055 320 18 walker walker PROPN cana-6055 320 19 , , PUNCT cana-6055 320 20 on on ADP cana-6055 320 21 distance distance NOUN cana-6055 320 22 magic magic ADJ cana-6055 320 23 labeling labeling NOUN cana-6055 320 24 of of ADP cana-6055 320 25 graphs graph NOUN cana-6055 320 26 , , PUNCT cana-6055 320 27 j. j. PROPN cana-6055 320 28 combin combin PROPN cana-6055 320 29 . . PUNCT cana-6055 320 30 math math NOUN cana-6055 320 31 . . PUNCT cana-6055 321 1 combin combin NOUN cana-6055 321 2 . . PUNCT cana-6055 322 1 comput comput NOUN cana-6055 322 2 . . PUNCT cana-6055 322 3 , , PUNCT cana-6055 322 4 71 71 NUM cana-6055 322 5 ( ( PUNCT cana-6055 322 6 2009 2009 NUM cana-6055 322 7 ) ) PUNCT cana-6055 322 8 , , PUNCT cana-6055 322 9 39 39 NUM cana-6055 322 10 - - SYM cana-6055 322 11 48 48 NUM cana-6055 322 12 . . PUNCT cana-6055 323 1 [ [ X cana-6055 323 2 8 8 NUM cana-6055 323 3 ] ] X cana-6055 323 4 resty resty NOUN cana-6055 323 5 d d PROPN cana-6055 323 6 and and CCONJ cana-6055 323 7 salman salman PROPN cana-6055 323 8 a a DET cana-6055 323 9 n n PROPN cana-6055 323 10 m m PROPN cana-6055 323 11 2015 2015 NUM cana-6055 323 12 the the DET cana-6055 323 13 rainbow rainbow NOUN cana-6055 323 14 connection connection NOUN cana-6055 323 15 number number NOUN cana-6055 323 16 of of ADP cana-6055 323 17 an an DET cana-6055 323 18 n n ADV cana-6055 323 19 - - PUNCT cana-6055 323 20 crossed cross VERB cana-6055 323 21 prism prism NOUN cana-6055 323 22 graph graph NOUN cana-6055 323 23 and and CCONJ cana-6055 323 24 its its PRON cana-6055 323 25 corona corona NOUN cana-6055 323 26 product product NOUN cana-6055 323 27 with with ADP cana-6055 323 28 a a DET cana-6055 323 29 trivial trivial ADJ cana-6055 323 30 graph graph NOUN cana-6055 323 31 , , PUNCT cana-6055 323 32 procedia procedia NOUN cana-6055 323 33 computer computer NOUN cana-6055 323 34 science science NOUN cana-6055 323 35 , , PUNCT cana-6055 323 36 2015 2015 NUM cana-6055 323 37 74 74 NUM cana-6055 323 38 143150 143150 NUM cana-6055 323 39 . . PUNCT cana-6055 324 1 [ [ X cana-6055 324 2 9 9 X cana-6055 324 3 ] ] PUNCT cana-6055 324 4 v. v. CCONJ cana-6055 324 5 vilfred vilfre VERB cana-6055 324 6 , , PUNCT cana-6055 324 7 𝛴-labelled 𝛴-labelled ADJ cana-6055 324 8 graph graph NOUN cana-6055 324 9 and and CCONJ cana-6055 324 10 circulant circulant ADJ cana-6055 324 11 graphs graph NOUN cana-6055 324 12 , , PUNCT cana-6055 324 13 ph.d ph.d PROPN cana-6055 324 14 . . PUNCT cana-6055 325 1 thesis thesis NOUN cana-6055 325 2 , , PUNCT cana-6055 325 3 university university NOUN cana-6055 325 4 of of ADP cana-6055 325 5 kerala kerala PROPN cana-6055 325 6 , , PUNCT cana-6055 325 7 trivandrum trivandrum PROPN cana-6055 325 8 , , PUNCT cana-6055 325 9 india india PROPN cana-6055 325 10 , , PUNCT cana-6055 325 11 1994 1994 NUM cana-6055 325 12 . . PUNCT