id sid tid token lemma pos cana-3483 1 1 communications communication NOUN cana-3483 1 2 on on ADP cana-3483 1 3 applied apply VERB cana-3483 1 4 nonlinear nonlinear ADJ cana-3483 1 5 analysis analysis NOUN cana-3483 1 6 issn issn NOUN cana-3483 1 7 : : PUNCT cana-3483 1 8 1074 1074 NUM cana-3483 1 9 - - PUNCT cana-3483 1 10 133x 133x NUM cana-3483 1 11 vol vol NOUN cana-3483 1 12 32 32 NUM cana-3483 1 13 no no NOUN cana-3483 1 14 . . PUNCT cana-3483 2 1 7s 7 NOUN cana-3483 2 2 ( ( PUNCT cana-3483 2 3 2025 2025 NUM cana-3483 2 4 ) ) PUNCT cana-3483 2 5 759 759 NUM cana-3483 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3483 2 7 some some PRON cana-3483 2 8 more more ADJ cana-3483 2 9 extensions extension NOUN cana-3483 2 10 on on ADP cana-3483 2 11 product product NOUN cana-3483 2 12 cordial cordial ADJ cana-3483 2 13 graphs graph NOUN cana-3483 2 14 varsha varsha PROPN cana-3483 2 15 rathia rathia PROPN cana-3483 2 16 , , PUNCT cana-3483 2 17 sweta sweta PROPN cana-3483 2 18 srivastavb srivastavb PROPN cana-3483 2 19 , , PUNCT cana-3483 2 20 sangeeta sangeeta PROPN cana-3483 2 21 guptac guptac PROPN cana-3483 2 22 a a PRON cana-3483 2 23 , , PUNCT cana-3483 2 24 b b NOUN cana-3483 2 25 , , PUNCT cana-3483 2 26 c c PROPN cana-3483 2 27 department department PROPN cana-3483 2 28 of of ADP cana-3483 2 29 mathematics mathematics PROPN cana-3483 2 30 , , PUNCT cana-3483 2 31 sharda sharda PROPN cana-3483 2 32 university university PROPN cana-3483 2 33 , , PUNCT cana-3483 2 34 greater great ADJ cana-3483 2 35 noida noida PROPN cana-3483 2 36 , , PUNCT cana-3483 2 37 uttar uttar PROPN cana-3483 2 38 pradesh pradesh PROPN cana-3483 2 39 201310 201310 NUM cana-3483 2 40 , , PUNCT cana-3483 2 41 india india PROPN cana-3483 2 42 ae ae PROPN cana-3483 2 43 - - NOUN cana-3483 2 44 mail mail NOUN cana-3483 2 45 : : PUNCT cana-3483 2 46 2022401996.varsha@dr.sharda.ac.in 2022401996.varsha@dr.sharda.ac.in NUM cana-3483 2 47 be be NOUN cana-3483 2 48 - - PUNCT cana-3483 2 49 mail mail NOUN cana-3483 2 50 : : PUNCT cana-3483 2 51 sweta.srivastav@sharda.ac.in sweta.srivastav@sharda.ac.in NUM cana-3483 2 52 ( ( PUNCT cana-3483 2 53 corresponding correspond VERB cana-3483 2 54 author author NOUN cana-3483 2 55 ) ) PUNCT cana-3483 2 56 ce ce PROPN cana-3483 2 57 - - NOUN cana-3483 2 58 mail mail NOUN cana-3483 2 59 : : PUNCT cana-3483 2 60 sangeeta.gupta@sharda.ac.in sangeeta.gupta@sharda.ac.in NUM cana-3483 2 61 article article NOUN cana-3483 2 62 history history NOUN cana-3483 2 63 : : PUNCT cana-3483 2 64 received receive VERB cana-3483 2 65 : : PUNCT cana-3483 2 66 28 28 NUM cana-3483 2 67 - - SYM cana-3483 2 68 10 10 NUM cana-3483 2 69 - - PUNCT cana-3483 2 70 2024 2024 NUM cana-3483 2 71 revised:12 revised:12 NOUN cana-3483 2 72 - - PUNCT cana-3483 2 73 11 11 NUM cana-3483 2 74 - - PUNCT cana-3483 2 75 2024 2024 NUM cana-3483 2 76 accepted:19 accepted:19 VERB cana-3483 2 77 - - PUNCT cana-3483 2 78 12 12 NUM cana-3483 2 79 - - PUNCT cana-3483 2 80 2024 2024 NUM cana-3483 2 81 abstract abstract NOUN cana-3483 2 82 : : PUNCT cana-3483 2 83 a a DET cana-3483 2 84 mapping mapping NOUN cana-3483 2 85 function function VERB cana-3483 2 86 𝑓 𝑓 PRON cana-3483 2 87 : : PUNCT cana-3483 2 88 𝛼(𝐺 𝛼(𝐺 PROPN cana-3483 2 89 ) ) PUNCT cana-3483 2 90 π‘‘π‘œ π‘‘π‘œ PROPN cana-3483 2 91 { { PUNCT cana-3483 2 92 0,1 0,1 NOUN cana-3483 2 93 } } PUNCT cana-3483 2 94 , , PUNCT cana-3483 2 95 causes cause VERB cana-3483 2 96 an an DET cana-3483 2 97 labeling labeling NOUN cana-3483 2 98 of of ADP cana-3483 2 99 edge edge NOUN cana-3483 2 100 𝑓:𝛽(𝐺 𝑓:𝛽(𝐺 ADV cana-3483 2 101 ) ) PUNCT cana-3483 2 102 β†’ β†’ SYM cana-3483 2 103 { { PUNCT cana-3483 2 104 0,1 0,1 NOUN cana-3483 2 105 } } PUNCT cana-3483 2 106 as as SCONJ cana-3483 2 107 specified specify VERB cana-3483 2 108 by by ADP cana-3483 2 109 𝑓 𝑓 DET cana-3483 2 110 βˆ— βˆ— NOUN cana-3483 2 111 ( ( PUNCT cana-3483 2 112 𝑑𝑦 𝑑𝑦 NOUN cana-3483 2 113 ) ) PUNCT cana-3483 2 114 = = PUNCT cana-3483 3 1 𝑓(𝑑 𝑓(𝑑 VERB cana-3483 3 2 ) ) PUNCT cana-3483 3 3 βˆ— βˆ— NOUN cana-3483 3 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-3483 3 5 ) ) PUNCT cana-3483 3 6 . . PUNCT cana-3483 4 1 this this DET cana-3483 4 2 type type NOUN cana-3483 4 3 of of ADP cana-3483 4 4 labeling labeling NOUN cana-3483 4 5 is be AUX cana-3483 4 6 called call VERB cana-3483 4 7 product product NOUN cana-3483 4 8 cordial cordial ADJ cana-3483 4 9 labeling labeling NOUN cana-3483 4 10 if if SCONJ cana-3483 4 11 it it PRON cana-3483 4 12 satisfies satisfy VERB cana-3483 4 13 certain certain ADJ cana-3483 4 14 circumstances circumstance NOUN cana-3483 4 15 regarding regard VERB cana-3483 4 16 the the DET cana-3483 4 17 differences difference NOUN cana-3483 4 18 in in ADP cana-3483 4 19 the the DET cana-3483 4 20 number number NOUN cana-3483 4 21 of of ADP cana-3483 4 22 vertices vertex NOUN cana-3483 4 23 and and CCONJ cana-3483 4 24 edges edge NOUN cana-3483 4 25 labeled label VERB cana-3483 4 26 with with ADP cana-3483 4 27 0 0 NUM cana-3483 4 28 and and CCONJ cana-3483 4 29 1 1 NUM cana-3483 4 30 . . PUNCT cana-3483 5 1 in in ADP cana-3483 5 2 this this DET cana-3483 5 3 investigation investigation NOUN cana-3483 5 4 , , PUNCT cana-3483 5 5 we we PRON cana-3483 5 6 prove prove VERB cana-3483 5 7 that that SCONJ cana-3483 5 8 the the DET cana-3483 5 9 copies copy NOUN cana-3483 5 10 of of ADP cana-3483 5 11 concentric concentric ADJ cana-3483 5 12 n n CCONJ cana-3483 5 13 - - PUNCT cana-3483 5 14 sunlet sunlet NOUN cana-3483 5 15 graphs graph NOUN cana-3483 5 16 , , PUNCT cana-3483 5 17 star star NOUN cana-3483 5 18 of of ADP cana-3483 5 19 concentric concentric ADJ cana-3483 5 20 n n CCONJ cana-3483 5 21 - - PUNCT cana-3483 5 22 sunlet sunlet NOUN cana-3483 5 23 graph graph NOUN cana-3483 5 24 and and CCONJ cana-3483 5 25 copies copy NOUN cana-3483 5 26 of of ADP cana-3483 5 27 shell shell ADJ cana-3483 5 28 graph graph NOUN cana-3483 5 29 satisfies satisfy VERB cana-3483 5 30 these these DET cana-3483 5 31 conditions condition NOUN cana-3483 5 32 , , PUNCT cana-3483 5 33 eventually eventually ADV cana-3483 5 34 they they PRON cana-3483 5 35 are be AUX cana-3483 5 36 product product NOUN cana-3483 5 37 cordial cordial ADJ cana-3483 5 38 graphs graph NOUN cana-3483 5 39 . . PUNCT cana-3483 6 1 keywords keyword NOUN cana-3483 6 2 : : PUNCT cana-3483 6 3 graph graph NOUN cana-3483 6 4 labeling labeling NOUN cana-3483 6 5 , , PUNCT cana-3483 6 6 product product NOUN cana-3483 6 7 cordial cordial ADJ cana-3483 6 8 labeling labeling NOUN cana-3483 6 9 , , PUNCT cana-3483 6 10 concentric concentric ADJ cana-3483 6 11 n n CCONJ cana-3483 6 12 - - PUNCT cana-3483 6 13 sunlet sunlet NOUN cana-3483 6 14 graph graph NOUN cana-3483 6 15 , , PUNCT cana-3483 6 16 star star NOUN cana-3483 6 17 of of ADP cana-3483 6 18 concentric concentric ADJ cana-3483 6 19 n n CCONJ cana-3483 6 20 - - PUNCT cana-3483 6 21 sunlet sunlet NOUN cana-3483 6 22 graph graph NOUN cana-3483 6 23 , , PUNCT cana-3483 6 24 shell shell ADJ cana-3483 6 25 graph graph NOUN cana-3483 6 26 . . PUNCT cana-3483 7 1 1 1 X cana-3483 7 2 . . X cana-3483 7 3 introduction introduction NOUN cana-3483 7 4 graph graph NOUN cana-3483 7 5 labeling labeling NOUN cana-3483 7 6 is be AUX cana-3483 7 7 a a DET cana-3483 7 8 fundamental fundamental ADJ cana-3483 7 9 concept concept NOUN cana-3483 7 10 in in ADP cana-3483 7 11 graph graph NOUN cana-3483 7 12 theory theory NOUN cana-3483 7 13 , , PUNCT cana-3483 7 14 involving involve VERB cana-3483 7 15 the the DET cana-3483 7 16 assignment assignment NOUN cana-3483 7 17 of of ADP cana-3483 7 18 integers integer NOUN cana-3483 7 19 , , PUNCT cana-3483 7 20 typically typically ADV cana-3483 7 21 0 0 NUM cana-3483 7 22 and and CCONJ cana-3483 7 23 1 1 NUM cana-3483 7 24 , , PUNCT cana-3483 7 25 to to ADP cana-3483 7 26 the the DET cana-3483 7 27 vertices vertex NOUN cana-3483 7 28 or or CCONJ cana-3483 7 29 edges edge NOUN cana-3483 7 30 of of ADP cana-3483 7 31 a a DET cana-3483 7 32 graph graph NOUN cana-3483 7 33 . . PUNCT cana-3483 8 1 this this DET cana-3483 8 2 area area NOUN cana-3483 8 3 of of ADP cana-3483 8 4 study study NOUN cana-3483 8 5 has have VERB cana-3483 8 6 diverse diverse ADJ cana-3483 8 7 applications application NOUN cana-3483 8 8 across across ADP cana-3483 8 9 various various ADJ cana-3483 8 10 domains domain NOUN cana-3483 8 11 , , PUNCT cana-3483 8 12 including include VERB cana-3483 8 13 network network NOUN cana-3483 8 14 modeling modeling NOUN cana-3483 8 15 , , PUNCT cana-3483 8 16 coding code VERB cana-3483 8 17 theory theory NOUN cana-3483 8 18 , , PUNCT cana-3483 8 19 circuit circuit NOUN cana-3483 8 20 layout layout PROPN cana-3483 8 21 optimization optimization NOUN cana-3483 8 22 , , PUNCT cana-3483 8 23 database database NOUN cana-3483 8 24 management management NOUN cana-3483 8 25 , , PUNCT cana-3483 8 26 protocol protocol NOUN cana-3483 8 27 design design NOUN cana-3483 8 28 , , PUNCT cana-3483 8 29 and and CCONJ cana-3483 8 30 resource resource NOUN cana-3483 8 31 allocation allocation NOUN cana-3483 8 32 . . PUNCT cana-3483 9 1 cordial cordial ADJ cana-3483 9 2 labeling labeling NOUN cana-3483 9 3 , , PUNCT cana-3483 9 4 introduced introduce VERB cana-3483 9 5 by by ADP cana-3483 9 6 i. i. PROPN cana-3483 9 7 cahit cahit PROPN cana-3483 10 1 [ [ X cana-3483 10 2 2 2 NUM cana-3483 10 3 ] ] PUNCT cana-3483 10 4 in in ADP cana-3483 10 5 1958 1958 NUM cana-3483 10 6 , , PUNCT cana-3483 10 7 is be AUX cana-3483 10 8 a a DET cana-3483 10 9 specific specific ADJ cana-3483 10 10 type type NOUN cana-3483 10 11 of of ADP cana-3483 10 12 graph graph NOUN cana-3483 10 13 labeling labeling NOUN cana-3483 10 14 where where SCONJ cana-3483 10 15 the the DET cana-3483 10 16 labels label NOUN cana-3483 10 17 assigned assign VERB cana-3483 10 18 to to ADP cana-3483 10 19 the the DET cana-3483 10 20 graph graph NOUN cana-3483 10 21 of of ADP cana-3483 10 22 vertices vertex NOUN cana-3483 10 23 or or CCONJ cana-3483 10 24 edges edge NOUN cana-3483 10 25 aim aim VERB cana-3483 10 26 to to PART cana-3483 10 27 balance balance VERB cana-3483 10 28 the the DET cana-3483 10 29 frequencies frequency NOUN cana-3483 10 30 of of ADP cana-3483 10 31 adjacent adjacent ADJ cana-3483 10 32 labels label NOUN cana-3483 10 33 . . PUNCT cana-3483 11 1 specifically specifically ADV cana-3483 11 2 , , PUNCT cana-3483 11 3 the the DET cana-3483 11 4 differences difference NOUN cana-3483 11 5 in in ADP cana-3483 11 6 the the DET cana-3483 11 7 frequencies frequency NOUN cana-3483 11 8 of of ADP cana-3483 11 9 labels label NOUN cana-3483 11 10 among among ADP cana-3483 11 11 adjacent adjacent ADJ cana-3483 11 12 elements element NOUN cana-3483 11 13 ( ( PUNCT cana-3483 11 14 vertices vertex NOUN cana-3483 11 15 or or CCONJ cana-3483 11 16 edges edge NOUN cana-3483 11 17 ) ) PUNCT cana-3483 11 18 should should AUX cana-3483 11 19 be be AUX cana-3483 11 20 either either CCONJ cana-3483 11 21 0 0 NUM cana-3483 11 22 or or CCONJ cana-3483 11 23 1 1 NUM cana-3483 11 24 . . PUNCT cana-3483 12 1 this this DET cana-3483 12 2 concept concept NOUN cana-3483 12 3 has have AUX cana-3483 12 4 significantly significantly ADV cana-3483 12 5 contributed contribute VERB cana-3483 12 6 to to ADP cana-3483 12 7 the the DET cana-3483 12 8 theoretical theoretical ADJ cana-3483 12 9 understanding understanding NOUN cana-3483 12 10 of of ADP cana-3483 12 11 graph graph NOUN cana-3483 12 12 theory theory NOUN cana-3483 12 13 and and CCONJ cana-3483 12 14 offers offer VERB cana-3483 12 15 a a DET cana-3483 12 16 framework framework NOUN cana-3483 12 17 for for ADP cana-3483 12 18 analyzing analyze VERB cana-3483 12 19 the the DET cana-3483 12 20 structural structural ADJ cana-3483 12 21 properties property NOUN cana-3483 12 22 of of ADP cana-3483 12 23 graphs graph NOUN cana-3483 12 24 . . PUNCT cana-3483 13 1 the the DET cana-3483 13 2 concept concept NOUN cana-3483 13 3 of of ADP cana-3483 13 4 product product NOUN cana-3483 13 5 cordial cordial ADJ cana-3483 13 6 labeling labeling NOUN cana-3483 13 7 was be AUX cana-3483 13 8 introduced introduce VERB cana-3483 13 9 by by ADP cana-3483 13 10 sundaram sundaram PROPN cana-3483 13 11 et et PROPN cana-3483 13 12 al al PROPN cana-3483 13 13 . . PUNCT cana-3483 14 1 [ [ X cana-3483 14 2 11 11 NUM cana-3483 14 3 ] ] PUNCT cana-3483 14 4 as as ADP cana-3483 14 5 an an DET cana-3483 14 6 extension extension NOUN cana-3483 14 7 of of ADP cana-3483 14 8 the the DET cana-3483 14 9 cordial cordial ADJ cana-3483 14 10 labeling labeling NOUN cana-3483 14 11 theory theory NOUN cana-3483 14 12 . . PUNCT cana-3483 15 1 gallian gallian ADJ cana-3483 16 1 [ [ X cana-3483 16 2 4 4 X cana-3483 16 3 ] ] PUNCT cana-3483 16 4 introduced introduce VERB cana-3483 16 5 a a DET cana-3483 16 6 dynamic dynamic ADJ cana-3483 16 7 survey survey NOUN cana-3483 16 8 of of ADP cana-3483 16 9 graph graph NOUN cana-3483 16 10 labeling labeling NOUN cana-3483 16 11 . . PUNCT cana-3483 17 1 vaidya vaidya PROPN cana-3483 17 2 and and CCONJ cana-3483 17 3 dani dani PROPN cana-3483 18 1 [ [ X cana-3483 18 2 14 14 NUM cana-3483 18 3 ] ] PUNCT cana-3483 18 4 demonstrated demonstrate VERB cana-3483 18 5 that that SCONJ cana-3483 18 6 joining join VERB cana-3483 18 7 apex apex NOUN cana-3483 18 8 vertices vertex NOUN cana-3483 18 9 of of ADP cana-3483 18 10 k k PROPN cana-3483 18 11 copies copy NOUN cana-3483 18 12 of of ADP cana-3483 18 13 stars star NOUN cana-3483 18 14 , , PUNCT cana-3483 18 15 shells shell NOUN cana-3483 18 16 , , PUNCT cana-3483 18 17 and and CCONJ cana-3483 18 18 wheels wheel NOUN cana-3483 18 19 to to ADP cana-3483 18 20 a a DET cana-3483 18 21 new new ADJ cana-3483 18 22 vertex vertex NOUN cana-3483 18 23 results result NOUN cana-3483 18 24 in in ADP cana-3483 18 25 product product NOUN cana-3483 18 26 cordial cordial ADJ cana-3483 18 27 graphs graph NOUN cana-3483 18 28 . . PUNCT cana-3483 19 1 some some DET cana-3483 19 2 graphs graph NOUN cana-3483 19 3 in in ADP cana-3483 19 4 the the DET cana-3483 19 5 context context NOUN cana-3483 19 6 of of ADP cana-3483 19 7 product product NOUN cana-3483 19 8 cordial cordial ADJ cana-3483 19 9 labeling labeling NOUN cana-3483 19 10 proved prove VERB cana-3483 19 11 by by ADP cana-3483 19 12 prajapati prajapati PROPN cana-3483 19 13 et et PROPN cana-3483 19 14 al al PROPN cana-3483 19 15 . . PUNCT cana-3483 20 1 [ [ X cana-3483 20 2 7 7 NUM cana-3483 20 3 ] ] PUNCT cana-3483 20 4 . . PUNCT cana-3483 20 5 srivastav srivastav PROPN cana-3483 20 6 and and CCONJ cana-3483 20 7 gupta gupta NOUN cana-3483 20 8 [ [ X cana-3483 20 9 10 10 NUM cana-3483 20 10 ] ] PUNCT cana-3483 20 11 divisor divisor NOUN cana-3483 20 12 3 3 NUM cana-3483 20 13 - - PUNCT cana-3483 20 14 equitable equitable ADJ cana-3483 20 15 labeling labeling NOUN cana-3483 20 16 of of ADP cana-3483 20 17 graphs graph NOUN cana-3483 20 18 . . PUNCT cana-3483 21 1 vaidya vaidya PROPN cana-3483 21 2 and and CCONJ cana-3483 21 3 kanani kanani PROPN cana-3483 22 1 [ [ X cana-3483 22 2 15 15 NUM cana-3483 22 3 ] ] X cana-3483 22 4 focused focus VERB cana-3483 22 5 on on ADP cana-3483 22 6 cycle cycle NOUN cana-3483 22 7 - - PUNCT cana-3483 22 8 related relate VERB cana-3483 22 9 graphs graph NOUN cana-3483 22 10 , , PUNCT cana-3483 22 11 while while SCONJ cana-3483 22 12 vaidya vaidya PROPN cana-3483 22 13 and and CCONJ cana-3483 22 14 barasara barasara ADJ cana-3483 22 15 [ [ X cana-3483 22 16 13 13 NUM cana-3483 22 17 ] ] PUNCT cana-3483 22 18 proved prove VERB cana-3483 22 19 that that SCONJ cana-3483 22 20 cycles cycle NOUN cana-3483 22 21 with with ADP cana-3483 22 22 chords chord NOUN cana-3483 22 23 , , PUNCT cana-3483 22 24 friendship friendship NOUN cana-3483 22 25 graphs graph NOUN cana-3483 22 26 , , PUNCT cana-3483 22 27 and and CCONJ cana-3483 22 28 middle middle ADJ cana-3483 22 29 graphs graph NOUN cana-3483 22 30 of of ADP cana-3483 22 31 paths path NOUN cana-3483 22 32 admit admit VERB cana-3483 22 33 product product NOUN cana-3483 22 34 cordial cordial ADJ cana-3483 22 35 labeling labeling NOUN cana-3483 22 36 , , PUNCT cana-3483 22 37 and and CCONJ cana-3483 22 38 also also ADV cana-3483 22 39 investigated investigate VERB cana-3483 22 40 graph graph NOUN cana-3483 22 41 operations operation NOUN cana-3483 22 42 . . PUNCT cana-3483 23 1 mathew mathew PROPN cana-3483 24 1 [ [ X cana-3483 24 2 5 5 NUM cana-3483 24 3 ] ] PUNCT cana-3483 24 4 demonstrated demonstrate VERB cana-3483 24 5 that that SCONJ cana-3483 24 6 vector vector NOUN cana-3483 24 7 switching switching NOUN cana-3483 24 8 of of ADP cana-3483 24 9 signed sign VERB cana-3483 24 10 graphs graph NOUN cana-3483 24 11 , , PUNCT cana-3483 24 12 vaidya vaidya PROPN cana-3483 24 13 and and CCONJ cana-3483 24 14 vyas vyas PROPN cana-3483 24 15 [ [ X cana-3483 24 16 12 12 NUM cana-3483 24 17 ] ] PUNCT cana-3483 24 18 explored explore VERB cana-3483 24 19 tensor tensor NOUN cana-3483 24 20 products product NOUN cana-3483 24 21 of of ADP cana-3483 24 22 graphs graph NOUN cana-3483 24 23 . . PUNCT cana-3483 25 1 acharya acharya PROPN cana-3483 25 2 and and CCONJ cana-3483 25 3 kureethara kureethara PROPN cana-3483 26 1 [ [ X cana-3483 26 2 1 1 NUM cana-3483 26 3 ] ] PUNCT cana-3483 26 4 focused focus VERB cana-3483 26 5 on on ADP cana-3483 26 6 dragon dragon NOUN cana-3483 26 7 graph graph NOUN cana-3483 26 8 . . PUNCT cana-3483 27 1 chartrand chartrand NOUN cana-3483 27 2 et et PROPN cana-3483 27 3 al al PROPN cana-3483 27 4 . . PUNCT cana-3483 28 1 [ [ X cana-3483 28 2 3 3 X cana-3483 28 3 ] ] PUNCT cana-3483 28 4 introduced introduce VERB cana-3483 28 5 on on ADP cana-3483 28 6 subset subset ADJ cana-3483 28 7 labelings labeling NOUN cana-3483 28 8 of of ADP cana-3483 28 9 trees tree NOUN cana-3483 28 10 . . PUNCT cana-3483 29 1 raja raja PROPN cana-3483 29 2 et et PROPN cana-3483 29 3 al al PROPN cana-3483 29 4 . . PUNCT cana-3483 30 1 [ [ X cana-3483 30 2 8 8 NUM cana-3483 30 3 ] ] PUNCT cana-3483 30 4 proved prove VERB cana-3483 30 5 that that SCONJ cana-3483 30 6 eccentric eccentric ADJ cana-3483 30 7 completion completion NOUN cana-3483 30 8 of of ADP cana-3483 30 9 a a DET cana-3483 30 10 graph graph NOUN cana-3483 30 11 . . PUNCT cana-3483 31 1 cordial cordial ADJ cana-3483 31 2 labeling labeling NOUN cana-3483 31 3 for for ADP cana-3483 31 4 different different ADJ cana-3483 31 5 types type NOUN cana-3483 31 6 of of ADP cana-3483 31 7 shell shell ADJ cana-3483 31 8 graph graph NOUN cana-3483 31 9 demonstrated demonstrate VERB cana-3483 31 10 by by ADP cana-3483 31 11 meena meena PROPN cana-3483 31 12 et et PROPN cana-3483 31 13 al al PROPN cana-3483 31 14 . . PUNCT cana-3483 32 1 [ [ X cana-3483 32 2 6 6 NUM cana-3483 32 3 ] ] PUNCT cana-3483 32 4 . . PUNCT cana-3483 33 1 sadawarte sadawarte PROPN cana-3483 33 2 and and CCONJ cana-3483 33 3 srivastav srivastav VERB cana-3483 34 1 [ [ X cana-3483 34 2 9 9 NUM cana-3483 34 3 ] ] PUNCT cana-3483 34 4 on on ADP cana-3483 34 5 sum sum NOUN cana-3483 34 6 3‐equitable 3‐equitable NUM cana-3483 34 7 labeling labeling NOUN cana-3483 34 8 of of ADP cana-3483 34 9 some some DET cana-3483 34 10 graphs graph NOUN cana-3483 34 11 . . PUNCT cana-3483 35 1 these these DET cana-3483 35 2 contributions contribution NOUN cana-3483 35 3 enhance enhance VERB cana-3483 35 4 the the DET cana-3483 35 5 understanding understanding NOUN cana-3483 35 6 and and CCONJ cana-3483 35 7 applicability applicability NOUN cana-3483 35 8 of of ADP cana-3483 35 9 product product NOUN cana-3483 35 10 cordial cordial ADJ cana-3483 35 11 labeling labeling NOUN cana-3483 35 12 in in ADP cana-3483 35 13 various various ADJ cana-3483 35 14 domains domain NOUN cana-3483 35 15 of of ADP cana-3483 35 16 graph graph NOUN cana-3483 35 17 theory theory NOUN cana-3483 35 18 . . PUNCT cana-3483 36 1 mailto:sweta.srivastav@sharda.ac.in mailto:sweta.srivastav@sharda.ac.in PROPN cana-3483 36 2 communications communication NOUN cana-3483 36 3 on on ADP cana-3483 36 4 applied apply VERB cana-3483 36 5 nonlinear nonlinear ADJ cana-3483 36 6 analysis analysis NOUN cana-3483 36 7 issn issn NOUN cana-3483 36 8 : : PUNCT cana-3483 36 9 1074 1074 NUM cana-3483 36 10 - - PUNCT cana-3483 36 11 133x 133x NUM cana-3483 36 12 vol vol NOUN cana-3483 36 13 32 32 NUM cana-3483 36 14 no no NOUN cana-3483 36 15 . . PUNCT cana-3483 37 1 7s 7 NOUN cana-3483 37 2 ( ( PUNCT cana-3483 37 3 2025 2025 NUM cana-3483 37 4 ) ) PUNCT cana-3483 37 5 760 760 NUM cana-3483 37 6 https://internationalpubls.com https://internationalpubls.com X cana-3483 37 7 in in ADP cana-3483 37 8 this this DET cana-3483 37 9 concern concern NOUN cana-3483 37 10 research research NOUN cana-3483 37 11 , , PUNCT cana-3483 37 12 we we PRON cana-3483 37 13 focus focus VERB cana-3483 37 14 on on ADP cana-3483 37 15 product product NOUN cana-3483 37 16 cordial cordial ADJ cana-3483 37 17 graph graph NOUN cana-3483 37 18 of of ADP cana-3483 37 19 specific specific ADJ cana-3483 37 20 types type NOUN cana-3483 37 21 of of ADP cana-3483 37 22 graphs graph NOUN cana-3483 37 23 , , PUNCT cana-3483 37 24 such such ADJ cana-3483 37 25 as as ADP cana-3483 37 26 copies copy NOUN cana-3483 37 27 of of ADP cana-3483 37 28 the the DET cana-3483 37 29 concentric concentric ADJ cana-3483 37 30 n n CCONJ cana-3483 37 31 - - PUNCT cana-3483 37 32 sunlet sunlet NOUN cana-3483 37 33 graph graph NOUN cana-3483 37 34 , , PUNCT cana-3483 37 35 the the DET cana-3483 37 36 star star NOUN cana-3483 37 37 of of ADP cana-3483 37 38 the the DET cana-3483 37 39 concentric concentric ADJ cana-3483 37 40 n n CCONJ cana-3483 37 41 - - PUNCT cana-3483 37 42 sunlet sunlet NOUN cana-3483 37 43 graph graph NOUN cana-3483 37 44 , , PUNCT cana-3483 37 45 and and CCONJ cana-3483 37 46 copies copy NOUN cana-3483 37 47 of of ADP cana-3483 37 48 the the DET cana-3483 37 49 shell shell NOUN cana-3483 37 50 graph graph NOUN cana-3483 37 51 . . PUNCT cana-3483 38 1 these these DET cana-3483 38 2 graphs graph NOUN cana-3483 38 3 exhibit exhibit VERB cana-3483 38 4 unique unique ADJ cana-3483 38 5 structural structural ADJ cana-3483 38 6 properties property NOUN cana-3483 38 7 and and CCONJ cana-3483 38 8 complexities complexity NOUN cana-3483 38 9 , , PUNCT cana-3483 38 10 making make VERB cana-3483 38 11 them they PRON cana-3483 38 12 ideal ideal ADJ cana-3483 38 13 subjects subject NOUN cana-3483 38 14 for for ADP cana-3483 38 15 studying study VERB cana-3483 38 16 labeling labeling NOUN cana-3483 38 17 properties property NOUN cana-3483 38 18 . . PUNCT cana-3483 39 1 by by ADP cana-3483 39 2 examining examine VERB cana-3483 39 3 their their PRON cana-3483 39 4 product product NOUN cana-3483 39 5 cordial cordial ADJ cana-3483 39 6 labeling labeling NOUN cana-3483 39 7 , , PUNCT cana-3483 39 8 we we PRON cana-3483 39 9 aim aim VERB cana-3483 39 10 to to PART cana-3483 39 11 gain gain VERB cana-3483 39 12 valuable valuable ADJ cana-3483 39 13 insights insight NOUN cana-3483 39 14 into into ADP cana-3483 39 15 their their PRON cana-3483 39 16 characteristics characteristic NOUN cana-3483 39 17 and and CCONJ cana-3483 39 18 relationships relationship NOUN cana-3483 39 19 , , PUNCT cana-3483 39 20 further far ADV cana-3483 39 21 advancing advance VERB cana-3483 39 22 the the DET cana-3483 39 23 field field NOUN cana-3483 39 24 of of ADP cana-3483 39 25 graph graph NOUN cana-3483 39 26 theory theory NOUN cana-3483 39 27 and and CCONJ cana-3483 39 28 its its PRON cana-3483 39 29 applications application NOUN cana-3483 39 30 . . PUNCT cana-3483 40 1 2 2 X cana-3483 40 2 . . X cana-3483 40 3 terminology terminology NOUN cana-3483 40 4 and and CCONJ cana-3483 40 5 notation notation NOUN cana-3483 40 6 definition definition NOUN cana-3483 40 7 2.1 2.1 NUM cana-3483 40 8 : : PUNCT cana-3483 40 9 the the DET cana-3483 40 10 definition definition NOUN cana-3483 40 11 of of ADP cana-3483 40 12 a a DET cana-3483 40 13 graph graph NOUN cana-3483 40 14 labeling labeling NOUN cana-3483 40 15 , , PUNCT cana-3483 40 16 it it PRON cana-3483 40 17 explains explain VERB cana-3483 40 18 that that SCONJ cana-3483 40 19 graph graph NOUN cana-3483 40 20 labeling labeling NOUN cana-3483 40 21 involves involve VERB cana-3483 40 22 giving give VERB cana-3483 40 23 the the DET cana-3483 40 24 vertices vertex NOUN cana-3483 40 25 , , PUNCT cana-3483 40 26 edges edge NOUN cana-3483 40 27 , , PUNCT cana-3483 40 28 or or CCONJ cana-3483 40 29 maybe maybe ADV cana-3483 40 30 both both CCONJ cana-3483 40 31 numeric numeric ADJ cana-3483 40 32 values value NOUN cana-3483 40 33 , , PUNCT cana-3483 40 34 based base VERB cana-3483 40 35 on on ADP cana-3483 40 36 the the DET cana-3483 40 37 specific specific ADJ cana-3483 40 38 demands demand NOUN cana-3483 40 39 of of ADP cana-3483 40 40 the the DET cana-3483 40 41 problem problem NOUN cana-3483 40 42 or or CCONJ cana-3483 40 43 application application NOUN cana-3483 40 44 . . PUNCT cana-3483 41 1 the the DET cana-3483 41 2 labeling labeling NOUN cana-3483 41 3 is be AUX cana-3483 41 4 called call VERB cana-3483 41 5 vertex vertex NOUN cana-3483 41 6 labeling labeling NOUN cana-3483 41 7 ( ( PUNCT cana-3483 41 8 or or CCONJ cana-3483 41 9 edge edge VERB cana-3483 41 10 labeling labeling NOUN cana-3483 41 11 ) ) PUNCT cana-3483 41 12 when when SCONJ cana-3483 41 13 the the DET cana-3483 41 14 set set NOUN cana-3483 41 15 of of ADP cana-3483 41 16 vertices vertex NOUN cana-3483 41 17 ( ( PUNCT cana-3483 41 18 or or CCONJ cana-3483 41 19 edges edge NOUN cana-3483 41 20 ) ) PUNCT cana-3483 41 21 serves serve VERB cana-3483 41 22 as as ADP cana-3483 41 23 the the DET cana-3483 41 24 domain domain NOUN cana-3483 41 25 for for ADP cana-3483 41 26 the the DET cana-3483 41 27 mapping mapping NOUN cana-3483 41 28 definition definition NOUN cana-3483 41 29 2.2 2.2 NUM cana-3483 41 30 : : PUNCT cana-3483 41 31 a a DET cana-3483 41 32 product product NOUN cana-3483 41 33 cordial cordial ADJ cana-3483 41 34 of of ADP cana-3483 41 35 a a DET cana-3483 41 36 graph graph NOUN cana-3483 41 37 𝐺 𝐺 NOUN cana-3483 41 38 involves involve VERB cana-3483 41 39 a a DET cana-3483 41 40 labelling labelling NOUN cana-3483 41 41 of of ADP cana-3483 41 42 vertex vertex NOUN cana-3483 41 43 𝑔 𝑔 NOUN cana-3483 41 44 : : PUNCT cana-3483 41 45 𝛼 𝛼 X cana-3483 41 46 β†’ β†’ SYM cana-3483 41 47 { { PUNCT cana-3483 41 48 0,1 0,1 NUM cana-3483 41 49 } } PUNCT cana-3483 41 50 , , PUNCT cana-3483 41 51 which which PRON cana-3483 41 52 induces induce VERB cana-3483 41 53 an an DET cana-3483 41 54 labelling labelling NOUN cana-3483 41 55 of of ADP cana-3483 41 56 edge edge NOUN cana-3483 41 57 𝑔:𝛽(𝐺 𝑔:𝛽(𝐺 NUM cana-3483 41 58 ) ) PUNCT cana-3483 41 59 β†’ β†’ SYM cana-3483 41 60 { { PUNCT cana-3483 41 61 0,1 0,1 NUM cana-3483 41 62 } } PUNCT cana-3483 41 63 , , PUNCT cana-3483 41 64 where where SCONJ cana-3483 41 65 𝑔 𝑔 PROPN cana-3483 41 66 βˆ— βˆ— NOUN cana-3483 41 67 ( ( PUNCT cana-3483 41 68 𝑑𝑦 𝑑𝑦 NOUN cana-3483 41 69 ) ) PUNCT cana-3483 41 70 = = SYM cana-3483 41 71 𝑔(𝑑)𝑔(𝑦 𝑔(𝑑)𝑔(𝑦 NOUN cana-3483 41 72 ) ) PUNCT cana-3483 41 73 . . PUNCT cana-3483 42 1 for for ADP cana-3483 42 2 the the DET cana-3483 42 3 graph graph NOUN cana-3483 42 4 𝐺 𝐺 PROPN cana-3483 42 5 to to PART cana-3483 42 6 be be AUX cana-3483 42 7 considered consider VERB cana-3483 42 8 a a DET cana-3483 42 9 product product NOUN cana-3483 42 10 cordial cordial ADJ cana-3483 42 11 graph graph NOUN cana-3483 42 12 , , PUNCT cana-3483 42 13 the the DET cana-3483 42 14 labeling labeling NOUN cana-3483 42 15 must must AUX cana-3483 42 16 meet meet VERB cana-3483 42 17 the the DET cana-3483 42 18 conditions condition NOUN cana-3483 42 19 : : PUNCT cana-3483 42 20 the the DET cana-3483 42 21 difference difference NOUN cana-3483 42 22 between between ADP cana-3483 42 23 the the DET cana-3483 42 24 number number NOUN cana-3483 42 25 of of ADP cana-3483 42 26 vertices vertex NOUN cana-3483 42 27 labeled label VERB cana-3483 42 28 0 0 PUNCT cana-3483 42 29 and and CCONJ cana-3483 42 30 those those PRON cana-3483 42 31 labeled label VERB cana-3483 42 32 1 1 NUM cana-3483 42 33 , , PUNCT cana-3483 42 34 |𝛼𝑔(0 |𝛼𝑔(0 ADV cana-3483 42 35 ) ) PUNCT cana-3483 42 36 βˆ’ βˆ’ PROPN cana-3483 42 37 𝛼𝑔(1)| 𝛼𝑔(1)| NOUN cana-3483 42 38 , , PUNCT cana-3483 42 39 must must AUX cana-3483 42 40 be be AUX cana-3483 42 41 less less ADJ cana-3483 42 42 than than ADP cana-3483 42 43 or or CCONJ cana-3483 42 44 equal equal ADJ cana-3483 42 45 to to ADP cana-3483 42 46 1 1 NUM cana-3483 42 47 , , PUNCT cana-3483 42 48 and and CCONJ cana-3483 42 49 similarly similarly ADV cana-3483 42 50 , , PUNCT cana-3483 42 51 the the DET cana-3483 42 52 distinction distinction NOUN cana-3483 42 53 between between ADP cana-3483 42 54 the the DET cana-3483 42 55 number number NOUN cana-3483 42 56 of of ADP cana-3483 42 57 edges edge NOUN cana-3483 42 58 labeled label VERB cana-3483 42 59 0 0 PUNCT cana-3483 43 1 and and CCONJ cana-3483 43 2 those those PRON cana-3483 43 3 labeled label VERB cana-3483 43 4 1 1 NUM cana-3483 43 5 , , PUNCT cana-3483 43 6 |𝛽𝑔(0 |𝛽𝑔(0 NUM cana-3483 43 7 ) ) PUNCT cana-3483 43 8 βˆ’ βˆ’ PROPN cana-3483 43 9 𝛽𝑔(1)|,must 𝛽𝑔(1)|,must PROPN cana-3483 43 10 also also ADV cana-3483 43 11 be be AUX cana-3483 43 12 less less ADJ cana-3483 43 13 than than ADP cana-3483 43 14 or or CCONJ cana-3483 43 15 equal equal ADJ cana-3483 43 16 to to ADP cana-3483 43 17 1 1 NUM cana-3483 43 18 . . PUNCT cana-3483 43 19 definition definition NOUN cana-3483 43 20 2.3 2.3 NUM cana-3483 43 21 : : PUNCT cana-3483 43 22 a a DET cana-3483 43 23 star star NOUN cana-3483 43 24 graph graph NOUN cana-3483 43 25 𝑆1,𝑛 𝑆1,𝑛 PROPN cana-3483 43 26 is be AUX cana-3483 43 27 a a DET cana-3483 43 28 tree tree NOUN cana-3483 43 29 graph graph NOUN cana-3483 43 30 consisting consist VERB cana-3483 43 31 of of ADP cana-3483 43 32 one one NUM cana-3483 43 33 central central ADJ cana-3483 43 34 node node NOUN cana-3483 43 35 𝛼1 𝛼1 NOUN cana-3483 43 36 and and CCONJ cana-3483 43 37 𝑛 𝑛 DET cana-3483 43 38 leaf leaf NOUN cana-3483 43 39 nodes node NOUN cana-3483 43 40 𝛼2 𝛼2 VERB cana-3483 43 41 , , PUNCT cana-3483 43 42 𝛼3 𝛼3 VERB cana-3483 43 43 , , PUNCT cana-3483 43 44 … … PUNCT cana-3483 43 45 , , PUNCT cana-3483 43 46 π›Όπ‘š+1 π›Όπ‘š+1 PROPN cana-3483 43 47 , , PUNCT cana-3483 43 48 where where SCONJ cana-3483 43 49 each each DET cana-3483 43 50 leaf leaf NOUN cana-3483 43 51 node node NOUN cana-3483 43 52 is be AUX cana-3483 43 53 connected connect VERB cana-3483 43 54 directly directly ADV cana-3483 43 55 to to ADP cana-3483 43 56 the the DET cana-3483 43 57 central central ADJ cana-3483 43 58 node node PROPN cana-3483 43 59 𝑣1 𝑣1 PROPN cana-3483 43 60 say say VERB cana-3483 43 61 ( ( PUNCT cana-3483 43 62 𝛼1 𝛼1 NOUN cana-3483 43 63 , , PUNCT cana-3483 43 64 𝛼𝑖 𝛼𝑖 NOUN cana-3483 43 65 ) ) PUNCT cana-3483 43 66 for for ADP cana-3483 43 67 𝑖 𝑖 NOUN cana-3483 43 68 = = SYM cana-3483 43 69 2,3 2,3 NUM cana-3483 43 70 , , PUNCT cana-3483 43 71 … … PUNCT cana-3483 43 72 , , PUNCT cana-3483 44 1 π‘š π‘š X cana-3483 44 2 + + NOUN cana-3483 44 3 1 1 X cana-3483 44 4 . . X cana-3483 44 5 there there PRON cana-3483 44 6 are be VERB cana-3483 44 7 no no DET cana-3483 44 8 edges edge NOUN cana-3483 44 9 between between ADP cana-3483 44 10 the the DET cana-3483 44 11 leaf leaf NOUN cana-3483 44 12 nodes node NOUN cana-3483 44 13 themselves themselves PRON cana-3483 44 14 . . PUNCT cana-3483 45 1 definition definition NOUN cana-3483 45 2 2.4 2.4 NUM cana-3483 45 3 : : PUNCT cana-3483 45 4 the the DET cana-3483 45 5 n n CCONJ cana-3483 45 6 - - PUNCT cana-3483 45 7 sunlet sunlet NOUN cana-3483 45 8 graph graph NOUN cana-3483 45 9 , , PUNCT cana-3483 45 10 indicated indicate VERB cana-3483 45 11 as as ADP cana-3483 45 12 sn sn PROPN cana-3483 45 13 , , PUNCT cana-3483 45 14 consisting consist VERB cana-3483 45 15 of of ADP cana-3483 45 16 2n 2n NUM cana-3483 45 17 vertices vertex NOUN cana-3483 45 18 , , PUNCT cana-3483 45 19 formed form VERB cana-3483 45 20 by by ADP cana-3483 45 21 connecting connect VERB cana-3483 45 22 n n CCONJ cana-3483 45 23 pendant pendant ADJ cana-3483 45 24 edges edge NOUN cana-3483 45 25 to to ADP cana-3483 45 26 the the DET cana-3483 45 27 cycle cycle NOUN cana-3483 45 28 cn cn PROPN cana-3483 45 29 . . PROPN cana-3483 45 30 definition definition NOUN cana-3483 45 31 2.5 2.5 NUM cana-3483 45 32 : : PUNCT cana-3483 45 33 the the DET cana-3483 45 34 g g PROPN cana-3483 45 35 graph graph NOUN cana-3483 45 36 created create VERB cana-3483 45 37 by by ADP cana-3483 45 38 connecting connect VERB cana-3483 45 39 three three NUM cana-3483 45 40 replicas replica NOUN cana-3483 45 41 of of ADP cana-3483 45 42 the the DET cana-3483 45 43 n n CCONJ cana-3483 45 44 - - PUNCT cana-3483 45 45 sunlet sunlet NOUN cana-3483 45 46 graph graph NOUN cana-3483 45 47 is be AUX cana-3483 45 48 3concentric 3concentric NUM cana-3483 45 49 n n CCONJ cana-3483 45 50 - - PUNCT cana-3483 45 51 sunlet sunlet NOUN cana-3483 45 52 graph graph NOUN cana-3483 45 53 denoted denote VERB cana-3483 45 54 by by ADP cana-3483 45 55 3 3 NUM cana-3483 45 56 βˆ— βˆ— NOUN cana-3483 45 57 𝑆𝑛. 𝑆𝑛. PROPN cana-3483 45 58 inner inner ADJ cana-3483 45 59 most most ADJ cana-3483 45 60 copy copy NOUN cana-3483 45 61 of of ADP cana-3483 45 62 n n CCONJ cana-3483 45 63 - - PUNCT cana-3483 45 64 sunlet sunlet NOUN cana-3483 45 65 graph graph NOUN cana-3483 45 66 as as ADP cana-3483 45 67 𝛼1 𝛼1 NOUN cana-3483 45 68 , , PUNCT cana-3483 45 69 𝛼2 𝛼2 ADJ cana-3483 45 70 , , PUNCT cana-3483 45 71 … … PUNCT cana-3483 45 72 𝛼𝑛 𝛼𝑛 PROPN cana-3483 45 73 and and CCONJ cana-3483 45 74 its its PRON cana-3483 45 75 spokes spoke NOUN cana-3483 45 76 denoted denote VERB cana-3483 45 77 as as ADP cana-3483 45 78 𝛼1 𝛼1 NOUN cana-3483 45 79 ’ ' PUNCT cana-3483 45 80 , , PUNCT cana-3483 45 81 𝛼2 𝛼2 PROPN cana-3483 45 82 ’ ' PUNCT cana-3483 45 83 , , PUNCT cana-3483 45 84 … … PUNCT cana-3483 45 85 , , PUNCT cana-3483 45 86 𝛼𝑛 𝛼𝑛 PROPN cana-3483 45 87 ’ ' PUNCT cana-3483 45 88 , , PUNCT cana-3483 45 89 the the DET cana-3483 45 90 spokes spoke NOUN cana-3483 45 91 of of ADP cana-3483 45 92 inner inner ADJ cana-3483 45 93 most most ADV cana-3483 45 94 sunlet sunlet NOUN cana-3483 45 95 graph graph NOUN cana-3483 45 96 is be AUX cana-3483 45 97 fused fuse VERB cana-3483 45 98 with with ADP cana-3483 45 99 the the DET cana-3483 45 100 vertices vertex NOUN cana-3483 45 101 of of ADP cana-3483 45 102 middle middle ADJ cana-3483 45 103 graph graph NOUN cana-3483 45 104 , , PUNCT cana-3483 45 105 the the DET cana-3483 45 106 vertices vertex NOUN cana-3483 45 107 of of ADP cana-3483 45 108 middle middle ADJ cana-3483 45 109 copy copy NOUN cana-3483 45 110 of of ADP cana-3483 45 111 n n CCONJ cana-3483 45 112 - - PUNCT cana-3483 45 113 sunlet sunlet NOUN cana-3483 45 114 graph graph NOUN cana-3483 45 115 is be AUX cana-3483 45 116 now now ADV cana-3483 45 117 denoted denote VERB cana-3483 45 118 as as ADP cana-3483 45 119 𝛼1 𝛼1 NOUN cana-3483 45 120 ’ ' PUNCT cana-3483 45 121 , , PUNCT cana-3483 45 122 𝛼2 𝛼2 PROPN cana-3483 45 123 ’ ' PUNCT cana-3483 45 124 , , PUNCT cana-3483 45 125 … … PUNCT cana-3483 45 126 , , PUNCT cana-3483 45 127 𝛼𝑛 𝛼𝑛 PROPN cana-3483 45 128 ’ ' PUNCT cana-3483 45 129 and and CCONJ cana-3483 45 130 its its PRON cana-3483 45 131 spokes spoke NOUN cana-3483 45 132 as as ADP cana-3483 45 133 π‘₯1 π‘₯1 NOUN cana-3483 45 134 ’ ' PUNCT cana-3483 45 135 , , PUNCT cana-3483 45 136 π‘₯2 π‘₯2 NOUN cana-3483 45 137 ’ ' PUNCT cana-3483 45 138 , , PUNCT cana-3483 45 139 … … PUNCT cana-3483 45 140 . . PUNCT cana-3483 46 1 π‘₯3 π‘₯3 NOUN cana-3483 46 2 ’ ' PUNCT cana-3483 46 3 and and CCONJ cana-3483 46 4 again again ADV cana-3483 46 5 its its PRON cana-3483 46 6 spokes spoke NOUN cana-3483 46 7 are be AUX cana-3483 46 8 fused fuse VERB cana-3483 46 9 with with ADP cana-3483 46 10 vertices vertex NOUN cana-3483 46 11 of of ADP cana-3483 46 12 outer outer ADJ cana-3483 46 13 sunlet sunlet NOUN cana-3483 46 14 graph graph NOUN cana-3483 46 15 . . PUNCT cana-3483 47 1 the the DET cana-3483 47 2 outer outer ADJ cana-3483 47 3 most most ADJ cana-3483 47 4 copy copy NOUN cana-3483 47 5 is be AUX cana-3483 47 6 denoted denote VERB cana-3483 47 7 by by ADP cana-3483 47 8 π‘₯1 π‘₯1 NOUN cana-3483 47 9 ’ ' PUNCT cana-3483 47 10 , , PUNCT cana-3483 47 11 π‘₯ π‘₯ NOUN cana-3483 47 12 ’ ' PUNCT cana-3483 47 13 , , PUNCT cana-3483 47 14 … … PUNCT cana-3483 47 15 . . PUNCT cana-3483 48 1 π‘₯3 π‘₯3 NOUN cana-3483 48 2 ’ ' PUNCT cana-3483 48 3 and and CCONJ cana-3483 48 4 spokes spoke NOUN cana-3483 48 5 are be AUX cana-3483 48 6 denoted denote VERB cana-3483 48 7 as as ADP cana-3483 48 8 𝑀1 𝑀1 VERB cana-3483 48 9 ’ ' PUNCT cana-3483 48 10 , , PUNCT cana-3483 48 11 𝑀2 𝑀2 NOUN cana-3483 48 12 ’ ' PUNCT cana-3483 48 13 , , PUNCT cana-3483 48 14 … … PUNCT cana-3483 48 15 . . PUNCT cana-3483 49 1 , , PUNCT cana-3483 49 2 𝑀𝑛 𝑀𝑛 VERB cana-3483 49 3 ’ ' PUNCT cana-3483 49 4 . . PUNCT cana-3483 50 1 in in ADP cana-3483 50 2 below below ADP cana-3483 50 3 figure figure NOUN cana-3483 50 4 1 1 NUM cana-3483 50 5 we we PRON cana-3483 50 6 have have AUX cana-3483 50 7 considered consider VERB cana-3483 50 8 3*s4 3*s4 NUM cana-3483 50 9 as as ADP cana-3483 50 10 an an DET cana-3483 50 11 example example NOUN cana-3483 50 12 . . PUNCT cana-3483 51 1 figure figure NOUN cana-3483 51 2 1 1 NUM cana-3483 51 3 : : SYM cana-3483 51 4 3 3 NUM cana-3483 51 5 βˆ— βˆ— NOUN cana-3483 51 6 𝑆4 𝑆4 PROPN cana-3483 51 7 definition definition NOUN cana-3483 51 8 2.6 2.6 NUM cana-3483 51 9 : : PUNCT cana-3483 51 10 the the DET cana-3483 51 11 shell shell NOUN cana-3483 51 12 graph graph NOUN cana-3483 51 13 c(n c(n PROPN cana-3483 51 14 , , PUNCT cana-3483 51 15 n-3 n-3 NOUN cana-3483 51 16 ) ) PUNCT cana-3483 51 17 , , PUNCT cana-3483 51 18 also also ADV cana-3483 51 19 known know VERB cana-3483 51 20 as as ADP cana-3483 51 21 the the DET cana-3483 51 22 fan fan NOUN cana-3483 51 23 graph graph NOUN cana-3483 51 24 π‘“π‘›βˆ’1 π‘“π‘›βˆ’1 PROPN cana-3483 51 25 , , PUNCT cana-3483 51 26 is be AUX cana-3483 51 27 constructed construct VERB cana-3483 51 28 by by ADP cana-3483 51 29 selecting select VERB cana-3483 51 30 𝑛 𝑛 DET cana-3483 51 31 βˆ’ βˆ’ NUM cana-3483 51 32 3 3 NUM cana-3483 51 33 simultaneous simultaneous ADJ cana-3483 51 34 chords chord NOUN cana-3483 51 35 in in ADP cana-3483 51 36 a a DET cana-3483 51 37 graph graph NOUN cana-3483 51 38 of of ADP cana-3483 51 39 cycle cycle NOUN cana-3483 52 1 𝐢𝑛 𝐢𝑛 PROPN cana-3483 52 2 , , PUNCT cana-3483 52 3 the the DET cana-3483 52 4 vertices vertex NOUN cana-3483 52 5 where where SCONJ cana-3483 52 6 every every DET cana-3483 52 7 chords chord NOUN cana-3483 52 8 intersect intersect NOUN cana-3483 52 9 referred refer VERB cana-3483 52 10 to to ADP cana-3483 52 11 as as ADP cana-3483 52 12 the the DET cana-3483 52 13 apex apex PROPN cana-3483 52 14 vertex vertex NOUN cana-3483 52 15 . . PUNCT cana-3483 53 1 the the DET cana-3483 53 2 shell shell NOUN cana-3483 53 3 𝑆𝑛 𝑆𝑛 PROPN cana-3483 53 4 is be AUX cana-3483 53 5 also also ADV cana-3483 53 6 known know VERB cana-3483 53 7 as as ADP cana-3483 53 8 the the DET cana-3483 53 9 fan fan PROPN cana-3483 53 10 π‘“π‘›βˆ’1 π‘“π‘›βˆ’1 PROPN cana-3483 53 11 . . PUNCT cana-3483 54 1 therefore therefore ADV cana-3483 54 2 , , PUNCT cana-3483 54 3 we we PRON cana-3483 54 4 can can AUX cana-3483 54 5 express express VERB cana-3483 54 6 this this PRON cana-3483 54 7 as as ADP cana-3483 54 8 𝑆𝑛 𝑆𝑛 PROPN cana-3483 54 9 = = SYM cana-3483 54 10 π‘“π‘›βˆ’1 π‘“π‘›βˆ’1 PROPN cana-3483 54 11 = = SYM cana-3483 54 12 π‘ƒπ‘›βˆ’1 π‘ƒπ‘›βˆ’1 PROPN cana-3483 54 13 + + CCONJ cana-3483 54 14 𝐾1 𝐾1 PROPN cana-3483 54 15 . . PUNCT cana-3483 55 1 communications communication NOUN cana-3483 55 2 on on ADP cana-3483 55 3 applied apply VERB cana-3483 55 4 nonlinear nonlinear ADJ cana-3483 55 5 analysis analysis NOUN cana-3483 55 6 issn issn NOUN cana-3483 55 7 : : PUNCT cana-3483 55 8 1074 1074 NUM cana-3483 55 9 - - PUNCT cana-3483 55 10 133x 133x NUM cana-3483 55 11 vol vol NOUN cana-3483 55 12 32 32 NUM cana-3483 55 13 no no NOUN cana-3483 55 14 . . PUNCT cana-3483 56 1 7s 7 NOUN cana-3483 56 2 ( ( PUNCT cana-3483 56 3 2025 2025 NUM cana-3483 56 4 ) ) PUNCT cana-3483 56 5 761 761 NUM cana-3483 56 6 https://internationalpubls.com https://internationalpubls.com X cana-3483 56 7 3 3 X cana-3483 56 8 . . X cana-3483 56 9 main main ADJ cana-3483 56 10 results result NOUN cana-3483 56 11 theorem theorem VERB cana-3483 56 12 3.1 3.1 NUM cana-3483 56 13 . . PUNCT cana-3483 57 1 a a DET cana-3483 57 2 graph graph NOUN cana-3483 57 3 acquired acquire VERB cana-3483 57 4 of of ADP cana-3483 57 5 two two NUM cana-3483 57 6 copies copy NOUN cana-3483 57 7 by by ADP cana-3483 57 8 joining join VERB cana-3483 57 9 the the DET cana-3483 57 10 3concentric 3concentric NUM cana-3483 57 11 n n CCONJ cana-3483 57 12 - - PUNCT cana-3483 57 13 sunlet sunlet NOUN cana-3483 57 14 graph graph NOUN cana-3483 57 15 , , PUNCT cana-3483 57 16 ο€’ ο€’ NOUN cana-3483 57 17 n n NOUN cana-3483 57 18 is be AUX cana-3483 57 19 product product NOUN cana-3483 57 20 cordial cordial ADJ cana-3483 57 21 . . PUNCT cana-3483 58 1 proof proof NOUN cana-3483 58 2 : : PUNCT cana-3483 58 3 the the DET cana-3483 58 4 graph graph NOUN cana-3483 58 5 obtained obtain VERB cana-3483 58 6 of of ADP cana-3483 58 7 two two NUM cana-3483 58 8 copies copy NOUN cana-3483 58 9 by by ADP cana-3483 58 10 joining join VERB cana-3483 58 11 the the DET cana-3483 58 12 3 3 NUM cana-3483 58 13 - - PUNCT cana-3483 58 14 concentric concentric ADJ cana-3483 58 15 n n CCONJ cana-3483 58 16 - - PUNCT cana-3483 58 17 sunlet sunlet NOUN cana-3483 58 18 graph graph NOUN cana-3483 58 19 . . PUNCT cana-3483 59 1 in in ADP cana-3483 59 2 this this DET cana-3483 59 3 graph graph NOUN cana-3483 59 4 we we PRON cana-3483 59 5 consider consider VERB cana-3483 59 6 inner inner ADJ cana-3483 59 7 most most ADJ cana-3483 59 8 copy copy NOUN cana-3483 59 9 of of ADP cana-3483 59 10 n n CCONJ cana-3483 59 11 - - PUNCT cana-3483 59 12 sunlet sunlet NOUN cana-3483 59 13 graph graph NOUN cana-3483 59 14 as as ADP cana-3483 59 15 𝛼1 𝛼1 NOUN cana-3483 59 16 , , PUNCT cana-3483 59 17 𝛼2 𝛼2 ADJ cana-3483 59 18 , , PUNCT cana-3483 59 19 … … PUNCT cana-3483 59 20 , , PUNCT cana-3483 59 21 𝛼𝑛 𝛼𝑛 PROPN cana-3483 59 22 and and CCONJ cana-3483 59 23 its its PRON cana-3483 59 24 spokes spoke NOUN cana-3483 59 25 denoted denote VERB cana-3483 59 26 as as ADP cana-3483 59 27 𝛼1 𝛼1 NOUN cana-3483 59 28 ’ ' PUNCT cana-3483 59 29 , , PUNCT cana-3483 59 30 𝛼2 𝛼2 PROPN cana-3483 59 31 ’ ' PUNCT cana-3483 59 32 , , PUNCT cana-3483 59 33 … … PUNCT cana-3483 59 34 , , PUNCT cana-3483 59 35 𝛼𝑛 𝛼𝑛 PROPN cana-3483 59 36 ’ ' PUNCT cana-3483 59 37 the the DET cana-3483 59 38 spokes spoke NOUN cana-3483 59 39 of of ADP cana-3483 59 40 inner inner ADJ cana-3483 59 41 most most ADV cana-3483 59 42 sunlet sunlet NOUN cana-3483 59 43 graph graph NOUN cana-3483 59 44 is be AUX cana-3483 59 45 fused fuse VERB cana-3483 59 46 with with ADP cana-3483 59 47 the the DET cana-3483 59 48 vertices vertex NOUN cana-3483 59 49 of of ADP cana-3483 59 50 middle middle ADJ cana-3483 59 51 graph graph NOUN cana-3483 59 52 , , PUNCT cana-3483 59 53 now now ADV cana-3483 59 54 the the DET cana-3483 59 55 vertices vertex NOUN cana-3483 59 56 of of ADP cana-3483 59 57 middle middle ADJ cana-3483 59 58 copy copy NOUN cana-3483 59 59 of of ADP cana-3483 59 60 n n CCONJ cana-3483 59 61 - - PUNCT cana-3483 59 62 sunlet sunlet NOUN cana-3483 59 63 graph graph NOUN cana-3483 59 64 is be AUX cana-3483 59 65 denoted denote VERB cana-3483 59 66 as as ADP cana-3483 59 67 𝛼1 𝛼1 NOUN cana-3483 59 68 ’ ' PUNCT cana-3483 59 69 , , PUNCT cana-3483 59 70 𝛼2 𝛼2 PROPN cana-3483 59 71 ’ ' PUNCT cana-3483 59 72 , , PUNCT cana-3483 59 73 … … PUNCT cana-3483 59 74 , , PUNCT cana-3483 59 75 𝛼𝑛 𝛼𝑛 PROPN cana-3483 59 76 ’ ' PUNCT cana-3483 59 77 and and CCONJ cana-3483 59 78 its its PRON cana-3483 59 79 spokes spoke NOUN cana-3483 59 80 as as ADP cana-3483 59 81 π‘₯1 π‘₯1 NOUN cana-3483 59 82 ’ ' PUNCT cana-3483 59 83 , , PUNCT cana-3483 59 84 π‘₯2 π‘₯2 NOUN cana-3483 59 85 ’ ' PUNCT cana-3483 59 86 , , PUNCT cana-3483 59 87 … … PUNCT cana-3483 59 88 , , PUNCT cana-3483 59 89 π‘₯𝑛 π‘₯𝑛 VERB cana-3483 59 90 ’ ' PUNCT cana-3483 59 91 and and CCONJ cana-3483 59 92 again again ADV cana-3483 59 93 its its PRON cana-3483 59 94 spokes spoke NOUN cana-3483 59 95 are be AUX cana-3483 59 96 fused fuse VERB cana-3483 59 97 with with ADP cana-3483 59 98 outer outer ADJ cana-3483 59 99 vertices vertex NOUN cana-3483 59 100 of of ADP cana-3483 59 101 the the DET cana-3483 59 102 graph graph NOUN cana-3483 59 103 , , PUNCT cana-3483 59 104 then then ADV cana-3483 59 105 the the DET cana-3483 59 106 outer outer ADJ cana-3483 59 107 most most ADJ cana-3483 59 108 copy copy NOUN cana-3483 59 109 is be AUX cana-3483 59 110 denoted denote VERB cana-3483 59 111 by by ADP cana-3483 59 112 π‘₯1 π‘₯1 NOUN cana-3483 59 113 ’ ' PUNCT cana-3483 59 114 , , PUNCT cana-3483 59 115 π‘₯2 π‘₯2 NOUN cana-3483 59 116 ’ ' PUNCT cana-3483 59 117 , , PUNCT cana-3483 59 118 … … PUNCT cana-3483 59 119 , , PUNCT cana-3483 59 120 π‘₯𝑛 π‘₯𝑛 VERB cana-3483 59 121 ’ ' PUNCT cana-3483 59 122 and and CCONJ cana-3483 59 123 spokes spoke NOUN cana-3483 59 124 are be AUX cana-3483 59 125 denoted denote VERB cana-3483 59 126 as as ADP cana-3483 59 127 𝑀1 𝑀1 VERB cana-3483 59 128 ’ ' PUNCT cana-3483 59 129 , , PUNCT cana-3483 59 130 𝑀2 𝑀2 NOUN cana-3483 59 131 ’ ' PUNCT cana-3483 59 132 , , PUNCT cana-3483 59 133 … … PUNCT cana-3483 59 134 , , PUNCT cana-3483 59 135 𝑀𝑛 𝑀𝑛 PROPN cana-3483 59 136 ’ ' PUNCT cana-3483 59 137 . . PUNCT cana-3483 60 1 the the DET cana-3483 60 2 two two NUM cana-3483 60 3 graphs graph NOUN cana-3483 60 4 are be AUX cana-3483 60 5 joined join VERB cana-3483 60 6 by by ADP cana-3483 60 7 a a DET cana-3483 60 8 vertex vertex NOUN cana-3483 60 9 obtained obtain VERB cana-3483 60 10 by by ADP cana-3483 60 11 fusing fuse VERB cana-3483 60 12 the the DET cana-3483 60 13 outer outer ADJ cana-3483 60 14 vertices vertex NOUN cana-3483 60 15 of of ADP cana-3483 60 16 both both CCONJ cana-3483 60 17 the the DET cana-3483 60 18 graphs graph NOUN cana-3483 60 19 . . PUNCT cana-3483 61 1 to to PART cana-3483 61 2 define define VERB cana-3483 61 3 𝑔 𝑔 PROPN cana-3483 61 4 ∢ ∢ PROPN cana-3483 61 5 𝛼 𝛼 NOUN cana-3483 61 6 ( ( PUNCT cana-3483 61 7 3 3 NUM cana-3483 61 8 βˆ— βˆ— NOUN cana-3483 61 9 𝑆𝑛 𝑆𝑛 NOUN cana-3483 61 10 ) ) PUNCT cana-3483 61 11 β†’ β†’ SYM cana-3483 61 12 { { PUNCT cana-3483 61 13 0 0 NUM cana-3483 61 14 , , PUNCT cana-3483 61 15 1 1 NUM cana-3483 61 16 } } PUNCT cana-3483 61 17 , , PUNCT cana-3483 61 18 we we PRON cana-3483 61 19 examine examine VERB cana-3483 61 20 the the DET cana-3483 61 21 following follow VERB cana-3483 61 22 cases case NOUN cana-3483 61 23 : : PUNCT cana-3483 61 24 𝐺(𝑑 𝐺(𝑑 X cana-3483 61 25 ) ) PUNCT cana-3483 61 26 = = SYM cana-3483 61 27 { { PUNCT cana-3483 61 28 1 1 NUM cana-3483 61 29 ; ; PUNCT cana-3483 61 30 𝑑 𝑑 PROPN cana-3483 61 31 = = SYM cana-3483 61 32 𝛼𝑖 𝛼𝑖 PROPN cana-3483 61 33 , , PUNCT cana-3483 61 34 𝑖 𝑖 PROPN cana-3483 61 35 = = SYM cana-3483 61 36 1,2 1,2 NUM cana-3483 61 37 , , PUNCT cana-3483 61 38 . . PUNCT cana-3483 61 39 . . PUNCT cana-3483 62 1 , , PUNCT cana-3483 62 2 𝑛 𝑛 PROPN cana-3483 62 3 1 1 NUM cana-3483 62 4 ; ; PUNCT cana-3483 62 5 𝑑 𝑑 PROPN cana-3483 62 6 = = SYM cana-3483 62 7 𝛼𝑖 𝛼𝑖 PROPN cana-3483 62 8 β€² β€² NOUN cana-3483 62 9 , , PUNCT cana-3483 62 10 𝑖 𝑖 X cana-3483 62 11 = = SYM cana-3483 62 12 1,2 1,2 NUM cana-3483 62 13 , , PUNCT cana-3483 62 14 . . PUNCT cana-3483 62 15 . . PUNCT cana-3483 63 1 , , PUNCT cana-3483 63 2 𝑛 𝑛 PROPN cana-3483 63 3 0 0 NUM cana-3483 63 4 ; ; PUNCT cana-3483 63 5 𝑑 𝑑 PROPN cana-3483 63 6 = = SYM cana-3483 63 7 π‘₯𝑖′ π‘₯𝑖′ ADJ cana-3483 63 8 , , PUNCT cana-3483 63 9 𝑖 𝑖 NOUN cana-3483 63 10 = = SYM cana-3483 63 11 1,2 1,2 NUM cana-3483 63 12 , , PUNCT cana-3483 63 13 . . PUNCT cana-3483 63 14 . . PUNCT cana-3483 64 1 , , PUNCT cana-3483 64 2 𝑛 𝑛 PROPN cana-3483 64 3 0 0 NUM cana-3483 64 4 ; ; PUNCT cana-3483 64 5 𝑑 𝑑 X cana-3483 64 6 = = VERB cana-3483 64 7 𝑀𝑖 𝑀𝑖 ADP cana-3483 64 8 β€² β€² NUM cana-3483 64 9 , , PUNCT cana-3483 64 10 𝑖 𝑖 X cana-3483 64 11 = = SYM cana-3483 64 12 1,2 1,2 NUM cana-3483 64 13 , , PUNCT cana-3483 64 14 . . PUNCT cana-3483 64 15 . . PUNCT cana-3483 65 1 , , PUNCT cana-3483 65 2 𝑛 𝑛 X cana-3483 65 3 with with ADP cana-3483 65 4 the the DET cana-3483 65 5 above above ADJ cana-3483 65 6 labeling labeling NOUN cana-3483 65 7 pattern pattern NOUN cana-3483 65 8 satisfies satisfy VERB cana-3483 65 9 the the DET cana-3483 65 10 conditions condition NOUN cana-3483 65 11 |𝛼𝑔(0 |𝛼𝑔(0 ADV cana-3483 65 12 ) ) PUNCT cana-3483 66 1 βˆ’ βˆ’ PROPN cana-3483 66 2 𝛼𝑔(1)| 𝛼𝑔(1)| NOUN cana-3483 66 3 ≀ ≀ NOUN cana-3483 66 4 1 1 NUM cana-3483 66 5 and and CCONJ cana-3483 66 6 |𝛽𝑔(0 |𝛽𝑔(0 NUM cana-3483 66 7 ) ) PUNCT cana-3483 66 8 βˆ’ βˆ’ NOUN cana-3483 66 9 𝛽𝑔(1)| 𝛽𝑔(1)| NOUN cana-3483 66 10 ≀ ≀ NOUN cana-3483 66 11 1 1 NUM cana-3483 66 12 . . PUNCT cana-3483 67 1 i.e. i.e. X cana-3483 67 2 , , PUNCT cana-3483 67 3 this this PRON cana-3483 67 4 means mean VERB cana-3483 67 5 g g PROPN cana-3483 67 6 admits admit VERB cana-3483 67 7 a a DET cana-3483 67 8 product product NOUN cana-3483 67 9 cordial cordial ADJ cana-3483 67 10 labeling labeling NOUN cana-3483 67 11 . . PUNCT cana-3483 68 1 to to PART cana-3483 68 2 better well ADV cana-3483 68 3 understand understand VERB cana-3483 68 4 this this DET cana-3483 68 5 labeling labeling NOUN cana-3483 68 6 pattern pattern NOUN cana-3483 68 7 , , PUNCT cana-3483 68 8 we we PRON cana-3483 68 9 have have AUX cana-3483 68 10 considered consider VERB cana-3483 68 11 below below ADP cana-3483 68 12 example example NOUN cana-3483 68 13 in in ADP cana-3483 68 14 figure figure NOUN cana-3483 68 15 2 2 NUM cana-3483 68 16 of of ADP cana-3483 68 17 even even ADV cana-3483 68 18 number number NOUN cana-3483 68 19 of of ADP cana-3483 68 20 vertices vertex NOUN cana-3483 68 21 and and CCONJ cana-3483 68 22 figure figure VERB cana-3483 68 23 3 3 NUM cana-3483 68 24 of of ADP cana-3483 68 25 odd odd ADJ cana-3483 68 26 number number NOUN cana-3483 68 27 of of ADP cana-3483 68 28 vertices vertex NOUN cana-3483 68 29 . . PUNCT cana-3483 69 1 1 1 X cana-3483 69 2 . . X cana-3483 69 3 n=8 n=8 NUM cana-3483 69 4 , , PUNCT cana-3483 69 5 i.e. i.e. X cana-3483 69 6 3*s8 3*s8 NUM cana-3483 69 7 . . PUNCT cana-3483 70 1 figure figure NOUN cana-3483 70 2 2 2 NUM cana-3483 70 3 : : PUNCT cana-3483 70 4 two two NUM cana-3483 70 5 copies copy NOUN cana-3483 70 6 of of ADP cana-3483 70 7 3 3 NUM cana-3483 70 8 βˆ— βˆ— NOUN cana-3483 70 9 𝑆8 𝑆8 ADJ cana-3483 70 10 2 2 NUM cana-3483 70 11 . . PUNCT cana-3483 71 1 n=7 n=7 PROPN cana-3483 71 2 , , PUNCT cana-3483 71 3 3*s7 3*s7 NUM cana-3483 71 4 figure figure VERB cana-3483 71 5 3 3 NUM cana-3483 71 6 : : PUNCT cana-3483 71 7 two two NUM cana-3483 71 8 copies copy NOUN cana-3483 71 9 of of ADP cana-3483 71 10 3 3 NUM cana-3483 71 11 βˆ— βˆ— NOUN cana-3483 71 12 𝑆7 𝑆7 PROPN cana-3483 71 13 communications communication NOUN cana-3483 71 14 on on ADP cana-3483 71 15 applied apply VERB cana-3483 71 16 nonlinear nonlinear ADJ cana-3483 71 17 analysis analysis NOUN cana-3483 71 18 issn issn NOUN cana-3483 71 19 : : PUNCT cana-3483 71 20 1074 1074 NUM cana-3483 71 21 - - PUNCT cana-3483 71 22 133x 133x NUM cana-3483 71 23 vol vol NOUN cana-3483 71 24 32 32 NUM cana-3483 71 25 no no NOUN cana-3483 71 26 . . PUNCT cana-3483 72 1 7s 7 NOUN cana-3483 72 2 ( ( PUNCT cana-3483 72 3 2025 2025 NUM cana-3483 72 4 ) ) PUNCT cana-3483 72 5 762 762 NUM cana-3483 72 6 https://internationalpubls.com https://internationalpubls.com X cana-3483 72 7 theorem theorem VERB cana-3483 72 8 3.2 3.2 NUM cana-3483 72 9 . . PUNCT cana-3483 73 1 star star NOUN cana-3483 73 2 of of ADP cana-3483 73 3 3concentric 3concentric NUM cana-3483 73 4 n n CCONJ cana-3483 73 5 - - PUNCT cana-3483 73 6 sunlet sunlet NOUN cana-3483 73 7 graph graph NOUN cana-3483 73 8 , , PUNCT cana-3483 73 9 ο€’n ο€’n PROPN cana-3483 73 10 is be AUX cana-3483 73 11 product product NOUN cana-3483 73 12 cordial cordial ADJ cana-3483 73 13 graph graph NOUN cana-3483 73 14 . . PUNCT cana-3483 74 1 proof proof NOUN cana-3483 74 2 : : PUNCT cana-3483 74 3 let let VERB cana-3483 74 4 π‘₯1 π‘₯1 NOUN cana-3483 74 5 , , PUNCT cana-3483 74 6 π‘₯2 π‘₯2 NOUN cana-3483 74 7 , , PUNCT cana-3483 74 8 … … PUNCT cana-3483 74 9 , , PUNCT cana-3483 74 10 π‘₯𝑛 π‘₯𝑛 PROPN cana-3483 74 11 be be AUX cana-3483 74 12 a a DET cana-3483 74 13 successive successive ADJ cana-3483 74 14 apex apex NOUN cana-3483 74 15 vertices vertex NOUN cana-3483 74 16 of of ADP cana-3483 74 17 inner inner ADJ cana-3483 74 18 most most ADV cana-3483 74 19 3concentric 3concentric NUM cana-3483 74 20 n n CCONJ cana-3483 74 21 - - PUNCT cana-3483 74 22 sunlet sunlet NOUN cana-3483 74 23 graph graph NOUN cana-3483 74 24 . . PUNCT cana-3483 75 1 its its PRON cana-3483 75 2 spokes spoke NOUN cana-3483 75 3 denoted denote VERB cana-3483 75 4 by by ADP cana-3483 75 5 π‘₯1β€² π‘₯1β€² NOUN cana-3483 75 6 , , PUNCT cana-3483 75 7 π‘₯2β€² π‘₯2β€² NOUN cana-3483 75 8 , , PUNCT cana-3483 75 9 … … PUNCT cana-3483 75 10 π‘₯𝑛′ π‘₯𝑛′ NOUN cana-3483 75 11 and and CCONJ cana-3483 75 12 immediate immediate ADJ cana-3483 75 13 next next ADJ cana-3483 75 14 spokes spoke NOUN cana-3483 75 15 denoted denote VERB cana-3483 75 16 by by ADP cana-3483 75 17 π‘₯1 π‘₯1 NOUN cana-3483 75 18 " " PUNCT cana-3483 75 19 , , PUNCT cana-3483 75 20 π‘₯2 π‘₯2 NOUN cana-3483 75 21 " " PUNCT cana-3483 75 22 , , PUNCT cana-3483 75 23 … … PUNCT cana-3483 75 24 π‘₯𝑛 π‘₯𝑛 NOUN cana-3483 75 25 " " PUNCT cana-3483 75 26 and and CCONJ cana-3483 75 27 π‘₯1β€²β€²β€² π‘₯1β€²β€²β€² PROPN cana-3483 75 28 , , PUNCT cana-3483 75 29 π‘₯2β€²β€²β€² π‘₯2β€²β€²β€² PROPN cana-3483 75 30 , , PUNCT cana-3483 75 31 … … PUNCT cana-3483 75 32 π‘₯𝑛′′′ π‘₯𝑛′′′ ADV cana-3483 75 33 respectively respectively ADV cana-3483 75 34 . . PUNCT cana-3483 76 1 the the DET cana-3483 76 2 outer outer ADJ cana-3483 76 3 spokes spoke NOUN cana-3483 76 4 of of ADP cana-3483 76 5 apex apex NOUN cana-3483 76 6 vertices vertex NOUN cana-3483 76 7 are be AUX cana-3483 76 8 attached attach VERB cana-3483 76 9 with with ADP cana-3483 76 10 another another DET cana-3483 76 11 3 3 NUM cana-3483 76 12 concentric concentric ADJ cana-3483 76 13 n n CCONJ cana-3483 76 14 – – PUNCT cana-3483 76 15 sunlet sunlet NOUN cana-3483 76 16 graph graph NOUN cana-3483 76 17 for for ADP cana-3483 76 18 all all DET cana-3483 76 19 the the DET cana-3483 76 20 outer outer ADJ cana-3483 76 21 graphs graph NOUN cana-3483 76 22 inner inner ADJ cana-3483 76 23 most most ADJ cana-3483 76 24 circle circle NOUN cana-3483 76 25 labelled label VERB cana-3483 76 26 as as ADP cana-3483 76 27 𝛼1 𝛼1 NOUN cana-3483 76 28 , , PUNCT cana-3483 76 29 𝛼2 𝛼2 ADJ cana-3483 76 30 , , PUNCT cana-3483 76 31 … … PUNCT cana-3483 76 32 , , PUNCT cana-3483 76 33 𝛼𝑛 𝛼𝑛 PROPN cana-3483 76 34 and and CCONJ cana-3483 76 35 immediate immediate ADJ cana-3483 76 36 successors successor NOUN cana-3483 76 37 are be AUX cana-3483 76 38 𝛼1β€² 𝛼1β€² ADJ cana-3483 76 39 , , PUNCT cana-3483 76 40 𝛼2β€² 𝛼2β€² VERB cana-3483 76 41 , , PUNCT cana-3483 76 42 … … PUNCT cana-3483 76 43 , , PUNCT cana-3483 76 44 𝛼𝑛′ 𝛼𝑛′ NUM cana-3483 76 45 , , PUNCT cana-3483 76 46 𝛼1 𝛼1 NOUN cana-3483 76 47 " " PUNCT cana-3483 76 48 , , PUNCT cana-3483 76 49 𝛼2 𝛼2 PROPN cana-3483 76 50 " " PUNCT cana-3483 76 51 , , PUNCT cana-3483 76 52 … … PUNCT cana-3483 76 53 , , PUNCT cana-3483 76 54 𝛼𝑛 𝛼𝑛 PROPN cana-3483 76 55 " " PUNCT cana-3483 76 56 and and CCONJ cana-3483 76 57 𝛼1β€²β€²β€² 𝛼1β€²β€²β€² PROPN cana-3483 76 58 , , PUNCT cana-3483 76 59 𝛼2β€²β€²β€² 𝛼2β€²β€²β€² PROPN cana-3483 76 60 , , PUNCT cana-3483 76 61 … … PUNCT cana-3483 76 62 , , PUNCT cana-3483 76 63 𝛼𝑛 𝛼𝑛 PROPN cana-3483 76 64 β€²β€²β€² β€²β€²β€² ADV cana-3483 76 65 respectively respectively ADV cana-3483 76 66 . . PUNCT cana-3483 77 1 to to PART cana-3483 77 2 define define VERB cana-3483 77 3 the the DET cana-3483 77 4 function function NOUN cana-3483 77 5 we we PRON cana-3483 77 6 consider consider VERB cana-3483 77 7 the the DET cana-3483 77 8 following follow VERB cana-3483 77 9 cases case NOUN cana-3483 77 10 for for ADP cana-3483 77 11 apex apex NOUN cana-3483 77 12 graph graph NOUN cana-3483 77 13 and and CCONJ cana-3483 77 14 outer outer ADJ cana-3483 77 15 graphs graph NOUN cana-3483 77 16 . . PUNCT cana-3483 78 1 type type NOUN cana-3483 78 2 equation equation NOUN cana-3483 78 3 here here ADV cana-3483 78 4 . . PUNCT cana-3483 79 1 𝑔(𝑑 𝑔(𝑑 VERB cana-3483 79 2 ) ) PUNCT cana-3483 80 1 = = PRON cana-3483 80 2 { { PUNCT cana-3483 80 3 1 1 NUM cana-3483 80 4 ; ; PUNCT cana-3483 80 5 𝑑 𝑑 X cana-3483 80 6 = = PUNCT cana-3483 80 7 π‘₯𝑖′′′ π‘₯𝑖′′′ PROPN cana-3483 80 8 , , PUNCT cana-3483 80 9 𝑖 𝑖 SYM cana-3483 80 10 = = SYM cana-3483 80 11 1,2 1,2 NUM cana-3483 80 12 , , PUNCT cana-3483 80 13 . . PUNCT cana-3483 80 14 . . PUNCT cana-3483 81 1 , , PUNCT cana-3483 81 2 𝑛 𝑛 PROPN cana-3483 81 3 1 1 NUM cana-3483 81 4 ; ; PUNCT cana-3483 81 5 𝑑 𝑑 X cana-3483 81 6 = = PUNCT cana-3483 81 7 π‘₯𝑖′′ π‘₯𝑖′′ PROPN cana-3483 81 8 , , PUNCT cana-3483 81 9 𝑖 𝑖 NOUN cana-3483 81 10 = = SYM cana-3483 81 11 1,2 1,2 NUM cana-3483 81 12 , , PUNCT cana-3483 81 13 . . PUNCT cana-3483 81 14 . . PUNCT cana-3483 82 1 , , PUNCT cana-3483 82 2 𝑛 𝑛 PROPN cana-3483 82 3 0 0 NUM cana-3483 82 4 ; ; PUNCT cana-3483 82 5 𝑑 𝑑 PROPN cana-3483 82 6 = = SYM cana-3483 82 7 π‘₯𝑖′ π‘₯𝑖′ ADJ cana-3483 82 8 , , PUNCT cana-3483 82 9 𝑖 𝑖 NOUN cana-3483 82 10 = = SYM cana-3483 82 11 1,2 1,2 NUM cana-3483 82 12 , , PUNCT cana-3483 82 13 . . PUNCT cana-3483 82 14 . . PUNCT cana-3483 83 1 , , PUNCT cana-3483 83 2 𝑛 𝑛 PROPN cana-3483 83 3 0 0 NUM cana-3483 83 4 ; ; PUNCT cana-3483 83 5 𝑑 𝑑 PROPN cana-3483 83 6 = = SYM cana-3483 83 7 π‘₯𝑖 π‘₯𝑖 PROPN cana-3483 83 8 , , PUNCT cana-3483 83 9 𝑖 𝑖 NOUN cana-3483 83 10 = = SYM cana-3483 83 11 1,2 1,2 NUM cana-3483 83 12 , , PUNCT cana-3483 83 13 . . PUNCT cana-3483 83 14 . . PUNCT cana-3483 84 1 , , PUNCT cana-3483 84 2 𝑛 𝑛 PROPN cana-3483 84 3 𝑔(𝑑 𝑔(𝑑 PROPN cana-3483 84 4 ) ) PUNCT cana-3483 84 5 = = PRON cana-3483 84 6 { { PUNCT cana-3483 84 7 1 1 NUM cana-3483 84 8 ; ; PUNCT cana-3483 84 9 𝑑 𝑑 PROPN cana-3483 84 10 = = VERB cana-3483 84 11 𝛼𝑖𝑗 𝛼𝑖𝑗 NUM cana-3483 84 12 β€²β€²β€² β€²β€²β€² ADV cana-3483 84 13 , , PUNCT cana-3483 84 14 𝑖 𝑖 NOUN cana-3483 84 15 = = SYM cana-3483 84 16 1,2 1,2 NUM cana-3483 84 17 , , PUNCT cana-3483 84 18 . . PUNCT cana-3483 85 1 . . PUNCT cana-3483 86 1 , , PUNCT cana-3483 86 2 𝑛 𝑛 PROPN cana-3483 86 3 , , PUNCT cana-3483 86 4 𝑗 𝑗 NOUN cana-3483 86 5 = = SYM cana-3483 86 6 1,2 1,2 NUM cana-3483 86 7 , , PUNCT cana-3483 86 8 . . PUNCT cana-3483 86 9 . . PUNCT cana-3483 87 1 , , PUNCT cana-3483 87 2 𝑛 𝑛 PROPN cana-3483 87 3 1 1 NUM cana-3483 87 4 ; ; PUNCT cana-3483 87 5 𝑑 𝑑 PROPN cana-3483 87 6 = = SYM cana-3483 87 7 𝛼𝑖𝑗 𝛼𝑖𝑗 NUM cana-3483 87 8 β€²β€² β€²β€² PROPN cana-3483 87 9 , , PUNCT cana-3483 87 10 𝑖 𝑖 NOUN cana-3483 87 11 = = SYM cana-3483 87 12 1,2 1,2 NUM cana-3483 87 13 , , PUNCT cana-3483 87 14 . . PUNCT cana-3483 87 15 . . PUNCT cana-3483 88 1 , , PUNCT cana-3483 88 2 𝑛 𝑛 PROPN cana-3483 88 3 , , PUNCT cana-3483 88 4 𝑗 𝑗 NOUN cana-3483 88 5 = = SYM cana-3483 88 6 1,2 1,2 NUM cana-3483 88 7 , , PUNCT cana-3483 88 8 . . PUNCT cana-3483 88 9 . . PUNCT cana-3483 89 1 , , PUNCT cana-3483 89 2 𝑛 𝑛 PROPN cana-3483 89 3 0 0 NUM cana-3483 90 1 ; ; PUNCT cana-3483 90 2 𝑑 𝑑 PROPN cana-3483 90 3 = = VERB cana-3483 90 4 𝛼𝑖𝑗 𝛼𝑖𝑗 NUM cana-3483 90 5 β€² β€² NOUN cana-3483 90 6 , , PUNCT cana-3483 90 7 𝑖 𝑖 PUNCT cana-3483 90 8 = = SYM cana-3483 90 9 1,2 1,2 NUM cana-3483 90 10 , , PUNCT cana-3483 90 11 . . PUNCT cana-3483 90 12 . . PUNCT cana-3483 91 1 , , PUNCT cana-3483 91 2 𝑛 𝑛 PROPN cana-3483 91 3 , , PUNCT cana-3483 91 4 𝑗 𝑗 NOUN cana-3483 91 5 = = SYM cana-3483 91 6 1,2 1,2 NUM cana-3483 91 7 , , PUNCT cana-3483 91 8 . . PUNCT cana-3483 91 9 . . PUNCT cana-3483 92 1 , , PUNCT cana-3483 92 2 𝑛 𝑛 PROPN cana-3483 92 3 0 0 NUM cana-3483 92 4 ; ; PUNCT cana-3483 92 5 𝑑 𝑑 PROPN cana-3483 92 6 = = VERB cana-3483 92 7 𝛼𝑖𝑗 𝛼𝑖𝑗 PROPN cana-3483 92 8 , , PUNCT cana-3483 92 9 𝑖 𝑖 NOUN cana-3483 92 10 = = SYM cana-3483 92 11 1,2 1,2 NUM cana-3483 92 12 , , PUNCT cana-3483 92 13 . . PUNCT cana-3483 92 14 . . PUNCT cana-3483 93 1 , , PUNCT cana-3483 93 2 𝑛 𝑛 PROPN cana-3483 93 3 , , PUNCT cana-3483 93 4 𝑗 𝑗 NOUN cana-3483 93 5 = = SYM cana-3483 93 6 1,2 1,2 NUM cana-3483 93 7 , , PUNCT cana-3483 93 8 . . PUNCT cana-3483 93 9 . . PUNCT cana-3483 94 1 , , PUNCT cana-3483 94 2 𝑛 𝑛 X cana-3483 94 3 with with ADP cana-3483 94 4 the the DET cana-3483 94 5 above above ADP cana-3483 94 6 the the DET cana-3483 94 7 label label NOUN cana-3483 94 8 pattern pattern NOUN cana-3483 94 9 satisfies satisfy VERB cana-3483 94 10 the the DET cana-3483 94 11 conditions condition NOUN cana-3483 94 12 |𝛼𝑔(0 |𝛼𝑔(0 ADV cana-3483 94 13 ) ) PUNCT cana-3483 95 1 βˆ’ βˆ’ PROPN cana-3483 95 2 𝛼𝑔(1)| 𝛼𝑔(1)| NOUN cana-3483 95 3 ≀ ≀ NOUN cana-3483 95 4 1 1 NUM cana-3483 95 5 and and CCONJ cana-3483 95 6 |𝛽𝑔(0 |𝛽𝑔(0 NUM cana-3483 95 7 ) ) PUNCT cana-3483 95 8 βˆ’ βˆ’ NOUN cana-3483 95 9 𝛽𝑔(1)| 𝛽𝑔(1)| NOUN cana-3483 95 10 ≀ ≀ NOUN cana-3483 95 11 1 1 NUM cana-3483 95 12 . . PUNCT cana-3483 96 1 i.e. i.e. X cana-3483 96 2 , , PUNCT cana-3483 96 3 this this PRON cana-3483 96 4 means mean VERB cana-3483 96 5 g g PROPN cana-3483 96 6 admits admit VERB cana-3483 96 7 a a DET cana-3483 96 8 product product NOUN cana-3483 96 9 cordial cordial ADJ cana-3483 96 10 labeling labeling NOUN cana-3483 96 11 . . PUNCT cana-3483 97 1 to to PART cana-3483 97 2 better well ADV cana-3483 97 3 understand understand VERB cana-3483 97 4 this this DET cana-3483 97 5 labeling labeling NOUN cana-3483 97 6 pattern pattern NOUN cana-3483 97 7 , , PUNCT cana-3483 97 8 we we PRON cana-3483 97 9 have have AUX cana-3483 97 10 considered consider VERB cana-3483 97 11 below below ADP cana-3483 97 12 example example NOUN cana-3483 97 13 in in ADP cana-3483 97 14 figure figure NOUN cana-3483 97 15 4 4 NUM cana-3483 97 16 of of ADP cana-3483 97 17 even even ADV cana-3483 97 18 number number NOUN cana-3483 97 19 of of ADP cana-3483 97 20 vertices vertex NOUN cana-3483 97 21 and and CCONJ cana-3483 97 22 figure figure VERB cana-3483 97 23 5 5 NUM cana-3483 97 24 of of ADP cana-3483 97 25 odd odd ADJ cana-3483 97 26 number number NOUN cana-3483 97 27 of of ADP cana-3483 97 28 vertices vertex NOUN cana-3483 97 29 . . PUNCT cana-3483 98 1 1 1 X cana-3483 98 2 . . X cana-3483 98 3 𝒏 𝒏 PROPN cana-3483 98 4 = = SYM cana-3483 98 5 πŸ– πŸ– PROPN cana-3483 98 6 , , PUNCT cana-3483 98 7 πŸ‘ πŸ‘ PROPN cana-3483 98 8 βˆ— βˆ— NOUN cana-3483 98 9 π‘ΊπŸ– π‘ΊπŸ– PROPN cana-3483 98 10 βˆ— βˆ— NOUN cana-3483 98 11 figure figure NOUN cana-3483 98 12 4 4 NUM cana-3483 98 13 : : SYM cana-3483 98 14 3 3 NUM cana-3483 98 15 βˆ— βˆ— NOUN cana-3483 98 16 𝑆8 𝑆8 PROPN cana-3483 98 17 βˆ— βˆ— NOUN cana-3483 98 18 communications communication NOUN cana-3483 98 19 on on ADP cana-3483 98 20 applied apply VERB cana-3483 98 21 nonlinear nonlinear ADJ cana-3483 98 22 analysis analysis NOUN cana-3483 98 23 issn issn NOUN cana-3483 98 24 : : PUNCT cana-3483 98 25 1074 1074 NUM cana-3483 98 26 - - PUNCT cana-3483 98 27 133x 133x NUM cana-3483 98 28 vol vol NOUN cana-3483 98 29 32 32 NUM cana-3483 98 30 no no NOUN cana-3483 98 31 . . PUNCT cana-3483 99 1 7s 7 NOUN cana-3483 99 2 ( ( PUNCT cana-3483 99 3 2025 2025 NUM cana-3483 99 4 ) ) PUNCT cana-3483 99 5 763 763 NUM cana-3483 99 6 https://internationalpubls.com https://internationalpubls.com X cana-3483 99 7 2 2 X cana-3483 99 8 . . PUNCT cana-3483 100 1 𝒏 𝒏 PROPN cana-3483 100 2 = = SYM cana-3483 100 3 πŸ• πŸ• PROPN cana-3483 100 4 , , PUNCT cana-3483 100 5 πŸ‘ πŸ‘ NUM cana-3483 100 6 βˆ— βˆ— NOUN cana-3483 100 7 π‘ΊπŸ• π‘ΊπŸ• NOUN cana-3483 100 8 βˆ— βˆ— NOUN cana-3483 100 9 figure figure NOUN cana-3483 100 10 5 5 NUM cana-3483 100 11 : : SYM cana-3483 100 12 3 3 NUM cana-3483 100 13 βˆ— βˆ— NOUN cana-3483 100 14 𝑆8 𝑆8 PROPN cana-3483 100 15 βˆ— βˆ— NOUN cana-3483 100 16 theorem theorem VERB cana-3483 100 17 3.3 3.3 NUM cana-3483 100 18 . . PUNCT cana-3483 101 1 a a DET cana-3483 101 2 graph graph NOUN cana-3483 101 3 obtained obtain VERB cana-3483 101 4 by by ADP cana-3483 101 5 joining join VERB cana-3483 101 6 two two NUM cana-3483 101 7 copies copy NOUN cana-3483 101 8 of of ADP cana-3483 101 9 shell shell ADJ cana-3483 101 10 graph graph NOUN cana-3483 101 11 2*𝐢(𝑛,π‘›βˆ’3 2*𝐢(𝑛,π‘›βˆ’3 NUM cana-3483 101 12 ) ) PUNCT cana-3483 101 13 , , PUNCT cana-3483 101 14 odd odd ADJ cana-3483 101 15 n n PRON cana-3483 101 16 is be AUX cana-3483 101 17 product product NOUN cana-3483 101 18 cordial cordial ADJ cana-3483 101 19 graph graph NOUN cana-3483 101 20 . . PUNCT cana-3483 102 1 proof proof NOUN cana-3483 102 2 : : PUNCT cana-3483 102 3 the the DET cana-3483 102 4 labeling labeling NOUN cana-3483 102 5 created create VERB cana-3483 102 6 by by ADP cana-3483 102 7 participating participate VERB cana-3483 102 8 two two NUM cana-3483 102 9 replicas replica NOUN cana-3483 102 10 by by ADP cana-3483 102 11 path path NOUN cana-3483 102 12 of of ADP cana-3483 102 13 shell shell ADJ cana-3483 102 14 graph graph NOUN cana-3483 102 15 . . PUNCT cana-3483 103 1 in in ADP cana-3483 103 2 this this DET cana-3483 103 3 graph graph NOUN cana-3483 103 4 we we PRON cana-3483 103 5 consider consider VERB cana-3483 103 6 1st 1st ADJ cana-3483 103 7 copy copy NOUN cana-3483 103 8 of of ADP cana-3483 103 9 shell shell ADJ cana-3483 103 10 graph graph NOUN cana-3483 103 11 as as ADP cana-3483 103 12 𝛼1 𝛼1 NOUN cana-3483 103 13 , , PUNCT cana-3483 103 14 𝛼2 𝛼2 ADJ cana-3483 103 15 , , PUNCT cana-3483 103 16 … … PUNCT cana-3483 103 17 , , PUNCT cana-3483 103 18 𝛼𝑛 𝛼𝑛 PROPN cana-3483 103 19 and and CCONJ cana-3483 103 20 the the DET cana-3483 103 21 vertices vertex NOUN cana-3483 103 22 of of ADP cana-3483 103 23 2nd 2nd ADJ cana-3483 103 24 copy copy NOUN cana-3483 103 25 of of ADP cana-3483 103 26 shell shell ADJ cana-3483 103 27 graph graph NOUN cana-3483 103 28 is be AUX cana-3483 103 29 π‘₯1 π‘₯1 NOUN cana-3483 103 30 , , PUNCT cana-3483 103 31 π‘₯2 π‘₯2 NOUN cana-3483 103 32 , , PUNCT cana-3483 103 33 … … PUNCT cana-3483 103 34 , , PUNCT cana-3483 103 35 π‘₯𝑛. π‘₯𝑛. VERB cana-3483 103 36 to to PART cana-3483 103 37 define define VERB cana-3483 103 38 𝑔 𝑔 PROPN cana-3483 103 39 ∢ ∢ PROPN cana-3483 103 40 𝛼 𝛼 X cana-3483 103 41 ( ( PUNCT cana-3483 103 42 𝐢(𝑛,π‘›βˆ’3 𝐢(𝑛,π‘›βˆ’3 PROPN cana-3483 103 43 ) ) PUNCT cana-3483 103 44 ) ) PUNCT cana-3483 103 45 β†’ β†’ PUNCT cana-3483 103 46 { { PUNCT cana-3483 103 47 0 0 NUM cana-3483 103 48 , , PUNCT cana-3483 103 49 1 1 NUM cana-3483 103 50 } } PUNCT cana-3483 103 51 , , PUNCT cana-3483 103 52 we we PRON cana-3483 103 53 consider consider VERB cana-3483 103 54 the the DET cana-3483 103 55 following follow VERB cana-3483 103 56 case case NOUN cana-3483 103 57 . . PUNCT cana-3483 104 1 𝑔(𝑑 𝑔(𝑑 VERB cana-3483 104 2 ) ) PUNCT cana-3483 105 1 = = PRON cana-3483 105 2 { { PUNCT cana-3483 105 3 1 1 NUM cana-3483 105 4 ; ; PUNCT cana-3483 105 5 𝑑 𝑑 PROPN cana-3483 105 6 = = SYM cana-3483 105 7 𝛼𝑖 𝛼𝑖 PROPN cana-3483 105 8 , , PUNCT cana-3483 105 9 𝑖 𝑖 PROPN cana-3483 105 10 = = SYM cana-3483 105 11 1,2 1,2 NUM cana-3483 105 12 , , PUNCT cana-3483 105 13 . . PUNCT cana-3483 105 14 . . PUNCT cana-3483 106 1 , , PUNCT cana-3483 106 2 𝑛 𝑛 PROPN cana-3483 106 3 0 0 NUM cana-3483 106 4 ; ; PUNCT cana-3483 106 5 𝑑 𝑑 PROPN cana-3483 106 6 = = SYM cana-3483 106 7 π‘₯𝑖 π‘₯𝑖 PROPN cana-3483 106 8 , , PUNCT cana-3483 106 9 𝑖 𝑖 NOUN cana-3483 106 10 = = SYM cana-3483 106 11 1,2 1,2 NUM cana-3483 106 12 , , PUNCT cana-3483 106 13 . . PUNCT cana-3483 106 14 . . PUNCT cana-3483 107 1 , , PUNCT cana-3483 107 2 𝑛 𝑛 DET cana-3483 107 3 the the DET cana-3483 107 4 conditions condition NOUN cana-3483 107 5 are be AUX cana-3483 107 6 satisfied satisfied ADJ cana-3483 107 7 with with ADP cana-3483 107 8 the the DET cana-3483 107 9 above above ADJ cana-3483 107 10 labeling labeling NOUN cana-3483 107 11 pattern pattern NOUN cana-3483 107 12 |𝛼𝑔(0 |𝛼𝑔(0 ADV cana-3483 107 13 ) ) PUNCT cana-3483 108 1 βˆ’ βˆ’ PROPN cana-3483 108 2 𝛼𝑔(1)| 𝛼𝑔(1)| NOUN cana-3483 108 3 ≀ ≀ NOUN cana-3483 108 4 1 1 NUM cana-3483 108 5 and and CCONJ cana-3483 108 6 |𝛽𝑔(0 |𝛽𝑔(0 NUM cana-3483 108 7 ) ) PUNCT cana-3483 108 8 βˆ’ βˆ’ NOUN cana-3483 108 9 𝛽𝑔(1)| 𝛽𝑔(1)| NOUN cana-3483 108 10 ≀ ≀ NOUN cana-3483 108 11 1 1 NUM cana-3483 108 12 . . PUNCT cana-3483 109 1 i.e. i.e. X cana-3483 109 2 , , PUNCT cana-3483 109 3 this this PRON cana-3483 109 4 shows show VERB cana-3483 109 5 g g PROPN cana-3483 109 6 admits admit VERB cana-3483 109 7 a a DET cana-3483 109 8 product product NOUN cana-3483 109 9 cordial cordial ADJ cana-3483 109 10 labeling labeling NOUN cana-3483 109 11 . . PUNCT cana-3483 110 1 to to PART cana-3483 110 2 better well ADV cana-3483 110 3 understand understand VERB cana-3483 110 4 this this DET cana-3483 110 5 labeling labeling NOUN cana-3483 110 6 pattern pattern NOUN cana-3483 110 7 , , PUNCT cana-3483 110 8 we we PRON cana-3483 110 9 have have AUX cana-3483 110 10 considered consider VERB cana-3483 110 11 below below ADP cana-3483 110 12 example example NOUN cana-3483 110 13 in in ADP cana-3483 110 14 figure figure NOUN cana-3483 110 15 6 6 NUM cana-3483 110 16 and and CCONJ cana-3483 110 17 figure figure VERB cana-3483 110 18 7 7 NUM cana-3483 110 19 of of ADP cana-3483 110 20 odd odd ADJ cana-3483 110 21 number number NOUN cana-3483 110 22 of of ADP cana-3483 110 23 vertices vertex NOUN cana-3483 110 24 . . PUNCT cana-3483 111 1 1 1 X cana-3483 111 2 . . X cana-3483 111 3 n=11 n=11 NOUN cana-3483 111 4 figure figure NOUN cana-3483 111 5 6 6 NUM cana-3483 111 6 : : PUNCT cana-3483 111 7 two two NUM cana-3483 111 8 copies copy NOUN cana-3483 111 9 of of ADP cana-3483 111 10 𝐢(11,8 𝐢(11,8 NOUN cana-3483 111 11 ) ) PUNCT cana-3483 111 12 communications communication NOUN cana-3483 111 13 on on ADP cana-3483 111 14 applied apply VERB cana-3483 111 15 nonlinear nonlinear ADJ cana-3483 111 16 analysis analysis NOUN cana-3483 111 17 issn issn NOUN cana-3483 111 18 : : PUNCT cana-3483 111 19 1074 1074 NUM cana-3483 111 20 - - PUNCT cana-3483 111 21 133x 133x NUM cana-3483 111 22 vol vol NOUN cana-3483 111 23 32 32 NUM cana-3483 111 24 no no NOUN cana-3483 111 25 . . PUNCT cana-3483 112 1 7s 7 NOUN cana-3483 112 2 ( ( PUNCT cana-3483 112 3 2025 2025 NUM cana-3483 112 4 ) ) PUNCT cana-3483 112 5 764 764 NUM cana-3483 112 6 https://internationalpubls.com https://internationalpubls.com X cana-3483 112 7 2 2 X cana-3483 112 8 . . PUNCT cana-3483 113 1 n=13 n=13 PROPN cana-3483 113 2 figure figure VERB cana-3483 113 3 7 7 NUM cana-3483 113 4 : : SYM cana-3483 113 5 two two NUM cana-3483 113 6 copies copy NOUN cana-3483 113 7 of of ADP cana-3483 113 8 𝐢(13,10 𝐢(13,10 NOUN cana-3483 113 9 ) ) PUNCT cana-3483 113 10 4 4 NUM cana-3483 113 11 . . X cana-3483 113 12 concluding conclude VERB cana-3483 113 13 remarks remark NOUN cana-3483 113 14 product product VERB cana-3483 113 15 cordial cordial ADJ cana-3483 113 16 labeling labeling NOUN cana-3483 113 17 in in ADP cana-3483 113 18 graph graph NOUN cana-3483 113 19 theory theory NOUN cana-3483 113 20 , , PUNCT cana-3483 113 21 involves involve VERB cana-3483 113 22 labeling labeling NOUN cana-3483 113 23 operations operation NOUN cana-3483 113 24 on on ADP cana-3483 113 25 graphs graph NOUN cana-3483 113 26 such such ADJ cana-3483 113 27 as as ADP cana-3483 113 28 duplication duplication NOUN cana-3483 113 29 , , PUNCT cana-3483 113 30 fusion fusion NOUN cana-3483 113 31 , , PUNCT cana-3483 113 32 or or CCONJ cana-3483 113 33 reconstruction reconstruction NOUN cana-3483 113 34 . . PUNCT cana-3483 114 1 these these DET cana-3483 114 2 operations operation NOUN cana-3483 114 3 are be AUX cana-3483 114 4 used use VERB cana-3483 114 5 to to PART cana-3483 114 6 manipulate manipulate VERB cana-3483 114 7 graphs graph NOUN cana-3483 114 8 in in ADP cana-3483 114 9 various various ADJ cana-3483 114 10 ways way NOUN cana-3483 114 11 to to PART cana-3483 114 12 potentially potentially ADV cana-3483 114 13 enhance enhance VERB cana-3483 114 14 outcomes outcome NOUN cana-3483 114 15 or or CCONJ cana-3483 114 16 achieve achieve VERB cana-3483 114 17 specific specific ADJ cana-3483 114 18 research research NOUN cana-3483 114 19 goals goal NOUN cana-3483 114 20 . . PUNCT cana-3483 115 1 references reference NOUN cana-3483 115 2 [ [ X cana-3483 115 3 1 1 NUM cana-3483 115 4 ] ] PUNCT cana-3483 115 5 acharya acharya NOUN cana-3483 115 6 , , PUNCT cana-3483 115 7 m. m. NOUN cana-3483 115 8 and and CCONJ cana-3483 115 9 kureethara kureethara PROPN cana-3483 115 10 , , PUNCT cana-3483 115 11 j.v j.v PROPN cana-3483 115 12 . . PUNCT cana-3483 115 13 ( ( PUNCT cana-3483 115 14 2024 2024 NUM cana-3483 115 15 ) ) PUNCT cana-3483 115 16 . . PUNCT cana-3483 116 1 characterization characterization NOUN cana-3483 116 2 of of ADP cana-3483 116 3 product product NOUN cana-3483 116 4 cordial cordial ADJ cana-3483 116 5 dragon dragon NOUN cana-3483 116 6 graphs graph NOUN cana-3483 116 7 . . PUNCT cana-3483 117 1 communications communication NOUN cana-3483 117 2 in in ADP cana-3483 117 3 combinatorics combinatoric NOUN cana-3483 117 4 and and CCONJ cana-3483 117 5 optimization optimization NOUN cana-3483 117 6 , , PUNCT cana-3483 117 7 9(4 9(4 NUM cana-3483 117 8 ) ) PUNCT cana-3483 117 9 , , PUNCT cana-3483 117 10 pp.805 pp.805 PROPN cana-3483 117 11 - - PUNCT cana-3483 117 12 811 811 NUM cana-3483 117 13 . . PUNCT cana-3483 118 1 [ [ X cana-3483 118 2 2 2 NUM cana-3483 118 3 ] ] X cana-3483 118 4 cahit cahit ADJ cana-3483 118 5 , , PUNCT cana-3483 118 6 i.(1958 i.(1958 NOUN cana-3483 118 7 ) ) PUNCT cana-3483 118 8 . . PUNCT cana-3483 119 1 a a DET cana-3483 119 2 short short ADJ cana-3483 119 3 proof proof NOUN cana-3483 119 4 of of ADP cana-3483 119 5 groetzsch groetzsch PROPN cana-3483 119 6 ’s ’s PART cana-3483 119 7 three three NUM cana-3483 119 8 color color NOUN cana-3483 119 9 theorem theorem VERB cana-3483 119 10 . . PUNCT cana-3483 120 1 [ [ X cana-3483 120 2 3 3 NUM cana-3483 120 3 ] ] SYM cana-3483 120 4 chartrand chartrand NOUN cana-3483 120 5 , , PUNCT cana-3483 120 6 g. g. PROPN cana-3483 120 7 , , PUNCT cana-3483 120 8 salehi salehi PROPN cana-3483 120 9 , , PUNCT cana-3483 120 10 e. e. PROPN cana-3483 120 11 and and CCONJ cana-3483 120 12 zhang zhang PROPN cana-3483 120 13 , , PUNCT cana-3483 120 14 p. p. PROPN cana-3483 120 15 , , PUNCT cana-3483 120 16 2024 2024 NUM cana-3483 120 17 . . PUNCT cana-3483 121 1 on on ADP cana-3483 121 2 subset subset ADJ cana-3483 121 3 labelings labeling NOUN cana-3483 121 4 of of ADP cana-3483 121 5 trees tree NOUN cana-3483 121 6 . . PUNCT cana-3483 122 1 akce akce PROPN cana-3483 122 2 international international PROPN cana-3483 122 3 journal journal NOUN cana-3483 122 4 of of ADP cana-3483 122 5 graphs graph NOUN cana-3483 122 6 and and CCONJ cana-3483 122 7 combinatorics combinatoric NOUN cana-3483 122 8 , , PUNCT cana-3483 122 9 21(1 21(1 NUM cana-3483 122 10 ) ) PUNCT cana-3483 122 11 , , PUNCT cana-3483 122 12 pp.84 pp.84 NOUN cana-3483 122 13 - - PUNCT cana-3483 122 14 90 90 NUM cana-3483 122 15 . . PUNCT cana-3483 123 1 [ [ X cana-3483 123 2 4 4 NUM cana-3483 123 3 ] ] PUNCT cana-3483 123 4 gallian gallian NOUN cana-3483 123 5 , , PUNCT cana-3483 123 6 j. j. PROPN cana-3483 123 7 a. a. PROPN cana-3483 123 8 ( ( PUNCT cana-3483 123 9 2018 2018 NUM cana-3483 123 10 ) ) PUNCT cana-3483 123 11 . . PUNCT cana-3483 124 1 a a DET cana-3483 124 2 dynamic dynamic ADJ cana-3483 124 3 survey survey NOUN cana-3483 124 4 of of ADP cana-3483 124 5 graph graph NOUN cana-3483 124 6 labeling labeling NOUN cana-3483 124 7 . . PUNCT cana-3483 125 1 electronic electronic ADJ cana-3483 125 2 journal journal NOUN cana-3483 125 3 of of ADP cana-3483 125 4 combinatorics combinatoric NOUN cana-3483 125 5 , , PUNCT cana-3483 125 6 1(dynamicsurveys 1(dynamicsurveys NUM cana-3483 125 7 ) ) PUNCT cana-3483 125 8 , , PUNCT cana-3483 125 9 ds6 ds6 NOUN cana-3483 125 10 . . PUNCT cana-3483 126 1 [ [ X cana-3483 126 2 5 5 NUM cana-3483 126 3 ] ] PUNCT cana-3483 126 4 mathew mathew PROPN cana-3483 126 5 , , PUNCT cana-3483 126 6 a. a. NOUN cana-3483 126 7 ( ( PUNCT cana-3483 126 8 2023 2023 NUM cana-3483 126 9 ) ) PUNCT cana-3483 126 10 . . PUNCT cana-3483 127 1 vector vector NOUN cana-3483 127 2 valued value VERB cana-3483 127 3 switching switch VERB cana-3483 127 4 in in ADP cana-3483 127 5 the the DET cana-3483 127 6 products product NOUN cana-3483 127 7 of of ADP cana-3483 127 8 signed sign VERB cana-3483 127 9 graphs graph NOUN cana-3483 127 10 . . PUNCT cana-3483 128 1 arxiv arxiv PROPN cana-3483 128 2 preprint preprint NOUN cana-3483 128 3 arxiv:2306.10132 arxiv:2306.10132 NOUN cana-3483 128 4 . . PUNCT cana-3483 129 1 [ [ X cana-3483 129 2 6 6 NUM cana-3483 129 3 ] ] X cana-3483 129 4 meena meena PROPN cana-3483 129 5 , , PUNCT cana-3483 129 6 s. s. PROPN cana-3483 129 7 , , PUNCT cana-3483 129 8 renugha renugha PROPN cana-3483 129 9 , , PUNCT cana-3483 129 10 m. m. NOUN cana-3483 129 11 and and CCONJ cana-3483 129 12 sivasakthi sivasakthi NOUN cana-3483 129 13 , , PUNCT cana-3483 129 14 m. m. NOUN cana-3483 129 15 ( ( PUNCT cana-3483 129 16 2015 2015 NUM cana-3483 129 17 ) ) PUNCT cana-3483 129 18 . . PUNCT cana-3483 130 1 cordial cordial ADJ cana-3483 130 2 labeling labeling NOUN cana-3483 130 3 for for ADP cana-3483 130 4 different different ADJ cana-3483 130 5 types type NOUN cana-3483 130 6 of of ADP cana-3483 130 7 shell shell ADJ cana-3483 130 8 graph graph NOUN cana-3483 130 9 . . PUNCT cana-3483 131 1 international international ADJ cana-3483 131 2 journal journal PROPN cana-3483 131 3 of of ADP cana-3483 131 4 scientific scientific ADJ cana-3483 131 5 and and CCONJ cana-3483 131 6 engineering engineering NOUN cana-3483 131 7 research research NOUN cana-3483 131 8 , , PUNCT cana-3483 131 9 6(9 6(9 NUM cana-3483 131 10 ) ) PUNCT cana-3483 131 11 , , PUNCT cana-3483 131 12 pp.1282 pp.1282 PROPN cana-3483 131 13 - - PUNCT cana-3483 131 14 1288 1288 NUM cana-3483 131 15 . . PUNCT cana-3483 132 1 [ [ X cana-3483 132 2 7 7 NUM cana-3483 132 3 ] ] X cana-3483 132 4 prajapati prajapati PROPN cana-3483 132 5 , , PUNCT cana-3483 132 6 u. u. PROPN cana-3483 132 7 m. m. PROPN cana-3483 132 8 , , PUNCT cana-3483 132 9 & & CCONJ cana-3483 132 10 raval raval NOUN cana-3483 132 11 , , PUNCT cana-3483 132 12 k. k. PROPN cana-3483 132 13 k. k. PROPN cana-3483 132 14 ( ( PUNCT cana-3483 132 15 2017 2017 NUM cana-3483 132 16 ) ) PUNCT cana-3483 132 17 . . PUNCT cana-3483 133 1 some some DET cana-3483 133 2 graphs graph NOUN cana-3483 133 3 in in ADP cana-3483 133 4 the the DET cana-3483 133 5 context context NOUN cana-3483 133 6 of of ADP cana-3483 133 7 product product NOUN cana-3483 133 8 cordial cordial ADJ cana-3483 133 9 labeling labeling NOUN cana-3483 133 10 . . PUNCT cana-3483 134 1 math math NOUN cana-3483 134 2 . . PUNCT cana-3483 135 1 today today NOUN cana-3483 135 2 , , PUNCT cana-3483 135 3 33 33 NUM cana-3483 135 4 , , PUNCT cana-3483 135 5 5866 5866 NUM cana-3483 135 6 . . PUNCT cana-3483 136 1 [ [ X cana-3483 136 2 8 8 NUM cana-3483 136 3 ] ] X cana-3483 136 4 raja raja PROPN cana-3483 136 5 , , PUNCT cana-3483 136 6 m.r m.r PROPN cana-3483 136 7 . . PROPN cana-3483 136 8 , , PUNCT cana-3483 136 9 mangam mangam PROPN cana-3483 136 10 , , PUNCT cana-3483 136 11 t.a t.a PROPN cana-3483 136 12 . . PROPN cana-3483 136 13 and and CCONJ cana-3483 136 14 naduvath naduvath NOUN cana-3483 136 15 , , PUNCT cana-3483 136 16 s. s. PROPN cana-3483 136 17 , , PUNCT cana-3483 136 18 2022 2022 NUM cana-3483 136 19 . . PUNCT cana-3483 137 1 eccentric eccentric ADJ cana-3483 137 2 completion completion NOUN cana-3483 137 3 of of ADP cana-3483 137 4 a a DET cana-3483 137 5 graph graph NOUN cana-3483 137 6 . . PUNCT cana-3483 138 1 communications communication NOUN cana-3483 138 2 in in ADP cana-3483 138 3 combinatorics combinatoric NOUN cana-3483 138 4 and and CCONJ cana-3483 138 5 optimization optimization NOUN cana-3483 138 6 , , PUNCT cana-3483 138 7 7(2 7(2 NUM cana-3483 138 8 ) ) PUNCT cana-3483 138 9 , , PUNCT cana-3483 138 10 pp.193 pp.193 NOUN cana-3483 138 11 - - PUNCT cana-3483 138 12 201 201 NUM cana-3483 138 13 . . PUNCT cana-3483 139 1 [ [ X cana-3483 139 2 9 9 NUM cana-3483 139 3 ] ] PUNCT cana-3483 139 4 sadawarte sadawarte NOUN cana-3483 139 5 , , PUNCT cana-3483 139 6 s. s. PROPN cana-3483 139 7 and and CCONJ cana-3483 139 8 srivastav srivastav PROPN cana-3483 139 9 , , PUNCT cana-3483 139 10 s. s. PROPN cana-3483 139 11 , , PUNCT cana-3483 139 12 2024 2024 NUM cana-3483 139 13 . . PUNCT cana-3483 140 1 on on ADP cana-3483 140 2 sum sum VERB cana-3483 140 3 3‐equitable 3‐equitable NUM cana-3483 140 4 labeling labeling NOUN cana-3483 140 5 of of ADP cana-3483 140 6 some some DET cana-3483 140 7 graphs graph NOUN cana-3483 140 8 . . PUNCT cana-3483 141 1 mathematical mathematical ADJ cana-3483 141 2 modeling modeling NOUN cana-3483 141 3 for for ADP cana-3483 141 4 computer computer NOUN cana-3483 141 5 applications application NOUN cana-3483 141 6 , , PUNCT cana-3483 141 7 pp.197 pp.197 NOUN cana-3483 141 8 - - PUNCT cana-3483 141 9 205 205 NUM cana-3483 141 10 . . PUNCT cana-3483 142 1 [ [ X cana-3483 142 2 10 10 NUM cana-3483 142 3 ] ] X cana-3483 142 4 srivastav srivastav NOUN cana-3483 142 5 , , PUNCT cana-3483 142 6 s. s. PROPN cana-3483 142 7 and and CCONJ cana-3483 142 8 gupta gupta PROPN cana-3483 142 9 , , PUNCT cana-3483 142 10 s. s. PROPN cana-3483 142 11 , , PUNCT cana-3483 142 12 2019 2019 NUM cana-3483 142 13 . . PUNCT cana-3483 143 1 divisor divisor NOUN cana-3483 143 2 3 3 NUM cana-3483 143 3 - - PUNCT cana-3483 143 4 equitable equitable ADJ cana-3483 143 5 labeling labeling NOUN cana-3483 143 6 of of ADP cana-3483 143 7 graphs graph NOUN cana-3483 143 8 . . PUNCT cana-3483 144 1 international international ADJ cana-3483 144 2 journal journal NOUN cana-3483 144 3 of of ADP cana-3483 144 4 computer computer NOUN cana-3483 144 5 science science NOUN cana-3483 144 6 and and CCONJ cana-3483 144 7 information information NOUN cana-3483 144 8 security security NOUN cana-3483 144 9 ( ( PUNCT cana-3483 144 10 ijcsis ijcsis NOUN cana-3483 144 11 ) ) PUNCT cana-3483 144 12 , , PUNCT cana-3483 144 13 17(2 17(2 NUM cana-3483 144 14 ) ) PUNCT cana-3483 144 15 . . PUNCT cana-3483 145 1 [ [ X cana-3483 145 2 11 11 NUM cana-3483 145 3 ] ] PUNCT cana-3483 145 4 sundaram sundaram PROPN cana-3483 145 5 , , PUNCT cana-3483 145 6 m. m. NOUN cana-3483 145 7 , , PUNCT cana-3483 145 8 ponraj ponraj VERB cana-3483 145 9 , , PUNCT cana-3483 145 10 r. r. PROPN cana-3483 145 11 , , PUNCT cana-3483 145 12 & & CCONJ cana-3483 145 13 somasundaram somasundaram PROPN cana-3483 145 14 , , PUNCT cana-3483 145 15 s. s. PROPN cana-3483 145 16 ( ( PUNCT cana-3483 145 17 2004 2004 NUM cana-3483 145 18 ) ) PUNCT cana-3483 145 19 . . PUNCT cana-3483 146 1 product product NOUN cana-3483 146 2 cordial cordial ADJ cana-3483 146 3 labeling labeling NOUN cana-3483 146 4 of of ADP cana-3483 146 5 graphs graph NOUN cana-3483 146 6 . . PUNCT cana-3483 147 1 bulletin bulletin NOUN cana-3483 147 2 of of ADP cana-3483 147 3 pure pure ADJ cana-3483 147 4 and and CCONJ cana-3483 147 5 applied apply VERB cana-3483 147 6 sciences science NOUN cana-3483 147 7 e e NOUN cana-3483 147 8 , , PUNCT cana-3483 147 9 23 23 NUM cana-3483 147 10 , , PUNCT cana-3483 147 11 155 155 NUM cana-3483 147 12 - - SYM cana-3483 147 13 163 163 NUM cana-3483 147 14 . . PUNCT cana-3483 148 1 [ [ X cana-3483 148 2 12 12 NUM cana-3483 148 3 ] ] X cana-3483 148 4 vaidya vaidya PROPN cana-3483 148 5 , , PUNCT cana-3483 148 6 s.k s.k PROPN cana-3483 148 7 . . PROPN cana-3483 148 8 and and CCONJ cana-3483 148 9 vyas vyas PROPN cana-3483 148 10 , , PUNCT cana-3483 148 11 n.b n.b PROPN cana-3483 148 12 . . PROPN cana-3483 148 13 ( ( PUNCT cana-3483 148 14 2011 2011 NUM cana-3483 148 15 ) ) PUNCT cana-3483 148 16 . . PUNCT cana-3483 149 1 product product NOUN cana-3483 149 2 cordial cordial ADJ cana-3483 149 3 labeling labeling NOUN cana-3483 149 4 in in ADP cana-3483 149 5 the the DET cana-3483 149 6 context context NOUN cana-3483 149 7 of of ADP cana-3483 149 8 tensor tensor NOUN cana-3483 149 9 product product NOUN cana-3483 149 10 of of ADP cana-3483 149 11 graphs graph NOUN cana-3483 149 12 . . PUNCT cana-3483 150 1 journal journal NOUN cana-3483 150 2 of of ADP cana-3483 150 3 mathematics mathematics PROPN cana-3483 150 4 research research NOUN cana-3483 150 5 , , PUNCT cana-3483 150 6 3(3 3(3 NUM cana-3483 150 7 ) ) PUNCT cana-3483 150 8 , , PUNCT cana-3483 150 9 pp.83 pp.83 PROPN cana-3483 150 10 - - PUNCT cana-3483 150 11 88 88 NUM cana-3483 150 12 . . PUNCT cana-3483 151 1 [ [ X cana-3483 151 2 13 13 NUM cana-3483 151 3 ] ] X cana-3483 151 4 vaidya vaidya PROPN cana-3483 151 5 , , PUNCT cana-3483 151 6 s.k s.k PROPN cana-3483 151 7 . . PROPN cana-3483 151 8 and and CCONJ cana-3483 151 9 barasara barasara PROPN cana-3483 151 10 , , PUNCT cana-3483 151 11 c.m c.m NOUN cana-3483 151 12 . . PROPN cana-3483 151 13 ( ( PUNCT cana-3483 151 14 2011 2011 NUM cana-3483 151 15 ) ) PUNCT cana-3483 151 16 . . PUNCT cana-3483 151 17 product product NOUN cana-3483 151 18 cordial cordial ADJ cana-3483 151 19 labeling labeling NOUN cana-3483 151 20 for for ADP cana-3483 151 21 some some DET cana-3483 151 22 new new ADJ cana-3483 151 23 graphs graph NOUN cana-3483 151 24 . . PUNCT cana-3483 152 1 journal journal NOUN cana-3483 152 2 of of ADP cana-3483 152 3 mathematics mathematics PROPN cana-3483 152 4 research research NOUN cana-3483 152 5 , , PUNCT cana-3483 152 6 3(2 3(2 NUM cana-3483 152 7 ) ) PUNCT cana-3483 152 8 , , PUNCT cana-3483 152 9 pp.206 pp.206 NOUN cana-3483 152 10 - - PUNCT cana-3483 152 11 211 211 NUM cana-3483 152 12 . . PUNCT cana-3483 153 1 [ [ X cana-3483 153 2 14 14 NUM cana-3483 153 3 ] ] X cana-3483 153 4 vaidya vaidya PROPN cana-3483 153 5 , , PUNCT cana-3483 153 6 s.k s.k PROPN cana-3483 153 7 . . PROPN cana-3483 153 8 and and CCONJ cana-3483 153 9 dani dani PROPN cana-3483 153 10 , , PUNCT cana-3483 153 11 n.a n.a PROPN cana-3483 153 12 . . PROPN cana-3483 153 13 ( ( PUNCT cana-3483 153 14 2010 2010 NUM cana-3483 153 15 ) ) PUNCT cana-3483 153 16 . . PUNCT cana-3483 154 1 some some DET cana-3483 154 2 new new ADJ cana-3483 154 3 product product NOUN cana-3483 154 4 cordial cordial ADJ cana-3483 154 5 graphs graph NOUN cana-3483 154 6 . . PUNCT cana-3483 155 1 journal journal NOUN cana-3483 155 2 of of ADP cana-3483 155 3 app app PROPN cana-3483 155 4 . . PUNCT cana-3483 156 1 comp comp PROPN cana-3483 156 2 . . PUNCT cana-3483 157 1 sci sci PROPN cana-3483 157 2 . . PROPN cana-3483 157 3 math math PROPN cana-3483 157 4 , , PUNCT cana-3483 157 5 8(4 8(4 NUM cana-3483 157 6 ) ) PUNCT cana-3483 157 7 , , PUNCT cana-3483 157 8 pp.6265 pp.6265 NOUN cana-3483 157 9 . . PUNCT cana-3483 158 1 [ [ X cana-3483 158 2 15 15 NUM cana-3483 158 3 ] ] X cana-3483 158 4 vaidya vaidya PROPN cana-3483 158 5 , , PUNCT cana-3483 158 6 s.k s.k PROPN cana-3483 158 7 . . PROPN cana-3483 158 8 and and CCONJ cana-3483 158 9 kanani kanani PROPN cana-3483 158 10 , , PUNCT cana-3483 158 11 k.k k.k PROPN cana-3483 158 12 . . PROPN cana-3483 158 13 ( ( PUNCT cana-3483 158 14 2010 2010 NUM cana-3483 158 15 ) ) PUNCT cana-3483 158 16 . . PUNCT cana-3483 159 1 some some DET cana-3483 159 2 cycle cycle NOUN cana-3483 159 3 related relate VERB cana-3483 159 4 product product NOUN cana-3483 159 5 cordial cordial ADJ cana-3483 159 6 graphs graph NOUN cana-3483 159 7 . . PUNCT cana-3483 160 1 int int NOUN cana-3483 160 2 . . PUNCT cana-3483 161 1 j. j. PROPN cana-3483 161 2 of of ADP cana-3483 161 3 algorithms algorithms PROPN cana-3483 161 4 , , PUNCT cana-3483 161 5 comp comp NOUN cana-3483 161 6 . . PUNCT cana-3483 162 1 and and CCONJ cana-3483 162 2 math math NOUN cana-3483 162 3 , , PUNCT cana-3483 162 4 3(1 3(1 NUM cana-3483 162 5 ) ) PUNCT cana-3483 162 6 , , PUNCT cana-3483 162 7 pp.109 pp.109 PROPN cana-3483 162 8 - - NOUN cana-3483 162 9 116 116 NUM cana-3483 162 10 . . PUNCT