id sid tid token lemma pos cana-1654 1 1 communications communication NOUN cana-1654 1 2 on on ADP cana-1654 1 3 applied apply VERB cana-1654 1 4 nonlinear nonlinear ADJ cana-1654 1 5 analysis analysis NOUN cana-1654 1 6 issn issn NOUN cana-1654 1 7 : : PUNCT cana-1654 1 8 1074 1074 NUM cana-1654 1 9 - - PUNCT cana-1654 1 10 133x 133x NUM cana-1654 1 11 vol vol NOUN cana-1654 1 12 32 32 NUM cana-1654 1 13 no no NOUN cana-1654 1 14 . . NOUN cana-1654 1 15 1 1 NUM cana-1654 1 16 ( ( PUNCT cana-1654 1 17 2025 2025 NUM cana-1654 1 18 ) ) PUNCT cana-1654 1 19 335 335 NUM cana-1654 1 20 https://internationalpubls.com https://internationalpubls.com X cana-1654 1 21 odd odd ADV cana-1654 1 22 - - PUNCT cana-1654 1 23 even even ADV cana-1654 1 24 congruence congruence NOUN cana-1654 1 25 labeling labeling NOUN cana-1654 1 26 of of ADP cana-1654 1 27 some some DET cana-1654 1 28 graphs graph NOUN cana-1654 1 29 rituraj1 rituraj1 NOUN cana-1654 1 30 , , PUNCT cana-1654 1 31 shambhu shambhu ADJ cana-1654 1 32 kumar kumar PROPN cana-1654 1 33 mishra2 mishra2 PROPN cana-1654 1 34 1research 1research NUM cana-1654 1 35 scholar scholar NOUN cana-1654 1 36 , , PUNCT cana-1654 1 37 department department NOUN cana-1654 1 38 of of ADP cana-1654 1 39 mathematics mathematics PROPN cana-1654 1 40 , , PUNCT cana-1654 1 41 patliputra patliputra PROPN cana-1654 1 42 university university PROPN cana-1654 1 43 , , PUNCT cana-1654 1 44 patna patna PROPN cana-1654 1 45 , , PUNCT cana-1654 1 46 bihar bihar PROPN cana-1654 1 47 , , PUNCT cana-1654 1 48 india india PROPN cana-1654 1 49 . . PUNCT cana-1654 1 50 rituraj6312@gmail.com rituraj6312@gmail.com PROPN cana-1654 2 1 2professor 2professor NUM cana-1654 2 2 , , PUNCT cana-1654 2 3 department department NOUN cana-1654 2 4 of of ADP cana-1654 2 5 mathematics mathematics PROPN cana-1654 2 6 , , PUNCT cana-1654 2 7 patliputra patliputra PROPN cana-1654 2 8 university university PROPN cana-1654 2 9 , , PUNCT cana-1654 2 10 patna patna PROPN cana-1654 2 11 , , PUNCT cana-1654 2 12 bihar bihar PROPN cana-1654 2 13 , , PUNCT cana-1654 2 14 india india PROPN cana-1654 2 15 shambhumishra5@gmail.com shambhumishra5@gmail.com PROPN cana-1654 2 16 article article NOUN cana-1654 2 17 history history NOUN cana-1654 2 18 : : PUNCT cana-1654 2 19 received receive VERB cana-1654 2 20 : : PUNCT cana-1654 2 21 15 15 NUM cana-1654 2 22 - - SYM cana-1654 2 23 07 07 NUM cana-1654 2 24 - - PUNCT cana-1654 2 25 2024 2024 NUM cana-1654 2 26 revised revise VERB cana-1654 2 27 : : PUNCT cana-1654 2 28 30 30 NUM cana-1654 2 29 - - SYM cana-1654 2 30 08 08 NUM cana-1654 2 31 - - PUNCT cana-1654 2 32 2024 2024 NUM cana-1654 2 33 accepted accept VERB cana-1654 2 34 : : PUNCT cana-1654 2 35 11 11 NUM cana-1654 2 36 - - SYM cana-1654 2 37 09 09 NUM cana-1654 2 38 - - PUNCT cana-1654 2 39 2024 2024 NUM cana-1654 2 40 abstract abstract NOUN cana-1654 2 41 : : PUNCT cana-1654 2 42 introduction introduction NOUN cana-1654 2 43 : : PUNCT cana-1654 2 44 graph graph NOUN cana-1654 2 45 labeling labeling NOUN cana-1654 2 46 is be AUX cana-1654 2 47 one one NUM cana-1654 2 48 of of ADP cana-1654 2 49 the the DET cana-1654 2 50 fastest fast ADJ cana-1654 2 51 growing grow VERB cana-1654 2 52 areas area NOUN cana-1654 2 53 in in ADP cana-1654 2 54 the the DET cana-1654 2 55 field field NOUN cana-1654 2 56 of of ADP cana-1654 2 57 graph graph NOUN cana-1654 2 58 theory theory NOUN cana-1654 2 59 . . PUNCT cana-1654 3 1 a a DET cana-1654 3 2 graph graph NOUN cana-1654 3 3 labeling labeling NOUN cana-1654 3 4 is be AUX cana-1654 3 5 basically basically ADV cana-1654 3 6 an an DET cana-1654 3 7 assignment assignment NOUN cana-1654 3 8 of of ADP cana-1654 3 9 integers integer NOUN cana-1654 3 10 to to ADP cana-1654 3 11 the the DET cana-1654 3 12 nodes node NOUN cana-1654 3 13 or or CCONJ cana-1654 3 14 edges edge NOUN cana-1654 3 15 or or CCONJ cana-1654 3 16 both both PRON cana-1654 3 17 , , PUNCT cana-1654 3 18 subject subject ADJ cana-1654 3 19 to to ADP cana-1654 3 20 certain certain ADJ cana-1654 3 21 conditions condition NOUN cana-1654 3 22 . . PUNCT cana-1654 4 1 different different ADJ cana-1654 4 2 types type NOUN cana-1654 4 3 of of ADP cana-1654 4 4 graph graph NOUN cana-1654 4 5 labeling labeling NOUN cana-1654 4 6 techniques technique NOUN cana-1654 4 7 have have AUX cana-1654 4 8 been be AUX cana-1654 4 9 developed develop VERB cana-1654 4 10 by by ADP cana-1654 4 11 several several ADJ cana-1654 4 12 authors author NOUN cana-1654 4 13 . . PUNCT cana-1654 5 1 objectives objective NOUN cana-1654 5 2 : : PUNCT cana-1654 5 3 the the DET cana-1654 5 4 applications application NOUN cana-1654 5 5 of of ADP cana-1654 5 6 labeling labeling NOUN cana-1654 5 7 of of ADP cana-1654 5 8 graphs graph NOUN cana-1654 5 9 has have VERB cana-1654 5 10 diverse diverse ADJ cana-1654 5 11 fields field NOUN cana-1654 5 12 in in ADP cana-1654 5 13 the the DET cana-1654 5 14 study study NOUN cana-1654 5 15 of of ADP cana-1654 5 16 data datum NOUN cana-1654 5 17 base base NOUN cana-1654 5 18 management management NOUN cana-1654 5 19 , , PUNCT cana-1654 5 20 secret secret ADJ cana-1654 5 21 sharing sharing NOUN cana-1654 5 22 schemes scheme NOUN cana-1654 5 23 , , PUNCT cana-1654 5 24 physical physical ADJ cana-1654 5 25 cosmology cosmology NOUN cana-1654 5 26 , , PUNCT cana-1654 5 27 debug debug NOUN cana-1654 5 28 circuit circuit NOUN cana-1654 5 29 design design NOUN cana-1654 5 30 , , PUNCT cana-1654 5 31 x x NOUN cana-1654 5 32 - - NOUN cana-1654 5 33 ray ray NOUN cana-1654 5 34 , , PUNCT cana-1654 5 35 crystallography crystallography NOUN cana-1654 5 36 , , PUNCT cana-1654 5 37 astronomy astronomy NOUN cana-1654 5 38 , , PUNCT cana-1654 5 39 radar radar NOUN cana-1654 5 40 , , PUNCT cana-1654 5 41 broadcasting broadcasting NOUN cana-1654 5 42 network network NOUN cana-1654 5 43 , , PUNCT cana-1654 5 44 secret secret ADJ cana-1654 5 45 , , PUNCT cana-1654 5 46 coding code VERB cana-1654 5 47 theory theory NOUN cana-1654 5 48 and and CCONJ cana-1654 5 49 much much ADV cana-1654 5 50 more more ADJ cana-1654 5 51 . . PUNCT cana-1654 6 1 the the DET cana-1654 6 2 graph graph NOUN cana-1654 6 3 labeling labeling NOUN cana-1654 6 4 techniques technique NOUN cana-1654 6 5 serve serve VERB cana-1654 6 6 as as ADP cana-1654 6 7 useful useful ADJ cana-1654 6 8 mathematical mathematical ADJ cana-1654 6 9 models model NOUN cana-1654 6 10 and and CCONJ cana-1654 6 11 solve solve VERB cana-1654 6 12 various various ADJ cana-1654 6 13 graph graph NOUN cana-1654 6 14 related relate VERB cana-1654 6 15 problems problem NOUN cana-1654 6 16 . . PUNCT cana-1654 7 1 methods method NOUN cana-1654 7 2 : : PUNCT cana-1654 7 3 the the DET cana-1654 7 4 strength strength NOUN cana-1654 7 5 of of ADP cana-1654 7 6 this this DET cana-1654 7 7 research research NOUN cana-1654 7 8 paper paper NOUN cana-1654 7 9 is be AUX cana-1654 7 10 based base VERB cana-1654 7 11 on on ADP cana-1654 7 12 odd odd ADJ cana-1654 7 13 - - PUNCT cana-1654 7 14 even even ADV cana-1654 7 15 congruence congruence NOUN cana-1654 7 16 labeling labeling NOUN cana-1654 7 17 of of ADP cana-1654 7 18 different different ADJ cana-1654 7 19 types type NOUN cana-1654 7 20 of of ADP cana-1654 7 21 graphs graph NOUN cana-1654 7 22 . . PUNCT cana-1654 8 1 assignment assignment NOUN cana-1654 8 2 of of ADP cana-1654 8 3 natural natural ADJ cana-1654 8 4 numbers number NOUN cana-1654 8 5 as as ADP cana-1654 8 6 labels label NOUN cana-1654 8 7 for for ADP cana-1654 8 8 the the DET cana-1654 8 9 edges edge NOUN cana-1654 8 10 and and CCONJ cana-1654 8 11 vertices vertex NOUN cana-1654 8 12 of of ADP cana-1654 8 13 a a DET cana-1654 8 14 graph graph NOUN cana-1654 8 15 . . PUNCT cana-1654 9 1 it it PRON cana-1654 9 2 is be AUX cana-1654 9 3 based base VERB cana-1654 9 4 on on ADP cana-1654 9 5 modular modular ADJ cana-1654 9 6 arithmetic arithmetic ADJ cana-1654 9 7 property property NOUN cana-1654 9 8 known know VERB cana-1654 9 9 as as ADP cana-1654 9 10 congruence congruence NOUN cana-1654 9 11 graph graph NOUN cana-1654 9 12 labelling labelling NOUN cana-1654 9 13 of of ADP cana-1654 9 14 a a DET cana-1654 9 15 graph.for graph.for ADP cana-1654 9 16 congruence congruence NOUN cana-1654 9 17 graph graph NOUN cana-1654 9 18 labeling labeling NOUN cana-1654 9 19 it it PRON cana-1654 9 20 entails entail VERB cana-1654 9 21 the the DET cana-1654 9 22 assignment assignment NOUN cana-1654 9 23 of of ADP cana-1654 9 24 odd odd ADJ cana-1654 9 25 integers integer NOUN cana-1654 9 26 to to PART cana-1654 9 27 vertices vertex NOUN cana-1654 9 28 and and CCONJ cana-1654 9 29 even even ADV cana-1654 9 30 integers integer NOUN cana-1654 9 31 to to PART cana-1654 9 32 edges edge NOUN cana-1654 9 33 with with ADP cana-1654 9 34 property property NOUN cana-1654 9 35 of of ADP cana-1654 9 36 congruence congruence NOUN cana-1654 9 37 graph graph NOUN cana-1654 9 38 labeling labeling NOUN cana-1654 9 39 . . PUNCT cana-1654 10 1 results result NOUN cana-1654 10 2 : : PUNCT cana-1654 10 3 this this DET cana-1654 10 4 labeling labeling NOUN cana-1654 10 5 method method NOUN cana-1654 10 6 has have AUX cana-1654 10 7 been be AUX cana-1654 10 8 identified identify VERB cana-1654 10 9 on on ADP cana-1654 10 10 friendship friendship NOUN cana-1654 10 11 graph graph NOUN cana-1654 10 12 , , PUNCT cana-1654 10 13 shell shell NOUN cana-1654 10 14 graph graph NOUN cana-1654 10 15 , , PUNCT cana-1654 10 16 generalized generalized ADJ cana-1654 10 17 butterfly butterfly NOUN cana-1654 10 18 graph graph NOUN cana-1654 10 19 , , PUNCT cana-1654 10 20 fan fan NOUN cana-1654 10 21 graph graph NOUN cana-1654 10 22 , , PUNCT cana-1654 10 23 𝑃2 𝑃2 NOUN cana-1654 10 24 + + CCONJ cana-1654 10 25 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 10 26 graph graph NOUN cana-1654 10 27 keywords keyword NOUN cana-1654 10 28 : : PUNCT cana-1654 10 29 labeling labeling NOUN cana-1654 10 30 , , PUNCT cana-1654 10 31 congruence congruence NOUN cana-1654 10 32 labeling labeling NOUN cana-1654 10 33 , , PUNCT cana-1654 10 34 odd odd ADV cana-1654 10 35 - - PUNCT cana-1654 10 36 even even ADV cana-1654 10 37 congruence congruence NOUN cana-1654 10 38 labeling labeling NOUN cana-1654 10 39 , , PUNCT cana-1654 10 40 friendship friendship NOUN cana-1654 10 41 graph graph NOUN cana-1654 10 42 , , PUNCT cana-1654 10 43 shell shell NOUN cana-1654 10 44 graph graph NOUN cana-1654 10 45 , , PUNCT cana-1654 10 46 generalized generalized ADJ cana-1654 10 47 butterfly butterfly NOUN cana-1654 10 48 graph graph NOUN cana-1654 10 49 , , PUNCT cana-1654 10 50 fan fan NOUN cana-1654 10 51 graph graph NOUN cana-1654 10 52 , , PUNCT cana-1654 10 53 𝑃2 𝑃2 NOUN cana-1654 10 54 + + CCONJ cana-1654 10 55 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 10 56 graph graph NOUN cana-1654 10 57 1 1 NUM cana-1654 10 58 . . PUNCT cana-1654 11 1 introduction introduction NOUN cana-1654 11 2 in in ADP cana-1654 11 3 this this DET cana-1654 11 4 paper paper NOUN cana-1654 11 5 all all DET cana-1654 11 6 graphs graph NOUN cana-1654 11 7 𝐺 𝐺 PROPN cana-1654 11 8 considered consider VERB cana-1654 11 9 as as ADP cana-1654 11 10 finite finite NOUN cana-1654 11 11 , , PUNCT cana-1654 11 12 undirected undirected ADJ cana-1654 11 13 , , PUNCT cana-1654 11 14 connected connected ADJ cana-1654 11 15 and and CCONJ cana-1654 11 16 without without ADP cana-1654 11 17 loops loop NOUN cana-1654 11 18 . . PUNCT cana-1654 12 1 suppose suppose VERB cana-1654 12 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 12 3 ) ) PUNCT cana-1654 12 4 and and CCONJ cana-1654 12 5 𝐸(𝐺 𝐸(𝐺 PROPN cana-1654 12 6 ) ) PUNCT cana-1654 12 7 be be VERB cana-1654 12 8 the the DET cana-1654 12 9 set set NOUN cana-1654 12 10 of of ADP cana-1654 12 11 vertices vertex NOUN cana-1654 12 12 and and CCONJ cana-1654 12 13 edges edge NOUN cana-1654 12 14 of of ADP cana-1654 12 15 a a DET cana-1654 12 16 graph graph NOUN cana-1654 12 17 𝐺 𝐺 NOUN cana-1654 12 18 respectively respectively ADV cana-1654 12 19 . . PUNCT cana-1654 13 1 the the DET cana-1654 13 2 cardinality cardinality NOUN cana-1654 13 3 of of ADP cana-1654 13 4 vertex vertex NOUN cana-1654 13 5 set set NOUN cana-1654 13 6 is be AUX cana-1654 13 7 denoted denote VERB cana-1654 13 8 by by ADP cana-1654 13 9 |𝑉(𝐺)| |𝑉(𝐺)| PROPN cana-1654 13 10 and and CCONJ cana-1654 13 11 the the DET cana-1654 13 12 edge edge NOUN cana-1654 13 13 set set NOUN cana-1654 13 14 is be AUX cana-1654 13 15 denoted denote VERB cana-1654 13 16 by by ADP cana-1654 13 17 |𝐸(𝐺)| |𝐸(𝐺)| INTJ cana-1654 13 18 are be AUX cana-1654 13 19 called call VERB cana-1654 13 20 the the DET cana-1654 13 21 order order NOUN cana-1654 13 22 and and CCONJ cana-1654 13 23 size size NOUN cana-1654 13 24 of of ADP cana-1654 13 25 the the DET cana-1654 13 26 graph graph NOUN cana-1654 13 27 𝐺. 𝐺. NOUN cana-1654 13 28 for for ADP cana-1654 13 29 standard standard ADJ cana-1654 13 30 terminology terminology NOUN cana-1654 13 31 of of ADP cana-1654 13 32 graph graph NOUN cana-1654 13 33 theory theory NOUN cana-1654 13 34 we we PRON cana-1654 13 35 used use VERB cana-1654 13 36 [ [ X cana-1654 13 37 1 1 NUM cana-1654 13 38 ] ] PUNCT cana-1654 13 39 . . PUNCT cana-1654 14 1 for for ADP cana-1654 14 2 all all DET cana-1654 14 3 detailed detailed ADJ cana-1654 14 4 survey survey NOUN cana-1654 14 5 of of ADP cana-1654 14 6 graph graph NOUN cana-1654 14 7 labeling labeling NOUN cana-1654 14 8 , , PUNCT cana-1654 14 9 we we PRON cana-1654 14 10 refer refer VERB cana-1654 14 11 [ [ X cana-1654 14 12 2 2 NUM cana-1654 14 13 ] ] PUNCT cana-1654 14 14 . . PUNCT cana-1654 15 1 while while SCONJ cana-1654 15 2 studying study VERB cana-1654 15 3 graph graph NOUN cana-1654 15 4 theory theory NOUN cana-1654 15 5 , , PUNCT cana-1654 15 6 one one NOUN cana-1654 15 7 that that PRON cana-1654 15 8 has have AUX cana-1654 15 9 gained gain VERB cana-1654 15 10 a a DET cana-1654 15 11 lot lot NOUN cana-1654 15 12 of of ADP cana-1654 15 13 popularity popularity NOUN cana-1654 15 14 during during ADP cana-1654 15 15 the the DET cana-1654 15 16 last last ADJ cana-1654 15 17 62 62 NUM cana-1654 15 18 years year NOUN cana-1654 15 19 is be AUX cana-1654 15 20 the the DET cana-1654 15 21 concept concept NOUN cana-1654 15 22 of of ADP cana-1654 15 23 labeling labeling NOUN cana-1654 15 24 of of ADP cana-1654 15 25 graphs graph NOUN cana-1654 15 26 due due ADP cana-1654 15 27 to to ADP cana-1654 15 28 its its PRON cana-1654 15 29 wide wide ADJ cana-1654 15 30 range range NOUN cana-1654 15 31 of of ADP cana-1654 15 32 applications application NOUN cana-1654 15 33 . . PUNCT cana-1654 16 1 a a DET cana-1654 16 2 labeling labeling NOUN cana-1654 16 3 of of ADP cana-1654 16 4 a a DET cana-1654 16 5 graph graph NOUN cana-1654 16 6 𝐺 𝐺 NOUN cana-1654 16 7 is be AUX cana-1654 16 8 one one NUM cana-1654 16 9 - - PUNCT cana-1654 16 10 to to ADP cana-1654 16 11 - - PUNCT cana-1654 16 12 one one NUM cana-1654 16 13 mapping mapping NOUN cana-1654 16 14 that that PRON cana-1654 16 15 carries carry VERB cana-1654 16 16 the the DET cana-1654 16 17 set set NOUN cana-1654 16 18 of of ADP cana-1654 16 19 graph graph NOUN cana-1654 16 20 elements element NOUN cana-1654 16 21 onto onto ADP cana-1654 16 22 a a DET cana-1654 16 23 set set NOUN cana-1654 16 24 of of ADP cana-1654 16 25 numbers number NOUN cana-1654 16 26 , , PUNCT cana-1654 16 27 called call VERB cana-1654 16 28 labels label NOUN cana-1654 16 29 . . PUNCT cana-1654 17 1 in in ADP cana-1654 17 2 1967 1967 NUM cana-1654 17 3 , , PUNCT cana-1654 17 4 rosa rosa PROPN cana-1654 17 5 [ [ X cana-1654 17 6 5 5 NUM cana-1654 17 7 ] ] PUNCT cana-1654 17 8 published publish VERB cana-1654 17 9 a a DET cana-1654 17 10 pioneering pioneering ADJ cana-1654 17 11 paper paper NOUN cana-1654 17 12 on on ADP cana-1654 17 13 graph graph NOUN cana-1654 17 14 labeling labeling NOUN cana-1654 17 15 problems problem NOUN cana-1654 17 16 . . PUNCT cana-1654 18 1 thereafter thereafter ADV cana-1654 18 2 many many ADJ cana-1654 18 3 types type NOUN cana-1654 18 4 of of ADP cana-1654 18 5 graph graph NOUN cana-1654 18 6 labeling labeling NOUN cana-1654 18 7 methods method NOUN cana-1654 18 8 have have AUX cana-1654 18 9 been be AUX cana-1654 18 10 studied study VERB cana-1654 18 11 by by ADP cana-1654 18 12 several several ADJ cana-1654 18 13 authors author NOUN cana-1654 18 14 . . PUNCT cana-1654 19 1 g.thamizhendhi g.thamizhendhi NUM cana-1654 19 2 and and CCONJ cana-1654 19 3 k.kanakambika k.kanakambika X cana-1654 19 4 [ [ X cana-1654 19 5 6 6 NUM cana-1654 19 6 ] ] PUNCT cana-1654 19 7 have have AUX cana-1654 19 8 introduced introduce VERB cana-1654 19 9 odd odd ADJ cana-1654 19 10 - - PUNCT cana-1654 19 11 even even ADV cana-1654 19 12 congruence congruence NOUN cana-1654 19 13 labeling labeling NOUN cana-1654 19 14 and and CCONJ cana-1654 19 15 they they PRON cana-1654 19 16 proved prove VERB cana-1654 19 17 behaviour behaviour NOUN cana-1654 19 18 of of ADP cana-1654 19 19 several several ADJ cana-1654 19 20 graphs graph NOUN cana-1654 19 21 like like ADP cana-1654 19 22 bipartite bipartite NOUN cana-1654 19 23 graph graph NOUN cana-1654 19 24 , , PUNCT cana-1654 19 25 comb comb NOUN cana-1654 19 26 graph graph NOUN cana-1654 19 27 , , PUNCT cana-1654 19 28 star star NOUN cana-1654 19 29 graph graph NOUN cana-1654 19 30 , , PUNCT cana-1654 19 31 graph graph NOUN cana-1654 19 32 acquired acquire VERB cana-1654 19 33 by by ADP cana-1654 19 34 connecting connect VERB cana-1654 19 35 two two NUM cana-1654 19 36 copies copy NOUN cana-1654 19 37 of of ADP cana-1654 19 38 even even ADV cana-1654 19 39 cycle cycle NOUN cana-1654 19 40 πΆπ‘Ÿ πΆπ‘Ÿ PROPN cana-1654 19 41 by by ADP cana-1654 19 42 a a DET cana-1654 19 43 path path NOUN cana-1654 19 44 𝑃𝑑 𝑃𝑑 PROPN cana-1654 19 45 , , PUNCT cana-1654 19 46 shadow shadow NOUN cana-1654 19 47 graph graph NOUN cana-1654 19 48 of of ADP cana-1654 19 49 the the DET cana-1654 19 50 path path NOUN cana-1654 19 51 𝑃𝑑 𝑃𝑑 PROPN cana-1654 19 52 and and CCONJ cana-1654 19 53 the the DET cana-1654 19 54 tensor tensor NOUN cana-1654 19 55 product product NOUN cana-1654 19 56 of of ADP cana-1654 19 57 𝐾1,𝑑 𝐾1,𝑑 PROPN cana-1654 19 58 & & CCONJ cana-1654 19 59 𝑃2 𝑃2 PROPN cana-1654 19 60 as as ADP cana-1654 19 61 odd odd ADJ cana-1654 19 62 - - PUNCT cana-1654 19 63 even even ADV cana-1654 19 64 congruence congruence NOUN cana-1654 19 65 graph graph NOUN cana-1654 19 66 . . PUNCT cana-1654 20 1 they they PRON cana-1654 20 2 defined define VERB cana-1654 20 3 an an DET cana-1654 20 4 odd odd ADV cana-1654 20 5 - - PUNCT cana-1654 20 6 even even ADV cana-1654 20 7 congruence congruence NOUN cana-1654 20 8 graph graph NOUN cana-1654 20 9 , , PUNCT cana-1654 20 10 if if SCONJ cana-1654 20 11 communications communication NOUN cana-1654 20 12 on on ADP cana-1654 20 13 applied apply VERB cana-1654 20 14 nonlinear nonlinear ADJ cana-1654 20 15 analysis analysis NOUN cana-1654 20 16 issn issn NOUN cana-1654 20 17 : : PUNCT cana-1654 20 18 1074 1074 NUM cana-1654 20 19 - - PUNCT cana-1654 20 20 133x 133x NUM cana-1654 20 21 vol vol NOUN cana-1654 20 22 32 32 NUM cana-1654 20 23 no no NOUN cana-1654 20 24 . . NOUN cana-1654 20 25 1 1 NUM cana-1654 20 26 ( ( PUNCT cana-1654 20 27 2025 2025 NUM cana-1654 20 28 ) ) PUNCT cana-1654 20 29 336 336 NUM cana-1654 20 30 https://internationalpubls.com https://internationalpubls.com X cana-1654 20 31 vertex vertex NOUN cana-1654 20 32 and and CCONJ cana-1654 20 33 edge edge NOUN cana-1654 20 34 set set VERB cana-1654 20 35 are be AUX cana-1654 20 36 assigned assign VERB cana-1654 20 37 by by ADP cana-1654 20 38 distinct distinct ADJ cana-1654 20 39 odd odd ADJ cana-1654 20 40 and and CCONJ cana-1654 20 41 even even ADV cana-1654 20 42 integers integer NOUN cana-1654 20 43 respectively respectively ADV cana-1654 20 44 , , PUNCT cana-1654 20 45 further far ADV cana-1654 20 46 𝑓(𝑒𝑝 𝑓(𝑒𝑝 NUM cana-1654 20 47 ) ) PUNCT cana-1654 20 48 ≑ ≑ PROPN cana-1654 20 49 𝑓(π‘’π‘ž 𝑓(π‘’π‘ž NUM cana-1654 20 50 ) ) PUNCT cana-1654 20 51 ( ( PUNCT cana-1654 20 52 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ PROPN cana-1654 20 53 𝑔(𝑒 𝑔(𝑒 PROPN cana-1654 20 54 ) ) PUNCT cana-1654 20 55 ) ) PUNCT cana-1654 20 56 , , PUNCT cana-1654 20 57 𝑒𝑝 𝑒𝑝 VERB cana-1654 20 58 and and CCONJ cana-1654 20 59 π‘’π‘ž π‘’π‘ž INTJ cana-1654 20 60 are be AUX cana-1654 20 61 adjacent adjacent ADJ cana-1654 20 62 vertices vertex NOUN cana-1654 20 63 in in ADP cana-1654 20 64 𝐺. 𝐺. NOUN cana-1654 20 65 in in ADP cana-1654 20 66 this this DET cana-1654 20 67 paper paper NOUN cana-1654 20 68 we we PRON cana-1654 20 69 investigate investigate VERB cana-1654 20 70 the the DET cana-1654 20 71 existence existence NOUN cana-1654 20 72 of of ADP cana-1654 20 73 odd odd ADV cana-1654 20 74 - - PUNCT cana-1654 20 75 even even ADV cana-1654 20 76 congruence congruence NOUN cana-1654 20 77 labeling labeling NOUN cana-1654 20 78 for for ADP cana-1654 20 79 vertex vertex NOUN cana-1654 20 80 switching switching NOUN cana-1654 20 81 graph graph NOUN cana-1654 20 82 , , PUNCT cana-1654 20 83 friendship friendship NOUN cana-1654 20 84 graph graph NOUN cana-1654 20 85 , , PUNCT cana-1654 20 86 shell shell NOUN cana-1654 20 87 graph graph NOUN cana-1654 20 88 , , PUNCT cana-1654 20 89 generalized generalized ADJ cana-1654 20 90 butterfly butterfly NOUN cana-1654 20 91 graph graph NOUN cana-1654 20 92 , , PUNCT cana-1654 20 93 fan fan NOUN cana-1654 20 94 graph graph NOUN cana-1654 20 95 , , PUNCT cana-1654 20 96 𝑃2 𝑃2 NOUN cana-1654 20 97 + + CCONJ cana-1654 20 98 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 20 99 graph graph NOUN cana-1654 20 100 . . PUNCT cana-1654 21 1 2 2 X cana-1654 21 2 . . X cana-1654 21 3 preliminaries preliminary NOUN cana-1654 21 4 definition definition NOUN cana-1654 21 5 2.1 2.1 NUM cana-1654 21 6 . . PUNCT cana-1654 22 1 [ [ X cana-1654 22 2 6 6 NUM cana-1654 22 3 , , PUNCT cana-1654 22 4 7 7 NUM cana-1654 22 5 ] ] PUNCT cana-1654 22 6 . . PUNCT cana-1654 23 1 a a DET cana-1654 23 2 graph graph NOUN cana-1654 23 3 obtained obtain VERB cana-1654 23 4 by by ADP cana-1654 23 5 fetching fetch VERB cana-1654 23 6 a a DET cana-1654 23 7 vertex vertex NOUN cana-1654 23 8 π‘₯ π‘₯ NOUN cana-1654 23 9 of of ADP cana-1654 23 10 𝐻 𝐻 PROPN cana-1654 23 11 , , PUNCT cana-1654 23 12 eliminating eliminate VERB cana-1654 23 13 the the DET cana-1654 23 14 adjacent adjacent ADJ cana-1654 23 15 edges edge NOUN cana-1654 23 16 of of ADP cana-1654 23 17 π‘₯ π‘₯ NOUN cana-1654 23 18 and and CCONJ cana-1654 23 19 by by ADP cana-1654 23 20 adding add VERB cana-1654 23 21 new new ADJ cana-1654 23 22 edges edge NOUN cana-1654 23 23 that that PRON cana-1654 23 24 are be AUX cana-1654 23 25 joining join VERB cana-1654 23 26 π‘₯ π‘₯ PRON cana-1654 23 27 to to ADP cana-1654 23 28 their their PRON cana-1654 23 29 non non ADJ cana-1654 23 30 - - ADJ cana-1654 23 31 adjacent adjacent ADJ cana-1654 23 32 vertices vertex NOUN cana-1654 23 33 in in ADP cana-1654 23 34 𝐻 𝐻 PROPN cana-1654 23 35 is be AUX cana-1654 23 36 called call VERB cana-1654 23 37 vertex vertex NOUN cana-1654 23 38 switching switch VERB cana-1654 23 39 𝐻π‘₯ 𝐻π‘₯ PROPN cana-1654 23 40 of of ADP cana-1654 23 41 𝐻. 𝐻. PROPN cana-1654 23 42 definition definition NOUN cana-1654 23 43 2.2 2.2 NUM cana-1654 23 44 . . PUNCT cana-1654 24 1 [ [ X cana-1654 24 2 3 3 NUM cana-1654 24 3 ] ] PUNCT cana-1654 24 4 . . PUNCT cana-1654 25 1 the the DET cana-1654 25 2 friendship friendship NOUN cana-1654 25 3 graph graph NOUN cana-1654 25 4 πΉπ‘Ÿπ‘› πΉπ‘Ÿπ‘› PROPN cana-1654 25 5 is be AUX cana-1654 25 6 a a DET cana-1654 25 7 collection collection NOUN cana-1654 25 8 of of ADP cana-1654 25 9 𝑛 𝑛 DET cana-1654 25 10 triangles triangle NOUN cana-1654 25 11 with with ADP cana-1654 25 12 a a DET cana-1654 25 13 common common ADJ cana-1654 25 14 vertex vertex NOUN cana-1654 25 15 . . PUNCT cana-1654 26 1 it it PRON cana-1654 26 2 is be AUX cana-1654 26 3 a a DET cana-1654 26 4 planar planar ADJ cana-1654 26 5 undirected undirected ADJ cana-1654 26 6 graph graph NOUN cana-1654 26 7 with with ADP cana-1654 26 8 2𝑛 2𝑛 PROPN cana-1654 26 9 + + CCONJ cana-1654 26 10 1 1 NUM cana-1654 26 11 vertices vertex NOUN cana-1654 26 12 and and CCONJ cana-1654 26 13 3𝑛 3𝑛 NUM cana-1654 26 14 edges edge NOUN cana-1654 26 15 constructed construct VERB cana-1654 26 16 by by ADP cana-1654 26 17 joining join VERB cana-1654 26 18 𝑛 𝑛 DET cana-1654 26 19 copies copy NOUN cana-1654 26 20 of of ADP cana-1654 26 21 the the DET cana-1654 26 22 cycle cycle NOUN cana-1654 26 23 graph graph NOUN cana-1654 26 24 𝐢3 𝐢3 NOUN cana-1654 26 25 with with ADP cana-1654 26 26 a a DET cana-1654 26 27 common common ADJ cana-1654 26 28 vertex vertex NOUN cana-1654 26 29 . . PUNCT cana-1654 27 1 definition definition NOUN cana-1654 27 2 2.3 2.3 NUM cana-1654 27 3 . . PUNCT cana-1654 28 1 [ [ X cana-1654 28 2 10 10 NUM cana-1654 28 3 ] ] PUNCT cana-1654 28 4 . . PUNCT cana-1654 29 1 a a DET cana-1654 29 2 shell shell NOUN cana-1654 29 3 𝑆𝑛 𝑆𝑛 PROPN cana-1654 29 4 is be AUX cana-1654 29 5 the the DET cana-1654 29 6 graph graph NOUN cana-1654 29 7 obtained obtain VERB cana-1654 29 8 as as ADP cana-1654 29 9 a a DET cana-1654 29 10 cycle cycle NOUN cana-1654 29 11 𝐢𝑛 𝐢𝑛 NOUN cana-1654 29 12 by by ADP cana-1654 29 13 taking take VERB cana-1654 29 14 ( ( PUNCT cana-1654 29 15 𝑛 𝑛 PRON cana-1654 29 16 βˆ’ βˆ’ NOUN cana-1654 29 17 3 3 NUM cana-1654 29 18 ) ) PUNCT cana-1654 29 19 concurrent concurrent ADJ cana-1654 29 20 chords chord NOUN cana-1654 29 21 sharing share VERB cana-1654 29 22 a a DET cana-1654 29 23 common common ADJ cana-1654 29 24 end end NOUN cana-1654 29 25 point point NOUN cana-1654 29 26 called call VERB cana-1654 29 27 the the DET cana-1654 29 28 apex apex NOUN cana-1654 29 29 . . PUNCT cana-1654 30 1 shell shell PROPN cana-1654 30 2 graph graph NOUN cana-1654 30 3 are be AUX cana-1654 30 4 denoted denote VERB cana-1654 30 5 as as ADP cana-1654 30 6 𝐢(𝑛,π‘›βˆ’3 𝐢(𝑛,π‘›βˆ’3 NOUN cana-1654 30 7 ) ) PUNCT cana-1654 30 8 . . PUNCT cana-1654 31 1 a a DET cana-1654 31 2 shell shell NOUN cana-1654 31 3 𝑆𝑛 𝑆𝑛 NOUN cana-1654 31 4 is be AUX cana-1654 31 5 also also ADV cana-1654 31 6 called call VERB cana-1654 31 7 fan fan PROPN cana-1654 31 8 π‘“π‘›βˆ’1 π‘“π‘›βˆ’1 PROPN cana-1654 31 9 definition definition NOUN cana-1654 31 10 2.4 2.4 NUM cana-1654 31 11 . . PUNCT cana-1654 32 1 [ [ X cana-1654 32 2 4 4 NUM cana-1654 32 3 ] ] PUNCT cana-1654 32 4 . . PUNCT cana-1654 33 1 the the DET cana-1654 33 2 generalized generalized ADJ cana-1654 33 3 butterfly butterfly NOUN cana-1654 33 4 graph graph NOUN cana-1654 33 5 denoted denote VERB cana-1654 33 6 by by ADP cana-1654 33 7 𝐡𝐹𝑛 𝐡𝐹𝑛 NOUN cana-1654 33 8 obtained obtain VERB cana-1654 33 9 by by ADP cana-1654 33 10 inserting insert VERB cana-1654 33 11 adjoining adjoining ADJ cana-1654 33 12 vertices vertex NOUN cana-1654 33 13 to to ADP cana-1654 33 14 every every DET cana-1654 33 15 wing wing NOUN cana-1654 33 16 with with ADP cana-1654 33 17 assumption assumption NOUN cana-1654 33 18 that that DET cana-1654 33 19 sum sum NOUN cana-1654 33 20 of of ADP cana-1654 33 21 inserting insert VERB cana-1654 33 22 vertices vertex NOUN cana-1654 33 23 to to ADP cana-1654 33 24 every every DET cana-1654 33 25 wing wing NOUN cana-1654 33 26 are be AUX cana-1654 33 27 same same ADJ cana-1654 33 28 . . PUNCT cana-1654 34 1 definition definition NOUN cana-1654 34 2 2.5 2.5 NUM cana-1654 34 3 . . PUNCT cana-1654 35 1 [ [ X cana-1654 35 2 9 9 NUM cana-1654 35 3 ] ] PUNCT cana-1654 35 4 . . PUNCT cana-1654 36 1 a a DET cana-1654 36 2 fan fan NOUN cana-1654 36 3 graph graph NOUN cana-1654 36 4 𝑓𝑛 𝑓𝑛 PROPN cana-1654 36 5 , , PUNCT cana-1654 36 6 𝑛 𝑛 DET cana-1654 36 7 β‰₯ β‰₯ NOUN cana-1654 36 8 2 2 NUM cana-1654 36 9 obtained obtain VERB cana-1654 36 10 by by ADP cana-1654 36 11 joining join VERB cana-1654 36 12 all all DET cana-1654 36 13 vertices vertex NOUN cana-1654 36 14 of of ADP cana-1654 36 15 a a DET cana-1654 36 16 path path NOUN cana-1654 36 17 𝑃𝑛 𝑃𝑛 NOUN cana-1654 36 18 to to ADP cana-1654 36 19 a a DET cana-1654 36 20 further further ADJ cana-1654 36 21 vertex vertex NOUN cana-1654 36 22 , , PUNCT cana-1654 36 23 called call VERB cana-1654 36 24 the the DET cana-1654 36 25 center center NOUN cana-1654 36 26 . . PUNCT cana-1654 37 1 𝑓𝑛 𝑓𝑛 NOUN cana-1654 37 2 = = SYM cana-1654 37 3 𝐾1 𝐾1 PROPN cana-1654 38 1 + + CCONJ cana-1654 38 2 𝑃𝑛 𝑃𝑛 NOUN cana-1654 38 3 , , PUNCT cana-1654 38 4 |𝑉(𝑓𝑛)| |𝑉(𝑓𝑛)| NOUN cana-1654 38 5 = = SYM cana-1654 38 6 𝑛 𝑛 NOUN cana-1654 38 7 + + NUM cana-1654 38 8 1 1 NUM cana-1654 38 9 and and CCONJ cana-1654 38 10 |𝐸(𝐺)| |𝐸(𝐺)| NOUN cana-1654 38 11 = = NOUN cana-1654 38 12 2𝑛 2𝑛 PROPN cana-1654 39 1 βˆ’ βˆ’ NUM cana-1654 39 2 1 1 NUM cana-1654 39 3 3 3 NUM cana-1654 39 4 . . PUNCT cana-1654 39 5 main main ADJ cana-1654 39 6 results result NOUN cana-1654 39 7 theorem theorem VERB cana-1654 39 8 3.1 3.1 NUM cana-1654 39 9 the the DET cana-1654 39 10 graph graph NOUN cana-1654 39 11 obtained obtain VERB cana-1654 39 12 from from ADP cana-1654 39 13 switching switching NOUN cana-1654 39 14 of of ADP cana-1654 39 15 any any DET cana-1654 39 16 vertex vertex NOUN cana-1654 39 17 in in ADP cana-1654 39 18 cycle cycle NOUN cana-1654 39 19 πΆπ‘Ÿ πΆπ‘Ÿ PROPN cana-1654 39 20 is be AUX cana-1654 39 21 an an DET cana-1654 39 22 odd odd ADJ cana-1654 39 23 - - PUNCT cana-1654 39 24 even even ADV cana-1654 39 25 congruence congruence NOUN cana-1654 39 26 graph graph NOUN cana-1654 39 27 . . PUNCT cana-1654 40 1 proof proof NOUN cana-1654 40 2 : : PUNCT cana-1654 40 3 suppose suppose VERB cana-1654 40 4 π‘₯1 π‘₯1 NOUN cana-1654 40 5 , , PUNCT cana-1654 40 6 π‘₯2 π‘₯2 NOUN cana-1654 40 7 , , PUNCT cana-1654 40 8 … … PUNCT cana-1654 40 9 … … PUNCT cana-1654 40 10 … … PUNCT cana-1654 40 11 … … PUNCT cana-1654 40 12 … … PUNCT cana-1654 40 13 , , PUNCT cana-1654 40 14 π‘₯π‘Ÿβˆ’1 π‘₯π‘Ÿβˆ’1 VERB cana-1654 40 15 be be AUX cana-1654 40 16 the the DET cana-1654 40 17 vertices vertex NOUN cana-1654 40 18 of of ADP cana-1654 40 19 a a DET cana-1654 40 20 cycle cycle NOUN cana-1654 40 21 πΆπ‘Ÿ. πΆπ‘Ÿ. PROPN cana-1654 40 22 let let VERB cana-1654 40 23 𝐻π‘₯1 𝐻π‘₯1 NOUN cana-1654 40 24 is be AUX cana-1654 40 25 the the DET cana-1654 40 26 graph graph NOUN cana-1654 40 27 obtained obtain VERB cana-1654 40 28 from from ADP cana-1654 40 29 switching switching NOUN cana-1654 40 30 of of ADP cana-1654 40 31 a a DET cana-1654 40 32 vertex vertex NOUN cana-1654 40 33 π‘₯1 π‘₯1 NOUN cana-1654 40 34 in in ADP cana-1654 40 35 πΆπ‘Ÿ. πΆπ‘Ÿ. PROPN cana-1654 40 36 in in ADP cana-1654 40 37 𝐻π‘₯1 𝐻π‘₯1 NOUN cana-1654 40 38 each each DET cana-1654 40 39 vertex vertex NOUN cana-1654 40 40 π‘₯𝑖 π‘₯𝑖 PRON cana-1654 40 41 other other ADJ cana-1654 40 42 than than ADP cana-1654 40 43 π‘₯2 π‘₯2 NOUN cana-1654 40 44 and and CCONJ cana-1654 40 45 π‘₯π‘Ÿ π‘₯π‘Ÿ PRON cana-1654 40 46 join join VERB cana-1654 40 47 to to ADP cana-1654 40 48 π‘₯1 π‘₯1 PROPN cana-1654 40 49 . . PUNCT cana-1654 41 1 here here ADV cana-1654 41 2 |𝑉(𝐻π‘₯1 |𝑉(𝐻π‘₯1 NOUN cana-1654 41 3 ) ) PUNCT cana-1654 42 1 | | ADV cana-1654 42 2 = = SYM cana-1654 42 3 π‘Ÿ π‘Ÿ NOUN cana-1654 42 4 and and CCONJ cana-1654 42 5 |𝐸(𝐻π‘₯1 |𝐸(𝐻π‘₯1 ADJ cana-1654 42 6 ) ) PUNCT cana-1654 42 7 | | ADV cana-1654 42 8 = = SYM cana-1654 42 9 2π‘Ÿ 2π‘Ÿ NUM cana-1654 43 1 βˆ’ βˆ’ NUM cana-1654 43 2 5 5 NUM cana-1654 43 3 we we PRON cana-1654 43 4 have have VERB cana-1654 43 5 𝑑 𝑑 NOUN cana-1654 43 6 = = SYM cana-1654 43 7 min{2π‘Ÿ min{2π‘Ÿ PROPN cana-1654 43 8 , , PUNCT cana-1654 43 9 2(2π‘Ÿ 2(2π‘Ÿ NUM cana-1654 43 10 βˆ’ βˆ’ NOUN cana-1654 43 11 5 5 NUM cana-1654 43 12 ) ) PUNCT cana-1654 43 13 } } PUNCT cana-1654 44 1 = = PUNCT cana-1654 44 2 2(2π‘Ÿ 2(2π‘Ÿ NUM cana-1654 44 3 βˆ’ βˆ’ NUM cana-1654 44 4 5 5 NUM cana-1654 44 5 ) ) PUNCT cana-1654 44 6 let let VERB cana-1654 44 7 the the DET cana-1654 44 8 edge edge NOUN cana-1654 44 9 set set NOUN cana-1654 44 10 of of ADP cana-1654 44 11 𝐻π‘₯1 𝐻π‘₯1 NOUN cana-1654 44 12 be be AUX cana-1654 44 13 𝐸(𝐻π‘₯1 𝐸(𝐻π‘₯1 ADJ cana-1654 44 14 ) ) PUNCT cana-1654 44 15 = = PRON cana-1654 44 16 { { PUNCT cana-1654 44 17 π‘₯𝑖π‘₯𝑖+1 π‘₯𝑖π‘₯𝑖+1 PROPN cana-1654 44 18 / / SYM cana-1654 44 19 2 2 NUM cana-1654 44 20 ≀ ≀ NOUN cana-1654 44 21 𝑖 𝑖 SYM cana-1654 44 22 ≀ ≀ NUM cana-1654 44 23 π‘Ÿ π‘Ÿ X cana-1654 44 24 βˆ’ βˆ’ PROPN cana-1654 44 25 1 1 NUM cana-1654 44 26 } } PUNCT cana-1654 44 27 βˆͺ βˆͺ ADJ cana-1654 44 28 { { PUNCT cana-1654 44 29 π‘₯1π‘₯𝑖+1/ π‘₯1π‘₯𝑖+1/ NOUN cana-1654 44 30 2 2 NUM cana-1654 44 31 ≀ ≀ NOUN cana-1654 44 32 𝑖 𝑖 SYM cana-1654 44 33 ≀ ≀ NUM cana-1654 44 34 π‘Ÿ π‘Ÿ X cana-1654 45 1 βˆ’ βˆ’ PROPN cana-1654 45 2 2 2 NUM cana-1654 45 3 } } PUNCT cana-1654 45 4 where where SCONJ cana-1654 45 5 𝑒𝑖 𝑒𝑖 ADP cana-1654 45 6 = = PROPN cana-1654 45 7 π‘₯𝑖π‘₯𝑖+1 π‘₯𝑖π‘₯𝑖+1 PROPN cana-1654 45 8 / / SYM cana-1654 45 9 2 2 NUM cana-1654 45 10 ≀ ≀ NOUN cana-1654 45 11 𝑖 𝑖 SYM cana-1654 45 12 ≀ ≀ NUM cana-1654 45 13 π‘Ÿ π‘Ÿ X cana-1654 46 1 βˆ’ βˆ’ PROPN cana-1654 46 2 1 1 NUM cana-1654 46 3 and and CCONJ cana-1654 46 4 𝑀𝑖 𝑀𝑖 ADP cana-1654 46 5 = = PUNCT cana-1654 46 6 π‘₯1π‘₯𝑖+1/ π‘₯1π‘₯𝑖+1/ NOUN cana-1654 46 7 2 2 NUM cana-1654 46 8 ≀ ≀ NOUN cana-1654 46 9 𝑖 𝑖 SYM cana-1654 46 10 ≀ ≀ NUM cana-1654 46 11 π‘Ÿ π‘Ÿ NOUN cana-1654 46 12 βˆ’ βˆ’ PROPN cana-1654 46 13 2 2 NUM cana-1654 46 14 define define VERB cana-1654 46 15 a a DET cana-1654 46 16 labeling labeling NOUN cana-1654 46 17 𝑓 𝑓 DET cana-1654 46 18 ∢ ∢ NOUN cana-1654 46 19 𝑉(𝐻π‘₯1 𝑉(𝐻π‘₯1 NOUN cana-1654 46 20 ) ) PUNCT cana-1654 46 21 β†’ β†’ SYM cana-1654 46 22 { { PUNCT cana-1654 46 23 1,3,7,15,31,63 1,3,7,15,31,63 NUM cana-1654 46 24 , , PUNCT cana-1654 46 25 … … PUNCT cana-1654 46 26 . . PUNCT cana-1654 46 27 . . PUNCT cana-1654 47 1 … … PUNCT cana-1654 47 2 . . PUNCT cana-1654 48 1 . . PUNCT cana-1654 49 1 , , PUNCT cana-1654 50 1 2𝑛+𝑖 2𝑛+𝑖 NUM cana-1654 50 2 βˆ’ βˆ’ NOUN cana-1654 50 3 1 1 NUM cana-1654 50 4 } } PUNCT cana-1654 50 5 is be AUX cana-1654 50 6 defined define VERB cana-1654 50 7 as as SCONJ cana-1654 50 8 follows follow VERB cana-1654 50 9 𝑓(π‘₯1 𝑓(π‘₯1 ADV cana-1654 50 10 ) ) PUNCT cana-1654 50 11 = = SYM cana-1654 50 12 1 1 NUM cana-1654 50 13 , , PUNCT cana-1654 50 14 communications communication NOUN cana-1654 50 15 on on ADP cana-1654 50 16 applied apply VERB cana-1654 50 17 nonlinear nonlinear ADJ cana-1654 50 18 analysis analysis NOUN cana-1654 50 19 issn issn NOUN cana-1654 50 20 : : PUNCT cana-1654 50 21 1074 1074 NUM cana-1654 50 22 - - PUNCT cana-1654 50 23 133x 133x NUM cana-1654 50 24 vol vol NOUN cana-1654 50 25 32 32 NUM cana-1654 50 26 no no NOUN cana-1654 50 27 . . NOUN cana-1654 50 28 1 1 NUM cana-1654 50 29 ( ( PUNCT cana-1654 50 30 2025 2025 NUM cana-1654 50 31 ) ) PUNCT cana-1654 51 1 337 337 NUM cana-1654 51 2 https://internationalpubls.com https://internationalpubls.com X cana-1654 51 3 𝑓(π‘₯𝑖+1 𝑓(π‘₯𝑖+1 NOUN cana-1654 51 4 ) ) PUNCT cana-1654 51 5 = = PUNCT cana-1654 52 1 2𝑖+1 2𝑖+1 NUM cana-1654 52 2 βˆ’ βˆ’ NOUN cana-1654 52 3 1 1 NUM cana-1654 52 4 for for ADP cana-1654 52 5 1 1 NUM cana-1654 52 6 ≀ ≀ NUM cana-1654 52 7 𝑖 𝑖 SYM cana-1654 52 8 ≀ ≀ NUM cana-1654 53 1 π‘Ÿ π‘Ÿ PRON cana-1654 53 2 βˆ’ βˆ’ PROPN cana-1654 53 3 1 1 NUM cana-1654 53 4 the the DET cana-1654 53 5 edge edge NOUN cana-1654 53 6 labeling label VERB cana-1654 53 7 π‘˜ π‘˜ DET cana-1654 53 8 ∢ ∢ NOUN cana-1654 53 9 𝐸(𝐻π‘₯1 𝐸(𝐻π‘₯1 ADJ cana-1654 53 10 ) ) PUNCT cana-1654 53 11 β†’ β†’ SYM cana-1654 53 12 { { PUNCT cana-1654 53 13 4,6,8,14,16 4,6,8,14,16 NUM cana-1654 53 14 , , PUNCT cana-1654 53 15 … … PUNCT cana-1654 53 16 . . PUNCT cana-1654 53 17 . . PUNCT cana-1654 54 1 … … PUNCT cana-1654 54 2 . . PUNCT cana-1654 55 1 . . PUNCT cana-1654 56 1 , , PUNCT cana-1654 56 2 4𝑖 4𝑖 NOUN cana-1654 56 3 + + CCONJ cana-1654 56 4 2 2 NUM cana-1654 56 5 } } PUNCT cana-1654 56 6 is be AUX cana-1654 56 7 defined define VERB cana-1654 56 8 as as ADP cana-1654 56 9 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NOUN cana-1654 56 10 ) ) PUNCT cana-1654 57 1 = = SYM cana-1654 57 2 2𝑖 2𝑖 NOUN cana-1654 57 3 for for ADP cana-1654 57 4 2 2 NUM cana-1654 57 5 ≀ ≀ NOUN cana-1654 57 6 𝑖 𝑖 SYM cana-1654 57 7 ≀ ≀ NUM cana-1654 57 8 π‘Ÿ π‘Ÿ X cana-1654 57 9 βˆ’ βˆ’ PROPN cana-1654 57 10 1 1 NUM cana-1654 57 11 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 PROPN cana-1654 57 12 ) ) PUNCT cana-1654 57 13 = = SYM cana-1654 58 1 2𝑖+1 2𝑖+1 NUM cana-1654 58 2 βˆ’ βˆ’ NOUN cana-1654 58 3 2 2 NUM cana-1654 58 4 for for ADP cana-1654 58 5 2 2 NUM cana-1654 58 6 ≀ ≀ NOUN cana-1654 58 7 𝑖 𝑖 SYM cana-1654 58 8 ≀ ≀ NUM cana-1654 59 1 π‘Ÿ π‘Ÿ NOUN cana-1654 59 2 βˆ’ βˆ’ PROPN cana-1654 59 3 2 2 NUM cana-1654 59 4 clearly clearly ADV cana-1654 59 5 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NOUN cana-1654 59 6 ) ) PUNCT cana-1654 59 7 divides divide VERB cana-1654 59 8 |(𝑓(π‘₯𝑖 |(𝑓(π‘₯𝑖 NOUN cana-1654 59 9 ) ) PUNCT cana-1654 59 10 βˆ’ βˆ’ PROPN cana-1654 59 11 𝑓(π‘₯𝑖+1))| 𝑓(π‘₯𝑖+1))| PROPN cana-1654 59 12 for for ADP cana-1654 59 13 2 2 NUM cana-1654 59 14 ≀ ≀ NOUN cana-1654 59 15 𝑖 𝑖 SYM cana-1654 59 16 ≀ ≀ NUM cana-1654 59 17 π‘Ÿ π‘Ÿ X cana-1654 59 18 βˆ’ βˆ’ PROPN cana-1654 59 19 1 1 NUM cana-1654 59 20 and and CCONJ cana-1654 59 21 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 NUM cana-1654 59 22 ) ) PUNCT cana-1654 59 23 divides divide VERB cana-1654 59 24 |(𝑓(π‘₯1 |(𝑓(π‘₯1 ADV cana-1654 59 25 ) ) PUNCT cana-1654 59 26 βˆ’ βˆ’ PROPN cana-1654 59 27 𝑓(π‘₯𝑖+1))| 𝑓(π‘₯𝑖+1))| PROPN cana-1654 59 28 for for ADP cana-1654 59 29 2 2 NUM cana-1654 59 30 ≀ ≀ NOUN cana-1654 59 31 𝑖 𝑖 SYM cana-1654 59 32 ≀ ≀ NUM cana-1654 59 33 π‘Ÿ π‘Ÿ NOUN cana-1654 59 34 βˆ’ βˆ’ PROPN cana-1654 59 35 2 2 NUM cana-1654 59 36 hence hence ADV cana-1654 59 37 the the DET cana-1654 59 38 graph graph NOUN cana-1654 59 39 obtained obtain VERB cana-1654 59 40 from from ADP cana-1654 59 41 switching switching NOUN cana-1654 59 42 of of ADP cana-1654 59 43 any any DET cana-1654 59 44 vertex vertex NOUN cana-1654 59 45 in in ADP cana-1654 59 46 cycle cycle NOUN cana-1654 59 47 πΆπ‘Ÿ πΆπ‘Ÿ PROPN cana-1654 59 48 is be AUX cana-1654 59 49 an an DET cana-1654 59 50 odd odd ADJ cana-1654 59 51 - - PUNCT cana-1654 59 52 even even ADV cana-1654 59 53 congruence congruence NOUN cana-1654 59 54 graph graph NOUN cana-1654 59 55 . . PUNCT cana-1654 60 1 example example NOUN cana-1654 60 2 : : PUNCT cana-1654 60 3 3.2 3.2 NUM cana-1654 60 4 consider consider VERB cana-1654 60 5 a a DET cana-1654 60 6 graph graph NOUN cana-1654 60 7 𝐺 𝐺 NOUN cana-1654 60 8 = = NOUN cana-1654 60 9 𝐻π‘₯1 𝐻π‘₯1 NOUN cana-1654 60 10 obtained obtain VERB cana-1654 60 11 from from ADP cana-1654 60 12 𝐢8 𝐢8 NOUN cana-1654 60 13 . . PUNCT cana-1654 61 1 figure-1 figure-1 PROPN cana-1654 61 2 fig-1 fig-1 PART cana-1654 61 3 shows show VERB cana-1654 61 4 that that SCONJ cana-1654 61 5 graph graph NOUN cana-1654 61 6 𝐺 𝐺 PROPN cana-1654 61 7 admits admit VERB cana-1654 61 8 odd odd ADJ cana-1654 61 9 - - PUNCT cana-1654 61 10 even even ADV cana-1654 61 11 congruence congruence ADJ cana-1654 61 12 labeling labeling NOUN cana-1654 61 13 . . PUNCT cana-1654 62 1 theorem theorem VERB cana-1654 62 2 3.3 3.3 NUM cana-1654 62 3 the the DET cana-1654 62 4 friendship friendship NOUN cana-1654 62 5 graph graph NOUN cana-1654 62 6 πΉπ‘Ÿπ‘› πΉπ‘Ÿπ‘› PROPN cana-1654 62 7 is be AUX cana-1654 62 8 an an DET cana-1654 62 9 odd odd ADJ cana-1654 62 10 - - PUNCT cana-1654 62 11 even even ADV cana-1654 62 12 congruence congruence NOUN cana-1654 62 13 graph graph NOUN cana-1654 62 14 . . PUNCT cana-1654 63 1 proof proof NOUN cana-1654 63 2 : : PUNCT cana-1654 63 3 let let VERB cana-1654 63 4 the the DET cana-1654 63 5 vertex vertex NOUN cana-1654 63 6 of of ADP cana-1654 63 7 πΉπ‘Ÿπ‘› πΉπ‘Ÿπ‘› PROPN cana-1654 63 8 be be AUX cana-1654 63 9 𝑉(πΉπ‘Ÿπ‘› 𝑉(πΉπ‘Ÿπ‘› NOUN cana-1654 63 10 ) ) PUNCT cana-1654 64 1 = = PRON cana-1654 64 2 { { PUNCT cana-1654 64 3 𝑣𝑖 𝑣𝑖 NOUN cana-1654 64 4 / / SYM cana-1654 64 5 𝑖 𝑖 SYM cana-1654 64 6 = = SYM cana-1654 64 7 0,1,2,3 0,1,2,3 NUM cana-1654 64 8 … … PUNCT cana-1654 64 9 … … PUNCT cana-1654 64 10 … … PUNCT cana-1654 64 11 … … PUNCT cana-1654 64 12 … … PUNCT cana-1654 64 13 2𝑛 2𝑛 NUM cana-1654 64 14 } } PUNCT cana-1654 64 15 with with ADP cana-1654 64 16 𝑣0 𝑣0 PROPN cana-1654 64 17 as as ADP cana-1654 64 18 the the DET cana-1654 64 19 central central ADJ cana-1654 64 20 vertex vertex NOUN cana-1654 64 21 and and CCONJ cana-1654 64 22 the the DET cana-1654 64 23 edge edge NOUN cana-1654 64 24 set set NOUN cana-1654 64 25 be be AUX cana-1654 64 26 𝐸(πΉπ‘Ÿπ‘› 𝐸(πΉπ‘Ÿπ‘› NOUN cana-1654 64 27 ) ) PUNCT cana-1654 64 28 = = SYM cana-1654 64 29 { { PUNCT cana-1654 64 30 𝑣0𝑣𝑖 𝑣0𝑣𝑖 NOUN cana-1654 64 31 , , PUNCT cana-1654 64 32 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-1654 64 33 , , PUNCT cana-1654 64 34 𝑖 𝑖 PUNCT cana-1654 64 35 = = SYM cana-1654 64 36 1,2 1,2 NUM cana-1654 64 37 … … PUNCT cana-1654 64 38 … … PUNCT cana-1654 64 39 … … PUNCT cana-1654 64 40 … … PUNCT cana-1654 64 41 … … PUNCT cana-1654 64 42 2𝑛 2𝑛 NUM cana-1654 64 43 } } PUNCT cana-1654 64 44 where where SCONJ cana-1654 64 45 𝑒𝑖 𝑒𝑖 X cana-1654 64 46 = = PRON cana-1654 64 47 { { PUNCT cana-1654 64 48 𝑣0𝑣𝑖 𝑣0𝑣𝑖 NOUN cana-1654 64 49 /1 /1 NOUN cana-1654 64 50 ≀ ≀ NUM cana-1654 64 51 𝑖 𝑖 SYM cana-1654 64 52 ≀ ≀ NOUN cana-1654 64 53 2𝑛 2𝑛 NUM cana-1654 64 54 } } PUNCT cana-1654 64 55 and and CCONJ cana-1654 64 56 𝑀𝑖 𝑀𝑖 ADP cana-1654 64 57 = = SYM cana-1654 64 58 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-1654 64 59 for for ADP cana-1654 64 60 𝑖 𝑖 NOUN cana-1654 64 61 = = SYM cana-1654 64 62 1,3,5 1,3,5 NUM cana-1654 64 63 , , PUNCT cana-1654 64 64 … … PUNCT cana-1654 64 65 … … PUNCT cana-1654 64 66 … … PUNCT cana-1654 64 67 2𝑛 2𝑛 NOUN cana-1654 64 68 βˆ’ βˆ’ NOUN cana-1654 64 69 1 1 NUM cana-1654 64 70 with with ADP cana-1654 64 71 |𝑉| |𝑉| NOUN cana-1654 64 72 = = SYM cana-1654 64 73 2𝑛 2𝑛 PROPN cana-1654 65 1 + + CCONJ cana-1654 65 2 1 1 NUM cana-1654 65 3 and and CCONJ cana-1654 65 4 |𝐸| |𝐸| NOUN cana-1654 65 5 = = SYM cana-1654 65 6 3𝑛 3𝑛 NUM cana-1654 65 7 for for ADP cana-1654 65 8 𝐺 𝐺 PROPN cana-1654 65 9 = = SYM cana-1654 65 10 πΉπ‘Ÿπ‘› πΉπ‘Ÿπ‘› PROPN cana-1654 65 11 , , PUNCT cana-1654 65 12 we we PRON cana-1654 65 13 have have VERB cana-1654 65 14 𝑑 𝑑 NOUN cana-1654 65 15 = = SYM cana-1654 65 16 π‘šπ‘–π‘› π‘šπ‘–π‘› NOUN cana-1654 65 17 { { PUNCT cana-1654 65 18 2(2𝑛 2(2𝑛 NUM cana-1654 65 19 + + CCONJ cana-1654 65 20 1 1 NUM cana-1654 65 21 ) ) PUNCT cana-1654 65 22 , , PUNCT cana-1654 65 23 2(3𝑛 2(3𝑛 NUM cana-1654 65 24 ) ) PUNCT cana-1654 65 25 } } PUNCT cana-1654 65 26 = = PUNCT cana-1654 66 1 2(2𝑛 2(2𝑛 NUM cana-1654 66 2 + + CCONJ cana-1654 66 3 1 1 X cana-1654 66 4 ) ) PUNCT cana-1654 66 5 the the DET cana-1654 66 6 vertices vertex NOUN cana-1654 66 7 of of ADP cana-1654 66 8 πΉπ‘Ÿπ‘› πΉπ‘Ÿπ‘› PROPN cana-1654 66 9 are be AUX cana-1654 66 10 labeled label VERB cana-1654 66 11 as as ADP cana-1654 66 12 given give VERB cana-1654 66 13 below below ADV cana-1654 66 14 . . PUNCT cana-1654 67 1 communications communication NOUN cana-1654 67 2 on on ADP cana-1654 67 3 applied apply VERB cana-1654 67 4 nonlinear nonlinear ADJ cana-1654 67 5 analysis analysis NOUN cana-1654 67 6 issn issn NOUN cana-1654 67 7 : : PUNCT cana-1654 67 8 1074 1074 NUM cana-1654 67 9 - - PUNCT cana-1654 67 10 133x 133x NUM cana-1654 67 11 vol vol NOUN cana-1654 67 12 32 32 NUM cana-1654 67 13 no no NOUN cana-1654 67 14 . . NOUN cana-1654 67 15 1 1 NUM cana-1654 67 16 ( ( PUNCT cana-1654 67 17 2025 2025 NUM cana-1654 67 18 ) ) PUNCT cana-1654 67 19 338 338 NUM cana-1654 67 20 https://internationalpubls.com https://internationalpubls.com X cana-1654 67 21 define define VERB cana-1654 67 22 the the DET cana-1654 67 23 bijection bijection NOUN cana-1654 67 24 𝑓 𝑓 DET cana-1654 67 25 ∢ ∢ NOUN cana-1654 67 26 𝑉(πΉπ‘Ÿπ‘› 𝑉(πΉπ‘Ÿπ‘› NOUN cana-1654 67 27 ) ) PUNCT cana-1654 67 28 β†’ β†’ SYM cana-1654 67 29 { { PUNCT cana-1654 67 30 1,3,7 1,3,7 NUM cana-1654 67 31 , , PUNCT cana-1654 67 32 . . PUNCT cana-1654 67 33 . . PUNCT cana-1654 67 34 . . PUNCT cana-1654 67 35 . . PUNCT cana-1654 67 36 . . PUNCT cana-1654 67 37 . . PUNCT cana-1654 67 38 . . PUNCT cana-1654 67 39 . . PUNCT cana-1654 67 40 . . PUNCT cana-1654 67 41 . . PUNCT cana-1654 67 42 . . PUNCT cana-1654 67 43 . . PUNCT cana-1654 67 44 . . PUNCT cana-1654 67 45 . . PUNCT cana-1654 67 46 . . PUNCT cana-1654 68 1 , , PUNCT cana-1654 68 2 2𝑖 2𝑖 NOUN cana-1654 68 3 βˆ’ βˆ’ NOUN cana-1654 68 4 1 1 NUM cana-1654 68 5 } } PUNCT cana-1654 68 6 as as ADP cana-1654 68 7 𝑓(π‘£π‘–βˆ’1 𝑓(π‘£π‘–βˆ’1 NOUN cana-1654 68 8 ) ) PUNCT cana-1654 69 1 = = SYM cana-1654 69 2 2𝑖 2𝑖 NOUN cana-1654 69 3 βˆ’ βˆ’ NOUN cana-1654 69 4 1 1 NUM cana-1654 69 5 for for ADP cana-1654 69 6 1 1 NUM cana-1654 69 7 ≀ ≀ NUM cana-1654 69 8 𝑖 𝑖 SYM cana-1654 69 9 ≀ ≀ NOUN cana-1654 69 10 2𝑛 2𝑛 NOUN cana-1654 70 1 + + CCONJ cana-1654 70 2 1 1 NUM cana-1654 70 3 the the DET cana-1654 70 4 edge edge NOUN cana-1654 70 5 labeling label VERB cana-1654 70 6 π‘˜ π‘˜ ADP cana-1654 70 7 ∢ ∢ NOUN cana-1654 70 8 𝐸(πΉπ‘Ÿπ‘› 𝐸(πΉπ‘Ÿπ‘› NOUN cana-1654 70 9 ) ) PUNCT cana-1654 70 10 β†’ β†’ SYM cana-1654 70 11 { { PUNCT cana-1654 70 12 2,4,6,8 2,4,6,8 NUM cana-1654 70 13 , , PUNCT cana-1654 70 14 … … PUNCT cana-1654 70 15 . . PUNCT cana-1654 70 16 . . PUNCT cana-1654 71 1 … … PUNCT cana-1654 71 2 . . PUNCT cana-1654 72 1 . . PUNCT cana-1654 73 1 , , PUNCT cana-1654 73 2 4𝑖 4𝑖 NOUN cana-1654 73 3 + + CCONJ cana-1654 73 4 2 2 NUM cana-1654 73 5 } } PUNCT cana-1654 73 6 is be AUX cana-1654 73 7 defined define VERB cana-1654 73 8 as as ADP cana-1654 73 9 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NOUN cana-1654 73 10 ) ) PUNCT cana-1654 74 1 = = PUNCT cana-1654 74 2 2𝑖+1 2𝑖+1 NUM cana-1654 74 3 βˆ’ βˆ’ NOUN cana-1654 74 4 2 2 NUM cana-1654 74 5 for for ADP cana-1654 74 6 1 1 NUM cana-1654 74 7 ≀ ≀ NUM cana-1654 74 8 𝑖 𝑖 SYM cana-1654 74 9 ≀ ≀ NOUN cana-1654 74 10 2𝑛 2𝑛 PROPN cana-1654 74 11 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 PROPN cana-1654 74 12 ) ) PUNCT cana-1654 75 1 = = SYM cana-1654 75 2 4𝑖 4𝑖 NOUN cana-1654 75 3 for for ADP cana-1654 75 4 𝑖 𝑖 PRON cana-1654 75 5 = = SYM cana-1654 75 6 1,3,5 1,3,5 NUM cana-1654 75 7 , , PUNCT cana-1654 75 8 … … PUNCT cana-1654 75 9 … … PUNCT cana-1654 75 10 … … PUNCT cana-1654 75 11 2𝑛 2𝑛 NOUN cana-1654 76 1 βˆ’ βˆ’ PROPN cana-1654 76 2 1 1 NUM cana-1654 76 3 clearly clearly ADV cana-1654 76 4 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NOUN cana-1654 76 5 ) ) PUNCT cana-1654 76 6 divides divide VERB cana-1654 76 7 |(𝑓(𝑣𝑖 |(𝑓(𝑣𝑖 PROPN cana-1654 76 8 ) ) PUNCT cana-1654 76 9 βˆ’ βˆ’ PROPN cana-1654 76 10 𝑓(𝑣0))| 𝑓(𝑣0))| PROPN cana-1654 76 11 for for ADP cana-1654 76 12 1 1 NUM cana-1654 76 13 ≀ ≀ NUM cana-1654 76 14 𝑖 𝑖 SYM cana-1654 76 15 ≀ ≀ NOUN cana-1654 76 16 2𝑛 2𝑛 NOUN cana-1654 76 17 and and CCONJ cana-1654 76 18 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 PROPN cana-1654 76 19 ) ) PUNCT cana-1654 76 20 divides divide VERB cana-1654 76 21 |(𝑓(𝑣𝑖 |(𝑓(𝑣𝑖 PROPN cana-1654 76 22 ) ) PUNCT cana-1654 76 23 βˆ’ βˆ’ NOUN cana-1654 76 24 𝑓(𝑣𝑖+1))| 𝑓(𝑣𝑖+1))| NOUN cana-1654 76 25 for for ADP cana-1654 76 26 𝑖 𝑖 NOUN cana-1654 76 27 = = SYM cana-1654 76 28 1,3,5 1,3,5 NUM cana-1654 76 29 , , PUNCT cana-1654 76 30 … … PUNCT cana-1654 76 31 … … PUNCT cana-1654 76 32 … … PUNCT cana-1654 76 33 2𝑛 2𝑛 NOUN cana-1654 76 34 βˆ’ βˆ’ NOUN cana-1654 76 35 1 1 NUM cana-1654 76 36 hence hence ADV cana-1654 76 37 the the DET cana-1654 76 38 graph graph NOUN cana-1654 76 39 friendship friendship NOUN cana-1654 76 40 graph graph NOUN cana-1654 76 41 πΉπ‘Ÿπ‘› πΉπ‘Ÿπ‘› PROPN cana-1654 76 42 is be AUX cana-1654 76 43 an an DET cana-1654 76 44 odd odd ADJ cana-1654 76 45 - - PUNCT cana-1654 76 46 even even ADV cana-1654 76 47 congruence congruence NOUN cana-1654 76 48 graph graph NOUN cana-1654 76 49 . . PUNCT cana-1654 77 1 example example NOUN cana-1654 77 2 : : PUNCT cana-1654 77 3 3.4 3.4 NUM cana-1654 77 4 consider consider VERB cana-1654 77 5 a a DET cana-1654 77 6 graph graph NOUN cana-1654 77 7 𝐺 𝐺 NOUN cana-1654 77 8 = = PUNCT cana-1654 77 9 πΉπ‘Ÿπ‘› πΉπ‘Ÿπ‘› PROPN cana-1654 77 10 with with ADP cana-1654 77 11 𝑛 𝑛 NOUN cana-1654 77 12 = = SYM cana-1654 77 13 7 7 NUM cana-1654 77 14 figure-2 figure-2 NUM cana-1654 77 15 fig-2 fig-2 PUNCT cana-1654 77 16 shows show VERB cana-1654 77 17 that that SCONJ cana-1654 77 18 friendship friendship NOUN cana-1654 77 19 graph graph NOUN cana-1654 77 20 πΉπ‘Ÿ7 πΉπ‘Ÿ7 NOUN cana-1654 77 21 admits admit VERB cana-1654 77 22 odd odd ADJ cana-1654 77 23 - - PUNCT cana-1654 77 24 even even ADV cana-1654 77 25 congruence congruence ADJ cana-1654 77 26 labeling labeling NOUN cana-1654 77 27 . . PUNCT cana-1654 78 1 theorem theorem VERB cana-1654 78 2 3.5 3.5 NUM cana-1654 78 3 the the DET cana-1654 78 4 shell shell NOUN cana-1654 78 5 graph graph NOUN cana-1654 78 6 𝑆𝑛 𝑆𝑛 PROPN cana-1654 78 7 is be AUX cana-1654 78 8 an an DET cana-1654 78 9 odd odd ADJ cana-1654 78 10 - - PUNCT cana-1654 78 11 even even ADV cana-1654 78 12 congruence congruence NOUN cana-1654 78 13 graph graph NOUN cana-1654 78 14 . . PUNCT cana-1654 79 1 proof proof NOUN cana-1654 79 2 : : PUNCT cana-1654 79 3 let let VERB cana-1654 79 4 𝐺 𝐺 PRON cana-1654 79 5 be be AUX cana-1654 79 6 a a DET cana-1654 79 7 shell shell ADJ cana-1654 79 8 graph graph NOUN cana-1654 79 9 . . PUNCT cana-1654 80 1 define define VERB cana-1654 80 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 80 3 ) ) PUNCT cana-1654 80 4 = = SYM cana-1654 80 5 { { PUNCT cana-1654 80 6 𝑒 𝑒 PROPN cana-1654 80 7 , , PUNCT cana-1654 80 8 𝑣𝑖 𝑣𝑖 ADV cana-1654 80 9 /1 /1 NOUN cana-1654 80 10 ≀ ≀ NUM cana-1654 80 11 𝑖 𝑖 SYM cana-1654 80 12 ≀ ≀ NOUN cana-1654 80 13 ( ( PUNCT cana-1654 80 14 𝑛 𝑛 PRON cana-1654 80 15 βˆ’ βˆ’ PROPN cana-1654 80 16 1 1 NUM cana-1654 80 17 ) ) PUNCT cana-1654 80 18 } } PUNCT cana-1654 80 19 and and CCONJ cana-1654 80 20 𝐸(𝐺 𝐸(𝐺 NUM cana-1654 80 21 ) ) PUNCT cana-1654 81 1 = = SYM cana-1654 81 2 { { PUNCT cana-1654 81 3 𝑒𝑖 𝑒𝑖 PART cana-1654 81 4 = = PUNCT cana-1654 81 5 𝑒𝑣𝑖 𝑒𝑣𝑖 ADJ cana-1654 81 6 ∢ ∢ NOUN cana-1654 81 7 1 1 NUM cana-1654 81 8 ≀ ≀ NUM cana-1654 81 9 𝑖 𝑖 SYM cana-1654 81 10 ≀ ≀ NUM cana-1654 81 11 𝑛 𝑛 PRON cana-1654 81 12 βˆ’ βˆ’ PROPN cana-1654 81 13 1 1 NUM cana-1654 81 14 ; ; PUNCT cana-1654 81 15 𝑒𝑖 𝑒𝑖 PRON cana-1654 81 16 β€² β€² NOUN cana-1654 81 17 = = SYM cana-1654 81 18 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 NOUN cana-1654 81 19 : : PUNCT cana-1654 81 20 1 1 NUM cana-1654 81 21 ≀ ≀ NUM cana-1654 81 22 𝑖 𝑖 SYM cana-1654 81 23 ≀ ≀ NUM cana-1654 81 24 𝑛 𝑛 PRON cana-1654 81 25 βˆ’ βˆ’ PROPN cana-1654 81 26 2 2 NUM cana-1654 81 27 } } PUNCT cana-1654 81 28 are be AUX cana-1654 81 29 the the DET cana-1654 81 30 vertices vertex NOUN cana-1654 81 31 and and CCONJ cana-1654 81 32 edges edge NOUN cana-1654 81 33 of of ADP cana-1654 81 34 a a DET cana-1654 81 35 graph graph NOUN cana-1654 81 36 𝐺 𝐺 NOUN cana-1654 81 37 also also ADV cana-1654 81 38 |𝑉(𝐺)| |𝑉(𝐺)| PROPN cana-1654 81 39 = = SYM cana-1654 81 40 𝑛 𝑛 PROPN cana-1654 81 41 and and CCONJ cana-1654 81 42 |𝐸(𝐺)| |𝐸(𝐺)| NOUN cana-1654 81 43 = = NOUN cana-1654 81 44 2𝑛 2𝑛 PROPN cana-1654 82 1 βˆ’ βˆ’ NOUN cana-1654 82 2 3 3 NUM cana-1654 82 3 then then ADV cana-1654 82 4 𝑑 𝑑 PROPN cana-1654 82 5 = = ADJ cana-1654 82 6 min min X cana-1654 82 7 { { PUNCT cana-1654 82 8 2𝑛 2𝑛 NUM cana-1654 82 9 , , PUNCT cana-1654 82 10 2(2𝑛 2(2𝑛 NUM cana-1654 82 11 βˆ’ βˆ’ NOUN cana-1654 82 12 3 3 NUM cana-1654 82 13 ) ) PUNCT cana-1654 82 14 } } PUNCT cana-1654 82 15 = = SYM cana-1654 82 16 2𝑛 2𝑛 PROPN cana-1654 82 17 communications communication NOUN cana-1654 82 18 on on ADP cana-1654 82 19 applied apply VERB cana-1654 82 20 nonlinear nonlinear ADJ cana-1654 82 21 analysis analysis NOUN cana-1654 82 22 issn issn NOUN cana-1654 82 23 : : PUNCT cana-1654 82 24 1074 1074 NUM cana-1654 82 25 - - PUNCT cana-1654 82 26 133x 133x NUM cana-1654 82 27 vol vol NOUN cana-1654 82 28 32 32 NUM cana-1654 82 29 no no NOUN cana-1654 82 30 . . NOUN cana-1654 82 31 1 1 NUM cana-1654 82 32 ( ( PUNCT cana-1654 82 33 2025 2025 NUM cana-1654 82 34 ) ) PUNCT cana-1654 82 35 339 339 NUM cana-1654 82 36 https://internationalpubls.com https://internationalpubls.com X cana-1654 83 1 define define VERB cana-1654 83 2 a a DET cana-1654 83 3 bijective bijective ADJ cana-1654 83 4 function function NOUN cana-1654 83 5 𝑓 𝑓 DET cana-1654 83 6 ∢ ∢ NOUN cana-1654 83 7 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 83 8 ) ) PUNCT cana-1654 83 9 β†’ β†’ SYM cana-1654 83 10 { { PUNCT cana-1654 83 11 1,3,7 1,3,7 NUM cana-1654 83 12 , , PUNCT cana-1654 83 13 … … PUNCT cana-1654 83 14 . . PUNCT cana-1654 83 15 . . PUNCT cana-1654 84 1 … … PUNCT cana-1654 84 2 . . PUNCT cana-1654 85 1 . . PUNCT cana-1654 86 1 , , PUNCT cana-1654 87 1 2𝑖+1 2𝑖+1 NUM cana-1654 87 2 βˆ’ βˆ’ NOUN cana-1654 88 1 1 1 X cana-1654 88 2 } } PUNCT cana-1654 88 3 is be AUX cana-1654 88 4 defined define VERB cana-1654 88 5 as as SCONJ cana-1654 88 6 follows follow VERB cana-1654 88 7 𝑓(𝑒 𝑓(𝑒 PROPN cana-1654 88 8 ) ) PUNCT cana-1654 88 9 = = SYM cana-1654 88 10 1 1 NUM cana-1654 88 11 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-1654 88 12 ) ) PUNCT cana-1654 88 13 = = PUNCT cana-1654 89 1 2𝑖+1 2𝑖+1 NUM cana-1654 89 2 βˆ’ βˆ’ NOUN cana-1654 89 3 1 1 NUM cana-1654 89 4 for for ADP cana-1654 89 5 1 1 NUM cana-1654 89 6 ≀ ≀ NUM cana-1654 89 7 𝑖 𝑖 SYM cana-1654 89 8 ≀ ≀ NUM cana-1654 89 9 𝑛 𝑛 DET cana-1654 89 10 βˆ’ βˆ’ PROPN cana-1654 89 11 1 1 NUM cana-1654 89 12 the the DET cana-1654 89 13 edge edge NOUN cana-1654 89 14 labeling labeling NOUN cana-1654 89 15 π‘˜ π‘˜ ADP cana-1654 89 16 ∢ ∢ NOUN cana-1654 89 17 𝐸(𝐺 𝐸(𝐺 ADJ cana-1654 89 18 ) ) PUNCT cana-1654 89 19 β†’ β†’ SYM cana-1654 89 20 { { PUNCT cana-1654 89 21 2,4,8 2,4,8 NUM cana-1654 89 22 … … PUNCT cana-1654 89 23 . . PUNCT cana-1654 89 24 . . PUNCT cana-1654 90 1 … … PUNCT cana-1654 90 2 . . PUNCT cana-1654 91 1 . . PUNCT cana-1654 92 1 , , PUNCT cana-1654 92 2 4𝑖 4𝑖 NOUN cana-1654 92 3 + + CCONJ cana-1654 92 4 2 2 NUM cana-1654 92 5 } } PUNCT cana-1654 92 6 is be AUX cana-1654 92 7 defined define VERB cana-1654 92 8 as as ADP cana-1654 92 9 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NOUN cana-1654 92 10 ) ) PUNCT cana-1654 93 1 = = PUNCT cana-1654 93 2 2𝑖+1 2𝑖+1 NUM cana-1654 93 3 βˆ’ βˆ’ NOUN cana-1654 93 4 2 2 NUM cana-1654 93 5 for for ADP cana-1654 93 6 1 1 NUM cana-1654 93 7 ≀ ≀ NUM cana-1654 93 8 𝑖 𝑖 SYM cana-1654 93 9 ≀ ≀ NUM cana-1654 93 10 𝑛 𝑛 PRON cana-1654 93 11 βˆ’ βˆ’ NUM cana-1654 93 12 1 1 NUM cana-1654 93 13 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NUM cana-1654 93 14 β€² β€² NUM cana-1654 93 15 ) ) PUNCT cana-1654 94 1 = = NOUN cana-1654 94 2 2𝑖 2𝑖 NOUN cana-1654 94 3 for for ADP cana-1654 94 4 1 1 NUM cana-1654 94 5 ≀ ≀ NUM cana-1654 94 6 𝑖 𝑖 SYM cana-1654 94 7 ≀ ≀ NUM cana-1654 94 8 𝑛 𝑛 PRON cana-1654 94 9 βˆ’ βˆ’ PROPN cana-1654 94 10 2 2 NUM cana-1654 94 11 clearly clearly ADV cana-1654 94 12 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NOUN cana-1654 94 13 ) ) PUNCT cana-1654 94 14 divides divide VERB cana-1654 94 15 |(𝑓(𝑒 |(𝑓(𝑒 NOUN cana-1654 94 16 ) ) PUNCT cana-1654 94 17 βˆ’ βˆ’ NOUN cana-1654 95 1 𝑓(𝑣𝑖))| 𝑓(𝑣𝑖))| VERB cana-1654 95 2 for for ADP cana-1654 95 3 1 1 NUM cana-1654 95 4 ≀ ≀ NUM cana-1654 95 5 𝑖 𝑖 SYM cana-1654 95 6 ≀ ≀ NUM cana-1654 95 7 𝑛 𝑛 DET cana-1654 95 8 βˆ’ βˆ’ PROPN cana-1654 95 9 1 1 NUM cana-1654 95 10 and and CCONJ cana-1654 95 11 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 PROPN cana-1654 95 12 β€² β€² NOUN cana-1654 95 13 ) ) PUNCT cana-1654 95 14 divides divide VERB cana-1654 95 15 |(𝑓(𝑣𝑖 |(𝑓(𝑣𝑖 PROPN cana-1654 95 16 ) ) PUNCT cana-1654 95 17 βˆ’ βˆ’ NOUN cana-1654 96 1 𝑓(𝑣𝑖+1))| 𝑓(𝑣𝑖+1))| NOUN cana-1654 96 2 for for ADP cana-1654 96 3 1 1 NUM cana-1654 96 4 ≀ ≀ NUM cana-1654 96 5 𝑖 𝑖 SYM cana-1654 96 6 ≀ ≀ NUM cana-1654 96 7 𝑛 𝑛 DET cana-1654 96 8 βˆ’ βˆ’ NUM cana-1654 96 9 2 2 NUM cana-1654 96 10 hence hence ADV cana-1654 96 11 the the DET cana-1654 96 12 shell shell NOUN cana-1654 96 13 graph graph NOUN cana-1654 96 14 𝑆𝑛 𝑆𝑛 PROPN cana-1654 96 15 is be AUX cana-1654 96 16 an an DET cana-1654 96 17 odd odd ADJ cana-1654 96 18 - - PUNCT cana-1654 96 19 even even ADV cana-1654 96 20 congruence congruence NOUN cana-1654 96 21 graph graph NOUN cana-1654 96 22 . . PUNCT cana-1654 97 1 example example NOUN cana-1654 97 2 : : PUNCT cana-1654 97 3 3.6 3.6 NUM cana-1654 97 4 consider consider VERB cana-1654 97 5 a a DET cana-1654 97 6 shell shell NOUN cana-1654 97 7 graph graph NOUN cana-1654 97 8 𝐺 𝐺 NOUN cana-1654 97 9 = = PUNCT cana-1654 97 10 𝑆𝑛 𝑆𝑛 PROPN cana-1654 97 11 with with ADP cana-1654 97 12 𝑛 𝑛 NOUN cana-1654 97 13 = = SYM cana-1654 97 14 8 8 NUM cana-1654 97 15 figure-3 figure-3 NUM cana-1654 97 16 fig-3 fig-3 NUM cana-1654 97 17 shows show VERB cana-1654 97 18 that that SCONJ cana-1654 97 19 shell shell NOUN cana-1654 97 20 graph graph NOUN cana-1654 97 21 𝑆8 𝑆8 PROPN cana-1654 97 22 admits admit VERB cana-1654 97 23 odd odd ADJ cana-1654 97 24 - - PUNCT cana-1654 97 25 even even ADV cana-1654 97 26 congruence congruence ADJ cana-1654 97 27 labeling labeling NOUN cana-1654 97 28 . . PUNCT cana-1654 98 1 theorem theorem VERB cana-1654 98 2 3.7 3.7 NUM cana-1654 98 3 a a DET cana-1654 98 4 generalized generalized ADJ cana-1654 98 5 butterfly butterfly NOUN cana-1654 98 6 graph graph NOUN cana-1654 98 7 𝐡𝐹𝑛 𝐡𝐹𝑛 PROPN cana-1654 98 8 admits admit VERB cana-1654 98 9 odd odd ADJ cana-1654 98 10 - - PUNCT cana-1654 98 11 even even ADV cana-1654 98 12 congruence congruence NOUN cana-1654 98 13 graph graph NOUN cana-1654 98 14 for for ADP cana-1654 98 15 𝑛 𝑛 PRON cana-1654 98 16 β‰₯ β‰₯ NUM cana-1654 98 17 2 2 NUM cana-1654 98 18 . . PUNCT cana-1654 99 1 proof proof NOUN cana-1654 99 2 : : PUNCT cana-1654 99 3 suppose suppose VERB cana-1654 99 4 the the DET cana-1654 99 5 vertex vertex NOUN cana-1654 99 6 set set NOUN cana-1654 99 7 of of ADP cana-1654 99 8 𝐡𝐹𝑛 𝐡𝐹𝑛 NOUN cana-1654 99 9 be be AUX cana-1654 99 10 define define VERB cana-1654 99 11 𝑉(𝐡𝐹𝑛 𝑉(𝐡𝐹𝑛 PROPN cana-1654 99 12 ) ) PUNCT cana-1654 100 1 = = PRON cana-1654 100 2 { { PUNCT cana-1654 100 3 𝑣𝑖 𝑣𝑖 NOUN cana-1654 100 4 /0 /0 PUNCT cana-1654 100 5 ≀ ≀ NUM cana-1654 100 6 𝑖 𝑖 SYM cana-1654 100 7 ≀ ≀ NOUN cana-1654 100 8 2𝑛 2𝑛 NUM cana-1654 100 9 } } PUNCT cana-1654 100 10 and and CCONJ cana-1654 100 11 the the DET cana-1654 100 12 edge edge NOUN cana-1654 100 13 set set NOUN cana-1654 100 14 of of ADP cana-1654 100 15 𝐡𝐹𝑛 𝐡𝐹𝑛 NOUN cana-1654 100 16 be be AUX cana-1654 100 17 communications communication NOUN cana-1654 100 18 on on ADP cana-1654 100 19 applied apply VERB cana-1654 100 20 nonlinear nonlinear ADJ cana-1654 100 21 analysis analysis NOUN cana-1654 100 22 issn issn NOUN cana-1654 100 23 : : PUNCT cana-1654 100 24 1074 1074 NUM cana-1654 100 25 - - PUNCT cana-1654 100 26 133x 133x NUM cana-1654 100 27 vol vol NOUN cana-1654 100 28 32 32 NUM cana-1654 100 29 no no NOUN cana-1654 100 30 . . NOUN cana-1654 100 31 1 1 NUM cana-1654 100 32 ( ( PUNCT cana-1654 100 33 2025 2025 NUM cana-1654 100 34 ) ) PUNCT cana-1654 100 35 340 340 NUM cana-1654 100 36 https://internationalpubls.com https://internationalpubls.com X cana-1654 100 37 𝐸(𝐡𝐹𝑛 𝐸(𝐡𝐹𝑛 PROPN cana-1654 100 38 ) ) PUNCT cana-1654 100 39 = = PRON cana-1654 100 40 { { PUNCT cana-1654 100 41 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-1654 100 42 /1,2 /1,2 NOUN cana-1654 100 43 , , PUNCT cana-1654 100 44 … … PUNCT cana-1654 100 45 … … PUNCT cana-1654 100 46 . . PUNCT cana-1654 101 1 , , PUNCT cana-1654 101 2 𝑛 𝑛 DET cana-1654 101 3 βˆ’ βˆ’ PROPN cana-1654 101 4 1 1 NUM cana-1654 101 5 , , PUNCT cana-1654 101 6 𝑛 𝑛 PRON cana-1654 101 7 + + NOUN cana-1654 101 8 1 1 NUM cana-1654 101 9 , , PUNCT cana-1654 101 10 … … PUNCT cana-1654 101 11 … … PUNCT cana-1654 101 12 2𝑛 2𝑛 NOUN cana-1654 102 1 βˆ’ βˆ’ NOUN cana-1654 102 2 1 1 NUM cana-1654 102 3 } } PUNCT cana-1654 102 4 βˆͺ βˆͺ X cana-1654 102 5 { { PUNCT cana-1654 102 6 𝑣0𝑣𝑖 𝑣0𝑣𝑖 NOUN cana-1654 102 7 /𝑖 /𝑖 SYM cana-1654 102 8 = = SYM cana-1654 102 9 1,2 1,2 NUM cana-1654 102 10 , , PUNCT cana-1654 102 11 … … PUNCT cana-1654 102 12 … … PUNCT cana-1654 102 13 . . PUNCT cana-1654 102 14 . . PUNCT cana-1654 103 1 , , PUNCT cana-1654 103 2 2𝑛 2𝑛 NUM cana-1654 103 3 } } PUNCT cana-1654 103 4 where where SCONJ cana-1654 103 5 |𝑉(𝐺)| |𝑉(𝐺)| ADP cana-1654 103 6 = = SYM cana-1654 103 7 2𝑛 2𝑛 PROPN cana-1654 104 1 + + CCONJ cana-1654 104 2 1 1 NUM cana-1654 104 3 and and CCONJ cana-1654 104 4 |𝐸(𝐺)| |𝐸(𝐺)| NOUN cana-1654 104 5 = = PUNCT cana-1654 104 6 4𝑛 4𝑛 PROPN cana-1654 104 7 βˆ’ βˆ’ NOUN cana-1654 105 1 2 2 NUM cana-1654 106 1 then then ADV cana-1654 106 2 𝑑 𝑑 PROPN cana-1654 106 3 = = ADJ cana-1654 106 4 min min X cana-1654 106 5 { { PUNCT cana-1654 106 6 2(2𝑛 2(2𝑛 NUM cana-1654 106 7 + + NOUN cana-1654 106 8 1),2(4𝑛 1),2(4𝑛 NUM cana-1654 106 9 βˆ’ βˆ’ NOUN cana-1654 106 10 2 2 NUM cana-1654 106 11 ) ) PUNCT cana-1654 106 12 } } PUNCT cana-1654 106 13 = = PUNCT cana-1654 106 14 2(2𝑛 2(2𝑛 NUM cana-1654 106 15 + + CCONJ cana-1654 106 16 1 1 X cana-1654 106 17 ) ) PUNCT cana-1654 106 18 define define VERB cana-1654 106 19 a a DET cana-1654 106 20 bijective bijective ADJ cana-1654 106 21 function function NOUN cana-1654 106 22 𝑓 𝑓 DET cana-1654 106 23 ∢ ∢ NOUN cana-1654 106 24 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 106 25 ) ) PUNCT cana-1654 106 26 β†’ β†’ SYM cana-1654 106 27 { { PUNCT cana-1654 106 28 1,3,7 1,3,7 NUM cana-1654 106 29 , , PUNCT cana-1654 106 30 … … PUNCT cana-1654 106 31 . . PUNCT cana-1654 106 32 . . PUNCT cana-1654 107 1 … … PUNCT cana-1654 107 2 . . PUNCT cana-1654 108 1 . . PUNCT cana-1654 109 1 , , PUNCT cana-1654 110 1 2𝑖+1 2𝑖+1 PROPN cana-1654 110 2 βˆ’ βˆ’ NOUN cana-1654 111 1 1 1 NUM cana-1654 111 2 } } PUNCT cana-1654 111 3 is be AUX cana-1654 111 4 defined define VERB cana-1654 111 5 as as SCONJ cana-1654 111 6 follows follow VERB cana-1654 111 7 𝑓(𝑒 𝑓(𝑒 PROPN cana-1654 111 8 ) ) PUNCT cana-1654 111 9 = = SYM cana-1654 111 10 1 1 NUM cana-1654 111 11 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-1654 111 12 ) ) PUNCT cana-1654 111 13 = = PUNCT cana-1654 112 1 2𝑖+1 2𝑖+1 NUM cana-1654 112 2 βˆ’ βˆ’ NOUN cana-1654 112 3 1 1 NUM cana-1654 112 4 for for ADP cana-1654 112 5 1 1 NUM cana-1654 112 6 ≀ ≀ NUM cana-1654 112 7 𝑖 𝑖 SYM cana-1654 112 8 ≀ ≀ NUM cana-1654 112 9 𝑛 𝑛 DET cana-1654 112 10 βˆ’ βˆ’ PROPN cana-1654 112 11 1 1 NUM cana-1654 112 12 the the DET cana-1654 112 13 edge edge NOUN cana-1654 112 14 labeling labeling NOUN cana-1654 112 15 π‘˜ π‘˜ ADP cana-1654 112 16 ∢ ∢ NOUN cana-1654 112 17 𝐸(𝐺 𝐸(𝐺 ADJ cana-1654 112 18 ) ) PUNCT cana-1654 112 19 β†’ β†’ SYM cana-1654 112 20 { { PUNCT cana-1654 112 21 2,4,8 2,4,8 NUM cana-1654 112 22 … … PUNCT cana-1654 112 23 . . PUNCT cana-1654 112 24 . . PUNCT cana-1654 113 1 … … PUNCT cana-1654 113 2 . . PUNCT cana-1654 114 1 . . PUNCT cana-1654 115 1 , , PUNCT cana-1654 115 2 4𝑖 4𝑖 NOUN cana-1654 115 3 + + CCONJ cana-1654 115 4 2 2 NUM cana-1654 115 5 } } PUNCT cana-1654 115 6 is be AUX cana-1654 115 7 defined define VERB cana-1654 115 8 as as ADP cana-1654 115 9 β„Ž(π‘£π‘–βˆ’1𝑣𝑖 β„Ž(π‘£π‘–βˆ’1𝑣𝑖 NOUN cana-1654 115 10 ) ) PUNCT cana-1654 116 1 = = SYM cana-1654 116 2 2𝑖 2𝑖 NOUN cana-1654 116 3 for for ADP cana-1654 116 4 2 2 NUM cana-1654 116 5 ≀ ≀ NOUN cana-1654 116 6 𝑖 𝑖 SYM cana-1654 116 7 ≀ ≀ NUM cana-1654 116 8 𝑛 𝑛 DET cana-1654 116 9 π‘˜(π‘£π‘–βˆ’1𝑣𝑖 π‘˜(π‘£π‘–βˆ’1𝑣𝑖 NOUN cana-1654 116 10 ) ) PUNCT cana-1654 117 1 = = SYM cana-1654 117 2 2𝑖 2𝑖 NOUN cana-1654 117 3 for for ADP cana-1654 117 4 𝑛 𝑛 PROPN cana-1654 117 5 + + CCONJ cana-1654 117 6 2 2 NUM cana-1654 117 7 ≀ ≀ NUM cana-1654 117 8 𝑖 𝑖 PRON cana-1654 117 9 ≀ ≀ PROPN cana-1654 117 10 2𝑛 2𝑛 PROPN cana-1654 117 11 𝑝(𝑣0𝑣𝑖 𝑝(𝑣0𝑣𝑖 NUM cana-1654 117 12 ) ) PUNCT cana-1654 117 13 = = PUNCT cana-1654 118 1 2𝑖+1 2𝑖+1 NUM cana-1654 118 2 βˆ’ βˆ’ NOUN cana-1654 118 3 2 2 NUM cana-1654 118 4 for for ADP cana-1654 118 5 1 1 NUM cana-1654 118 6 ≀ ≀ NUM cana-1654 118 7 𝑖 𝑖 PRON cana-1654 118 8 ≀ ≀ NOUN cana-1654 118 9 2𝑛 2𝑛 PROPN cana-1654 118 10 clearly clearly ADV cana-1654 118 11 β„Ž(π‘£π‘–βˆ’1𝑣𝑖 β„Ž(π‘£π‘–βˆ’1𝑣𝑖 NOUN cana-1654 118 12 ) ) PUNCT cana-1654 118 13 divides divide VERB cana-1654 118 14 |(𝑓(π‘£π‘–βˆ’1 |(𝑓(π‘£π‘–βˆ’1 NOUN cana-1654 118 15 ) ) PUNCT cana-1654 118 16 βˆ’ βˆ’ PROPN cana-1654 119 1 𝑓(𝑣𝑖))| 𝑓(𝑣𝑖))| VERB cana-1654 119 2 for for ADP cana-1654 119 3 2 2 NUM cana-1654 119 4 ≀ ≀ NOUN cana-1654 119 5 𝑖 𝑖 SYM cana-1654 119 6 ≀ ≀ NUM cana-1654 119 7 𝑛 𝑛 PRON cana-1654 119 8 , , PUNCT cana-1654 119 9 π‘˜(π‘£π‘–βˆ’1𝑣𝑖 π‘˜(π‘£π‘–βˆ’1𝑣𝑖 NOUN cana-1654 119 10 ) ) PUNCT cana-1654 119 11 divides divide VERB cana-1654 119 12 |(𝑓(𝑣𝑖 |(𝑓(𝑣𝑖 PROPN cana-1654 119 13 ) ) PUNCT cana-1654 119 14 βˆ’ βˆ’ NOUN cana-1654 120 1 𝑓(𝑣𝑖+1))| 𝑓(𝑣𝑖+1))| NOUN cana-1654 120 2 for for ADP cana-1654 120 3 𝑛 𝑛 DET cana-1654 120 4 + + CCONJ cana-1654 120 5 2 2 NUM cana-1654 120 6 ≀ ≀ NUM cana-1654 120 7 𝑖 𝑖 PRON cana-1654 120 8 ≀ ≀ NOUN cana-1654 120 9 2𝑛 2𝑛 PROPN cana-1654 120 10 and and CCONJ cana-1654 120 11 𝑝(𝑣0𝑣𝑖 𝑝(𝑣0𝑣𝑖 NUM cana-1654 120 12 ) ) PUNCT cana-1654 120 13 divides divide VERB cana-1654 120 14 |(𝑓(𝑣𝑖 |(𝑓(𝑣𝑖 PROPN cana-1654 120 15 ) ) PUNCT cana-1654 120 16 βˆ’ βˆ’ PROPN cana-1654 120 17 𝑓(𝑣0))| 𝑓(𝑣0))| PROPN cana-1654 120 18 for for ADP cana-1654 120 19 1 1 NUM cana-1654 120 20 ≀ ≀ NUM cana-1654 120 21 𝑖 𝑖 PRON cana-1654 120 22 ≀ ≀ NOUN cana-1654 120 23 2𝑛 2𝑛 NOUN cana-1654 120 24 hence hence ADV cana-1654 120 25 the the DET cana-1654 120 26 generalized generalized ADJ cana-1654 120 27 butterfly butterfly NOUN cana-1654 120 28 graph graph NOUN cana-1654 120 29 𝐡𝐹𝑛 𝐡𝐹𝑛 NOUN cana-1654 120 30 is be AUX cana-1654 120 31 an an DET cana-1654 120 32 odd odd ADJ cana-1654 120 33 - - PUNCT cana-1654 120 34 even even ADV cana-1654 120 35 congruence congruence NOUN cana-1654 120 36 graph graph NOUN cana-1654 120 37 . . PUNCT cana-1654 121 1 example example NOUN cana-1654 121 2 : : PUNCT cana-1654 121 3 3.8 3.8 NUM cana-1654 121 4 consider consider VERB cana-1654 121 5 a a DET cana-1654 121 6 generalized generalized ADJ cana-1654 121 7 butterfly butterfly NOUN cana-1654 121 8 graph graph NOUN cana-1654 121 9 𝐡𝐹𝑛 𝐡𝐹𝑛 NOUN cana-1654 121 10 with with ADP cana-1654 121 11 𝑛 𝑛 PROPN cana-1654 121 12 = = SYM cana-1654 121 13 5 5 NUM cana-1654 121 14 figure-4 figure-4 NUM cana-1654 121 15 fig-4 fig-4 NUM cana-1654 121 16 shows show VERB cana-1654 121 17 that that SCONJ cana-1654 121 18 friendship friendship NOUN cana-1654 121 19 graph graph NOUN cana-1654 121 20 πΉπ‘Ÿ6 πΉπ‘Ÿ6 NOUN cana-1654 121 21 admits admit VERB cana-1654 121 22 odd odd ADJ cana-1654 121 23 - - PUNCT cana-1654 121 24 even even ADV cana-1654 121 25 congruence congruence ADJ cana-1654 121 26 labeling labeling NOUN cana-1654 121 27 . . PUNCT cana-1654 122 1 theorem theorem VERB cana-1654 122 2 3.9 3.9 NUM cana-1654 122 3 the the DET cana-1654 122 4 fan fan NOUN cana-1654 122 5 graph graph NOUN cana-1654 122 6 𝐹𝑛 𝐹𝑛 PROPN cana-1654 122 7 admits admit VERB cana-1654 122 8 odd odd ADJ cana-1654 122 9 - - PUNCT cana-1654 122 10 even even ADV cana-1654 122 11 congruence congruence NOUN cana-1654 122 12 graph graph NOUN cana-1654 122 13 . . PUNCT cana-1654 123 1 proof proof NOUN cana-1654 123 2 : : PUNCT cana-1654 123 3 let let VERB cana-1654 123 4 𝐺 𝐺 PRON cana-1654 123 5 be be AUX cana-1654 123 6 a a DET cana-1654 123 7 graph graph NOUN cana-1654 123 8 of of ADP cana-1654 123 9 the the DET cana-1654 123 10 fan fan NOUN cana-1654 123 11 graph graph NOUN cana-1654 123 12 𝐹𝑛 𝐹𝑛 PROPN cana-1654 123 13 communications communication NOUN cana-1654 123 14 on on ADP cana-1654 123 15 applied apply VERB cana-1654 123 16 nonlinear nonlinear ADJ cana-1654 123 17 analysis analysis NOUN cana-1654 123 18 issn issn NOUN cana-1654 123 19 : : PUNCT cana-1654 123 20 1074 1074 NUM cana-1654 123 21 - - PUNCT cana-1654 123 22 133x 133x NUM cana-1654 123 23 vol vol NOUN cana-1654 123 24 32 32 NUM cana-1654 123 25 no no NOUN cana-1654 123 26 . . NOUN cana-1654 123 27 1 1 NUM cana-1654 123 28 ( ( PUNCT cana-1654 123 29 2025 2025 NUM cana-1654 123 30 ) ) PUNCT cana-1654 123 31 341 341 NUM cana-1654 123 32 https://internationalpubls.com https://internationalpubls.com X cana-1654 123 33 let let VERB cana-1654 123 34 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 123 35 ) ) PUNCT cana-1654 123 36 = = PRON cana-1654 123 37 { { PUNCT cana-1654 123 38 𝑣𝑖 𝑣𝑖 ADP cana-1654 123 39 : : PUNCT cana-1654 123 40 1 1 NUM cana-1654 123 41 ≀ ≀ NUM cana-1654 123 42 𝑖 𝑖 SYM cana-1654 123 43 ≀ ≀ NUM cana-1654 123 44 𝑛 𝑛 ADP cana-1654 123 45 + + NOUN cana-1654 123 46 1 1 NUM cana-1654 123 47 } } PUNCT cana-1654 123 48 be be AUX cana-1654 123 49 the the DET cana-1654 123 50 vertices vertex NOUN cana-1654 123 51 of of ADP cana-1654 123 52 𝐺 𝐺 PROPN cana-1654 123 53 and and CCONJ cana-1654 123 54 𝐸(𝐺 𝐸(𝐺 PROPN cana-1654 123 55 ) ) PUNCT cana-1654 123 56 = = SYM cana-1654 123 57 { { PUNCT cana-1654 123 58 𝑣1𝑣𝑖+1 𝑣1𝑣𝑖+1 PROPN cana-1654 123 59 : : PUNCT cana-1654 123 60 1 1 NUM cana-1654 123 61 ≀ ≀ NUM cana-1654 123 62 𝑖 𝑖 SYM cana-1654 123 63 ≀ ≀ NUM cana-1654 123 64 𝑛 𝑛 PRON cana-1654 123 65 } } PUNCT cana-1654 123 66 βˆͺ βˆͺ X cana-1654 123 67 { { PUNCT cana-1654 123 68 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-1654 123 69 ∢ ∢ NOUN cana-1654 123 70 2 2 NUM cana-1654 123 71 ≀ ≀ NOUN cana-1654 123 72 𝑖 𝑖 SYM cana-1654 123 73 ≀ ≀ NUM cana-1654 123 74 𝑛 𝑛 PRON cana-1654 123 75 } } PUNCT cana-1654 123 76 be be VERB cana-1654 123 77 the the DET cana-1654 123 78 edges edge NOUN cana-1654 123 79 of of ADP cana-1654 123 80 𝐺. 𝐺. NOUN cana-1654 123 81 where where SCONJ cana-1654 123 82 𝑒𝑖 𝑒𝑖 X cana-1654 123 83 = = SYM cana-1654 123 84 { { PUNCT cana-1654 123 85 𝑣1𝑣𝑖+1 𝑣1𝑣𝑖+1 PROPN cana-1654 123 86 : : PUNCT cana-1654 123 87 1 1 NUM cana-1654 123 88 ≀ ≀ NUM cana-1654 123 89 𝑖 𝑖 SYM cana-1654 123 90 ≀ ≀ NUM cana-1654 123 91 𝑛 𝑛 NOUN cana-1654 123 92 } } PUNCT cana-1654 123 93 and and CCONJ cana-1654 123 94 𝑀𝑖 𝑀𝑖 ADP cana-1654 123 95 = = SYM cana-1654 123 96 { { PUNCT cana-1654 123 97 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN cana-1654 123 98 /2 /2 PUNCT cana-1654 124 1 ≀ ≀ NUM cana-1654 124 2 𝑖 𝑖 SYM cana-1654 124 3 ≀ ≀ NUM cana-1654 124 4 𝑛 𝑛 NOUN cana-1654 124 5 } } PUNCT cana-1654 124 6 now now ADV cana-1654 124 7 |𝑉(𝐺)| |𝑉(𝐺)| ADP cana-1654 124 8 = = SYM cana-1654 124 9 𝑛 𝑛 PROPN cana-1654 125 1 + + NUM cana-1654 125 2 1 1 NUM cana-1654 125 3 and and CCONJ cana-1654 125 4 |𝐸(𝐺)| |𝐸(𝐺)| NOUN cana-1654 125 5 = = NOUN cana-1654 125 6 2𝑛 2𝑛 PROPN cana-1654 125 7 βˆ’ βˆ’ NOUN cana-1654 126 1 1 1 NUM cana-1654 126 2 then then ADV cana-1654 126 3 𝑑 𝑑 PROPN cana-1654 126 4 = = ADJ cana-1654 126 5 min min PROPN cana-1654 126 6 ( ( PUNCT cana-1654 126 7 2(𝑛 2(𝑛 NUM cana-1654 126 8 + + CCONJ cana-1654 126 9 1 1 NUM cana-1654 126 10 ) ) PUNCT cana-1654 126 11 , , PUNCT cana-1654 126 12 2(2𝑛 2(2𝑛 NUM cana-1654 126 13 βˆ’ βˆ’ NOUN cana-1654 126 14 1 1 NUM cana-1654 126 15 ) ) PUNCT cana-1654 126 16 ) ) PUNCT cana-1654 127 1 = = PUNCT cana-1654 127 2 2(𝑛 2(𝑛 NUM cana-1654 127 3 + + CCONJ cana-1654 127 4 1 1 X cana-1654 127 5 ) ) PUNCT cana-1654 127 6 define define VERB cana-1654 127 7 the the DET cana-1654 127 8 bijection bijection NOUN cana-1654 127 9 𝑓 𝑓 DET cana-1654 127 10 ∢ ∢ NOUN cana-1654 127 11 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 127 12 ) ) PUNCT cana-1654 127 13 β†’ β†’ SYM cana-1654 127 14 { { PUNCT cana-1654 127 15 1,3,7 1,3,7 NUM cana-1654 127 16 , , PUNCT cana-1654 127 17 … … PUNCT cana-1654 127 18 . . PUNCT cana-1654 127 19 . . PUNCT cana-1654 128 1 … … PUNCT cana-1654 128 2 . . PUNCT cana-1654 129 1 . . PUNCT cana-1654 130 1 , , PUNCT cana-1654 130 2 2𝑝 2𝑝 NOUN cana-1654 130 3 βˆ’ βˆ’ NOUN cana-1654 130 4 1 1 NUM cana-1654 130 5 } } PUNCT cana-1654 130 6 is be AUX cana-1654 130 7 defined define VERB cana-1654 130 8 as as SCONJ cana-1654 130 9 follows follow VERB cana-1654 130 10 𝑓(𝑣𝑖 𝑓(𝑣𝑖 NOUN cana-1654 130 11 ) ) PUNCT cana-1654 131 1 = = SYM cana-1654 132 1 2𝑖 2𝑖 NOUN cana-1654 132 2 βˆ’ βˆ’ NOUN cana-1654 132 3 1 1 NUM cana-1654 132 4 for for ADP cana-1654 132 5 1 1 NUM cana-1654 132 6 ≀ ≀ NUM cana-1654 132 7 𝑖 𝑖 SYM cana-1654 132 8 ≀ ≀ NUM cana-1654 132 9 𝑛 𝑛 PRON cana-1654 133 1 + + CCONJ cana-1654 133 2 1 1 NUM cana-1654 133 3 the the DET cana-1654 133 4 edge edge NOUN cana-1654 133 5 labeling labeling NOUN cana-1654 133 6 π‘˜ π‘˜ ADP cana-1654 133 7 ∢ ∢ NOUN cana-1654 133 8 𝐸(𝐺 𝐸(𝐺 ADJ cana-1654 133 9 ) ) PUNCT cana-1654 133 10 β†’ β†’ SYM cana-1654 133 11 { { PUNCT cana-1654 133 12 2,4,6,8,14 2,4,6,8,14 ADV cana-1654 133 13 , , PUNCT cana-1654 133 14 … … PUNCT cana-1654 133 15 . . PUNCT cana-1654 133 16 . . PUNCT cana-1654 134 1 … … PUNCT cana-1654 134 2 . . PUNCT cana-1654 135 1 . . PUNCT cana-1654 136 1 , , PUNCT cana-1654 137 1 2𝑖+1 2𝑖+1 PROPN cana-1654 137 2 βˆ’ βˆ’ NOUN cana-1654 138 1 2 2 X cana-1654 138 2 } } PUNCT cana-1654 138 3 is be AUX cana-1654 138 4 defined define VERB cana-1654 138 5 as as SCONJ cana-1654 138 6 follows follow VERB cana-1654 138 7 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 PROPN cana-1654 138 8 ) ) PUNCT cana-1654 139 1 = = PUNCT cana-1654 139 2 2𝑖+1 2𝑖+1 NUM cana-1654 139 3 βˆ’ βˆ’ NOUN cana-1654 139 4 2 2 NUM cana-1654 139 5 for for ADP cana-1654 139 6 1 1 NUM cana-1654 139 7 ≀ ≀ NUM cana-1654 139 8 𝑖 𝑖 SYM cana-1654 139 9 ≀ ≀ NUM cana-1654 139 10 𝑛 𝑛 PRON cana-1654 139 11 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 PROPN cana-1654 139 12 ) ) PUNCT cana-1654 139 13 = = SYM cana-1654 140 1 2𝑖+1 2𝑖+1 PROPN cana-1654 140 2 for for ADP cana-1654 140 3 2 2 NUM cana-1654 140 4 ≀ ≀ NOUN cana-1654 140 5 𝑖 𝑖 SYM cana-1654 140 6 ≀ ≀ NOUN cana-1654 140 7 𝑛 𝑛 PRON cana-1654 140 8 clearly clearly ADV cana-1654 140 9 π‘˜(𝑒𝑖 π‘˜(𝑒𝑖 NOUN cana-1654 140 10 ) ) PUNCT cana-1654 140 11 divides divide VERB cana-1654 140 12 |(𝑓(𝑣𝑖+1 |(𝑓(𝑣𝑖+1 PROPN cana-1654 140 13 ) ) PUNCT cana-1654 140 14 βˆ’ βˆ’ PROPN cana-1654 140 15 𝑓(𝑣1))| 𝑓(𝑣1))| NOUN cana-1654 140 16 for for ADP cana-1654 140 17 1 1 NUM cana-1654 140 18 ≀ ≀ NUM cana-1654 140 19 𝑖 𝑖 SYM cana-1654 140 20 ≀ ≀ NUM cana-1654 140 21 𝑛 𝑛 PRON cana-1654 140 22 and and CCONJ cana-1654 140 23 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 PROPN cana-1654 140 24 ) ) PUNCT cana-1654 140 25 divides divide VERB cana-1654 140 26 |(𝑓(𝑣𝑖+1 |(𝑓(𝑣𝑖+1 PROPN cana-1654 140 27 ) ) PUNCT cana-1654 141 1 βˆ’ βˆ’ PROPN cana-1654 142 1 𝑓(𝑣𝑖))| 𝑓(𝑣𝑖))| VERB cana-1654 142 2 for for ADP cana-1654 142 3 2 2 NUM cana-1654 142 4 ≀ ≀ NOUN cana-1654 142 5 𝑖 𝑖 SYM cana-1654 142 6 ≀ ≀ NUM cana-1654 142 7 𝑛 𝑛 PRON cana-1654 142 8 hence hence ADV cana-1654 142 9 the the DET cana-1654 142 10 fan fan NOUN cana-1654 142 11 𝐹𝑛 𝐹𝑛 PROPN cana-1654 142 12 is be AUX cana-1654 142 13 an an DET cana-1654 142 14 odd odd ADJ cana-1654 142 15 - - PUNCT cana-1654 142 16 even even ADV cana-1654 142 17 congruence congruence NOUN cana-1654 142 18 graph graph NOUN cana-1654 142 19 . . PUNCT cana-1654 143 1 example example NOUN cana-1654 143 2 : : PUNCT cana-1654 143 3 3.10 3.10 NUM cana-1654 143 4 consider consider VERB cana-1654 143 5 a a DET cana-1654 143 6 generalized generalized ADJ cana-1654 143 7 fan fan NOUN cana-1654 143 8 graph graph NOUN cana-1654 143 9 𝐹𝑛 𝐹𝑛 PROPN cana-1654 143 10 with with ADP cana-1654 143 11 𝑛 𝑛 NOUN cana-1654 143 12 = = SYM cana-1654 143 13 6 6 NUM cana-1654 143 14 figure-5 figure-5 NUM cana-1654 143 15 fig-5 fig-5 NUM cana-1654 143 16 shows show VERB cana-1654 143 17 that that SCONJ cana-1654 143 18 friendship friendship NOUN cana-1654 143 19 graph graph NOUN cana-1654 143 20 𝐹6 𝐹6 NOUN cana-1654 143 21 admits admit VERB cana-1654 143 22 odd odd ADJ cana-1654 143 23 - - PUNCT cana-1654 143 24 even even ADV cana-1654 143 25 congruence congruence ADJ cana-1654 143 26 labeling labeling NOUN cana-1654 143 27 . . PUNCT cana-1654 144 1 theorem theorem VERB cana-1654 144 2 3.11 3.11 NUM cana-1654 144 3 the the DET cana-1654 144 4 graph graph NOUN cana-1654 144 5 𝑃2 𝑃2 NOUN cana-1654 145 1 + + CCONJ cana-1654 145 2 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 145 3 admits admit VERB cana-1654 145 4 odd odd ADJ cana-1654 145 5 - - PUNCT cana-1654 145 6 even even ADV cana-1654 145 7 congruence congruence NOUN cana-1654 145 8 graph graph NOUN cana-1654 145 9 . . PUNCT cana-1654 146 1 proof proof NOUN cana-1654 146 2 : : PUNCT cana-1654 146 3 suppose suppose VERB cana-1654 146 4 𝑃2 𝑃2 PROPN cana-1654 146 5 is be AUX cana-1654 146 6 a a DET cana-1654 146 7 path path NOUN cana-1654 146 8 having have VERB cana-1654 146 9 two two NUM cana-1654 146 10 vertices vertex NOUN cana-1654 146 11 𝑒1 𝑒1 NOUN cana-1654 146 12 , , PUNCT cana-1654 146 13 𝑒2 𝑒2 PROPN cana-1654 146 14 . . PUNCT cana-1654 147 1 communications communication NOUN cana-1654 147 2 on on ADP cana-1654 147 3 applied apply VERB cana-1654 147 4 nonlinear nonlinear ADJ cana-1654 147 5 analysis analysis NOUN cana-1654 147 6 issn issn NOUN cana-1654 147 7 : : PUNCT cana-1654 147 8 1074 1074 NUM cana-1654 147 9 - - PUNCT cana-1654 147 10 133x 133x NUM cana-1654 147 11 vol vol NOUN cana-1654 147 12 32 32 NUM cana-1654 147 13 no no NOUN cana-1654 147 14 . . NOUN cana-1654 147 15 1 1 NUM cana-1654 147 16 ( ( PUNCT cana-1654 147 17 2025 2025 NUM cana-1654 147 18 ) ) PUNCT cana-1654 147 19 342 342 NUM cana-1654 147 20 https://internationalpubls.com https://internationalpubls.com X cana-1654 147 21 let let VERB cana-1654 147 22 𝑒3 𝑒3 NOUN cana-1654 147 23 , , PUNCT cana-1654 147 24 𝑒4 𝑒4 NOUN cana-1654 147 25 , , PUNCT cana-1654 147 26 … … PUNCT cana-1654 147 27 … … PUNCT cana-1654 147 28 … … PUNCT cana-1654 147 29 . . PUNCT cana-1654 147 30 . . PUNCT cana-1654 148 1 , , PUNCT cana-1654 148 2 π‘’π‘š+2 π‘’π‘š+2 X cana-1654 148 3 be be AUX cana-1654 148 4 the the DET cana-1654 148 5 π‘š π‘š NOUN cana-1654 148 6 isolated isolate VERB cana-1654 148 7 vertices vertex NOUN cana-1654 148 8 connecting connect VERB cana-1654 148 9 𝑒1 𝑒1 NOUN cana-1654 148 10 , , PUNCT cana-1654 148 11 𝑒2 𝑒2 PROPN cana-1654 148 12 with with ADP cana-1654 148 13 𝑒𝑖 𝑒𝑖 NOUN cana-1654 148 14 , , PUNCT cana-1654 148 15 3 3 NUM cana-1654 148 16 ≀ ≀ NUM cana-1654 148 17 𝑖 𝑖 SYM cana-1654 148 18 ≀ ≀ NOUN cana-1654 148 19 π‘š π‘š X cana-1654 148 20 + + SYM cana-1654 148 21 2 2 NUM cana-1654 148 22 we we PRON cana-1654 148 23 obtain obtain VERB cana-1654 148 24 𝑃2 𝑃2 NOUN cana-1654 148 25 + + CCONJ cana-1654 148 26 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 148 27 suppose suppose VERB cana-1654 148 28 𝐺 𝐺 PROPN cana-1654 148 29 = = NOUN cana-1654 148 30 𝑃2 𝑃2 NOUN cana-1654 148 31 + + CCONJ cana-1654 148 32 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 148 33 let let VERB cana-1654 148 34 the the DET cana-1654 148 35 vertex vertex NOUN cana-1654 148 36 set set NOUN cana-1654 148 37 of of ADP cana-1654 148 38 𝐺 𝐺 PROPN cana-1654 148 39 be be AUX cana-1654 148 40 let let VERB cana-1654 148 41 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 148 42 ) ) PUNCT cana-1654 148 43 = = PRON cana-1654 148 44 { { PUNCT cana-1654 148 45 𝑒𝑖 𝑒𝑖 NOUN cana-1654 148 46 : : PUNCT cana-1654 148 47 1 1 NUM cana-1654 148 48 ≀ ≀ NUM cana-1654 148 49 𝑖 𝑖 SYM cana-1654 148 50 ≀ ≀ NOUN cana-1654 148 51 π‘š π‘š ADP cana-1654 148 52 + + NOUN cana-1654 148 53 2 2 NUM cana-1654 148 54 } } PUNCT cana-1654 148 55 be be AUX cana-1654 148 56 the the DET cana-1654 148 57 vertices vertex NOUN cana-1654 148 58 of of ADP cana-1654 148 59 𝐺 𝐺 PROPN cana-1654 148 60 and and CCONJ cana-1654 148 61 𝐸(𝐺 𝐸(𝐺 PROPN cana-1654 148 62 ) ) PUNCT cana-1654 148 63 = = PRON cana-1654 148 64 { { PUNCT cana-1654 148 65 𝑒1𝑒2 𝑒1𝑒2 NOUN cana-1654 148 66 } } PUNCT cana-1654 148 67 βˆͺ βˆͺ X cana-1654 148 68 { { PUNCT cana-1654 148 69 𝑒1𝑒𝑖 𝑒1𝑒𝑖 X cana-1654 148 70 /3 /3 PUNCT cana-1654 148 71 ≀ ≀ NUM cana-1654 148 72 𝑖 𝑖 SYM cana-1654 148 73 ≀ ≀ NOUN cana-1654 148 74 π‘š π‘š ADP cana-1654 148 75 + + ADJ cana-1654 148 76 2 2 NUM cana-1654 148 77 } } PUNCT cana-1654 148 78 βˆͺ βˆͺ X cana-1654 148 79 { { PUNCT cana-1654 148 80 𝑒2𝑒𝑖 𝑒2𝑒𝑖 X cana-1654 148 81 /3 /3 SYM cana-1654 149 1 ≀ ≀ NUM cana-1654 149 2 𝑖 𝑖 SYM cana-1654 149 3 ≀ ≀ NOUN cana-1654 149 4 π‘š π‘š ADP cana-1654 149 5 + + NOUN cana-1654 149 6 2 2 NUM cana-1654 149 7 } } PUNCT cana-1654 149 8 be be AUX cana-1654 149 9 the the DET cana-1654 149 10 edges edge NOUN cana-1654 149 11 of of ADP cana-1654 149 12 𝐺 𝐺 NOUN cana-1654 149 13 where where SCONJ cana-1654 149 14 𝑀𝑖 𝑀𝑖 ADV cana-1654 149 15 = = SYM cana-1654 149 16 𝑒1𝑒𝑖 𝑒1𝑒𝑖 X cana-1654 149 17 and and CCONJ cana-1654 149 18 π‘žπ‘– π‘žπ‘– ADJ cana-1654 149 19 = = ADJ cana-1654 149 20 𝑒2𝑒𝑖. 𝑒2𝑒𝑖. PROPN cana-1654 149 21 also also ADV cana-1654 149 22 |𝑉(𝐺)| |𝑉(𝐺)| VERB cana-1654 150 1 = = SYM cana-1654 150 2 2 2 NUM cana-1654 150 3 + + NUM cana-1654 150 4 π‘š π‘š X cana-1654 150 5 and and CCONJ cana-1654 150 6 |𝐸(𝐺)| |𝐸(𝐺)| NOUN cana-1654 150 7 = = NOUN cana-1654 150 8 2π‘š 2π‘š NOUN cana-1654 150 9 + + CCONJ cana-1654 150 10 1 1 NUM cana-1654 150 11 then then ADV cana-1654 150 12 𝑑 𝑑 PROPN cana-1654 150 13 = = ADJ cana-1654 150 14 min min PROPN cana-1654 150 15 ( ( PUNCT cana-1654 150 16 2(2 2(2 NUM cana-1654 150 17 + + CCONJ cana-1654 150 18 π‘š π‘š NOUN cana-1654 150 19 ) ) PUNCT cana-1654 150 20 , , PUNCT cana-1654 150 21 2(2π‘š 2(2π‘š PROPN cana-1654 151 1 + + CCONJ cana-1654 151 2 1 1 NUM cana-1654 151 3 ) ) PUNCT cana-1654 151 4 ) ) PUNCT cana-1654 152 1 = = PUNCT cana-1654 153 1 2(2 2(2 NUM cana-1654 153 2 + + CCONJ cana-1654 153 3 π‘š π‘š X cana-1654 153 4 ) ) PUNCT cana-1654 153 5 define define VERB cana-1654 153 6 a a DET cana-1654 153 7 bijective bijective ADJ cana-1654 153 8 function function NOUN cana-1654 153 9 β„Ž β„Ž X cana-1654 153 10 ∢ ∢ NOUN cana-1654 153 11 𝑉(𝐺 𝑉(𝐺 NOUN cana-1654 153 12 ) ) PUNCT cana-1654 153 13 β†’ β†’ SYM cana-1654 153 14 { { PUNCT cana-1654 153 15 1,3,7 1,3,7 NUM cana-1654 153 16 , , PUNCT cana-1654 153 17 … … PUNCT cana-1654 153 18 . . PUNCT cana-1654 153 19 . . PUNCT cana-1654 154 1 … … PUNCT cana-1654 154 2 . . PUNCT cana-1654 155 1 . . PUNCT cana-1654 156 1 , , PUNCT cana-1654 156 2 4𝑖 4𝑖 NOUN cana-1654 156 3 + + CCONJ cana-1654 156 4 3 3 X cana-1654 156 5 } } PUNCT cana-1654 156 6 is be AUX cana-1654 156 7 defined define VERB cana-1654 156 8 as as SCONJ cana-1654 156 9 follows follow VERB cana-1654 156 10 𝑓(𝑒1 𝑓(𝑒1 NOUN cana-1654 156 11 ) ) PUNCT cana-1654 157 1 = = SYM cana-1654 157 2 1 1 NUM cana-1654 157 3 , , PUNCT cana-1654 157 4 𝑓(𝑒2 𝑓(𝑒2 NOUN cana-1654 157 5 ) ) PUNCT cana-1654 157 6 = = SYM cana-1654 157 7 3 3 NUM cana-1654 157 8 β„Ž(𝑒𝑖+2 β„Ž(𝑒𝑖+2 NUM cana-1654 157 9 ) ) PUNCT cana-1654 157 10 = = SYM cana-1654 157 11 4𝑖 4𝑖 NOUN cana-1654 157 12 + + CCONJ cana-1654 157 13 3 3 NUM cana-1654 157 14 for for ADP cana-1654 157 15 1 1 NUM cana-1654 157 16 ≀ ≀ NUM cana-1654 157 17 𝑖 𝑖 SYM cana-1654 157 18 ≀ ≀ NOUN cana-1654 157 19 π‘š π‘š ADP cana-1654 157 20 the the DET cana-1654 157 21 edge edge NOUN cana-1654 157 22 labeling labeling NOUN cana-1654 157 23 π‘˜ π‘˜ ADP cana-1654 157 24 ∢ ∢ NOUN cana-1654 157 25 𝐸(𝐺 𝐸(𝐺 ADJ cana-1654 157 26 ) ) PUNCT cana-1654 157 27 β†’ β†’ SYM cana-1654 157 28 { { PUNCT cana-1654 157 29 2,4,6,8 2,4,6,8 NUM cana-1654 157 30 , , PUNCT cana-1654 157 31 … … PUNCT cana-1654 157 32 . . PUNCT cana-1654 157 33 . . PUNCT cana-1654 158 1 … … PUNCT cana-1654 158 2 . . PUNCT cana-1654 159 1 . . PUNCT cana-1654 160 1 , , PUNCT cana-1654 160 2 4𝑖 4𝑖 NOUN cana-1654 160 3 + + CCONJ cana-1654 160 4 2 2 NUM cana-1654 160 5 } } PUNCT cana-1654 160 6 is be AUX cana-1654 160 7 defined define VERB cana-1654 160 8 as as ADP cana-1654 160 9 𝑓(𝑀 𝑓(𝑀 NOUN cana-1654 160 10 ) ) PUNCT cana-1654 160 11 = = SYM cana-1654 160 12 2 2 NUM cana-1654 160 13 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 PROPN cana-1654 160 14 ) ) PUNCT cana-1654 161 1 = = SYM cana-1654 161 2 4𝑖 4𝑖 NOUN cana-1654 161 3 + + CCONJ cana-1654 161 4 2 2 NUM cana-1654 161 5 for for ADP cana-1654 161 6 1 1 NUM cana-1654 161 7 ≀ ≀ NUM cana-1654 161 8 𝑖 𝑖 SYM cana-1654 161 9 ≀ ≀ NOUN cana-1654 161 10 π‘š π‘š ADP cana-1654 161 11 π‘˜(π‘žπ‘– π‘˜(π‘žπ‘– NOUN cana-1654 161 12 ) ) PUNCT cana-1654 162 1 = = SYM cana-1654 162 2 4𝑖 4𝑖 NOUN cana-1654 162 3 for for ADP cana-1654 162 4 1 1 NUM cana-1654 162 5 ≀ ≀ NUM cana-1654 162 6 𝑖 𝑖 SYM cana-1654 162 7 ≀ ≀ NOUN cana-1654 162 8 π‘š π‘š ADP cana-1654 162 9 clearly clearly ADV cana-1654 162 10 π‘˜(𝑀 π‘˜(𝑀 PROPN cana-1654 162 11 ) ) PUNCT cana-1654 162 12 divides divide VERB cana-1654 162 13 |𝑓(𝑒1 |𝑓(𝑒1 NOUN cana-1654 162 14 ) ) PUNCT cana-1654 163 1 βˆ’ βˆ’ PROPN cana-1654 164 1 𝑓(𝑒2)| 𝑓(𝑒2)| PROPN cana-1654 164 2 π‘˜(𝑀𝑖 π‘˜(𝑀𝑖 PROPN cana-1654 164 3 ) ) PUNCT cana-1654 164 4 divides divide VERB cana-1654 164 5 |(𝑓(𝑒1 |(𝑓(𝑒1 NOUN cana-1654 164 6 ) ) PUNCT cana-1654 165 1 βˆ’ βˆ’ PROPN cana-1654 166 1 β„Ž(𝑒𝑖))| β„Ž(𝑒𝑖))| PROPN cana-1654 166 2 for for ADP cana-1654 166 3 1 1 NUM cana-1654 166 4 ≀ ≀ NUM cana-1654 166 5 𝑖 𝑖 SYM cana-1654 166 6 ≀ ≀ NOUN cana-1654 166 7 π‘š π‘š X cana-1654 166 8 and and CCONJ cana-1654 166 9 π‘˜(π‘žπ‘– π‘˜(π‘žπ‘– NUM cana-1654 166 10 ) ) PUNCT cana-1654 166 11 divides divide VERB cana-1654 166 12 |(𝑓(𝑒2 |(𝑓(𝑒2 NOUN cana-1654 166 13 ) ) PUNCT cana-1654 166 14 βˆ’ βˆ’ PROPN cana-1654 167 1 β„Ž(𝑒𝑖))| β„Ž(𝑒𝑖))| PROPN cana-1654 167 2 for for ADP cana-1654 167 3 1 1 NUM cana-1654 167 4 ≀ ≀ NUM cana-1654 167 5 𝑖 𝑖 SYM cana-1654 167 6 ≀ ≀ NOUN cana-1654 167 7 π‘š π‘š ADP cana-1654 167 8 hence hence ADV cana-1654 167 9 the the DET cana-1654 167 10 graph graph NOUN cana-1654 167 11 𝑃2 𝑃2 NOUN cana-1654 167 12 + + CCONJ cana-1654 167 13 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 167 14 is be AUX cana-1654 167 15 an an DET cana-1654 167 16 odd odd ADJ cana-1654 167 17 - - PUNCT cana-1654 167 18 even even ADV cana-1654 167 19 congruence congruence NOUN cana-1654 167 20 graph graph NOUN cana-1654 167 21 . . PUNCT cana-1654 168 1 example example NOUN cana-1654 168 2 : : PUNCT cana-1654 168 3 3.12 3.12 NUM cana-1654 168 4 consider consider VERB cana-1654 168 5 a a DET cana-1654 168 6 graph graph NOUN cana-1654 168 7 𝑃2 𝑃2 NOUN cana-1654 168 8 + + CCONJ cana-1654 168 9 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 168 10 with with ADP cana-1654 168 11 π‘š π‘š NOUN cana-1654 168 12 = = SYM cana-1654 168 13 6 6 NUM cana-1654 168 14 figure-6 figure-6 PROPN cana-1654 168 15 communications communication NOUN cana-1654 168 16 on on ADP cana-1654 168 17 applied apply VERB cana-1654 168 18 nonlinear nonlinear ADJ cana-1654 168 19 analysis analysis NOUN cana-1654 168 20 issn issn NOUN cana-1654 168 21 : : PUNCT cana-1654 168 22 1074 1074 NUM cana-1654 168 23 - - PUNCT cana-1654 168 24 133x 133x NUM cana-1654 168 25 vol vol NOUN cana-1654 168 26 32 32 NUM cana-1654 168 27 no no NOUN cana-1654 168 28 . . NOUN cana-1654 168 29 1 1 NUM cana-1654 168 30 ( ( PUNCT cana-1654 168 31 2025 2025 NUM cana-1654 168 32 ) ) PUNCT cana-1654 168 33 343 343 NUM cana-1654 168 34 https://internationalpubls.com https://internationalpubls.com X cana-1654 168 35 fig-6 fig-6 PRON cana-1654 168 36 shows show VERB cana-1654 168 37 that that SCONJ cana-1654 168 38 a a DET cana-1654 168 39 graph graph NOUN cana-1654 168 40 𝑃2 𝑃2 NOUN cana-1654 169 1 + + CCONJ cana-1654 169 2 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 169 3 with with ADP cana-1654 169 4 π‘š π‘š NOUN cana-1654 169 5 = = SYM cana-1654 169 6 6 6 NUM cana-1654 169 7 admits admit VERB cana-1654 169 8 odd odd ADJ cana-1654 169 9 - - PUNCT cana-1654 169 10 even even ADV cana-1654 169 11 congruence congruence ADJ cana-1654 169 12 labeling labeling NOUN cana-1654 169 13 . . PUNCT cana-1654 170 1 4 4 X cana-1654 170 2 . . X cana-1654 170 3 conclusion conclusion VERB cana-1654 170 4 the the DET cana-1654 170 5 labelling labelling NOUN cana-1654 170 6 of of ADP cana-1654 170 7 graphs graph NOUN cana-1654 170 8 is be AUX cana-1654 170 9 an an DET cana-1654 170 10 interesting interesting ADJ cana-1654 170 11 and and CCONJ cana-1654 170 12 vast vast ADJ cana-1654 170 13 research research NOUN cana-1654 170 14 area area NOUN cana-1654 170 15 which which PRON cana-1654 170 16 is be AUX cana-1654 170 17 very very ADV cana-1654 170 18 useful useful ADJ cana-1654 170 19 and and CCONJ cana-1654 170 20 it it PRON cana-1654 170 21 is be AUX cana-1654 170 22 extended extend VERB cana-1654 170 23 in in ADP cana-1654 170 24 various various ADJ cana-1654 170 25 topics topic NOUN cana-1654 170 26 by by ADP cana-1654 170 27 several several ADJ cana-1654 170 28 people people NOUN cana-1654 170 29 . . PUNCT cana-1654 171 1 in in ADP cana-1654 171 2 this this DET cana-1654 171 3 paper paper NOUN cana-1654 171 4 we we PRON cana-1654 171 5 have have AUX cana-1654 171 6 studied study VERB cana-1654 171 7 odd odd ADJ cana-1654 171 8 - - PUNCT cana-1654 171 9 even even ADV cana-1654 171 10 congruence congruence ADJ cana-1654 171 11 labeling labeling NOUN cana-1654 171 12 behaviour behaviour NOUN cana-1654 171 13 of of ADP cana-1654 171 14 odd odd ADJ cana-1654 171 15 - - PUNCT cana-1654 171 16 even even ADV cana-1654 171 17 congruence congruence NOUN cana-1654 171 18 labeling labeling NOUN cana-1654 171 19 for for ADP cana-1654 171 20 friendship friendship NOUN cana-1654 171 21 graph graph NOUN cana-1654 171 22 , , PUNCT cana-1654 171 23 shell shell NOUN cana-1654 171 24 graph graph NOUN cana-1654 171 25 , , PUNCT cana-1654 171 26 generalized generalized ADJ cana-1654 171 27 butterfly butterfly NOUN cana-1654 171 28 graphs graph NOUN cana-1654 171 29 , , PUNCT cana-1654 171 30 fan fan NOUN cana-1654 171 31 graph graph NOUN cana-1654 171 32 , , PUNCT cana-1654 171 33 𝑃2 𝑃2 NOUN cana-1654 171 34 + + CCONJ cana-1654 171 35 π‘šπΎ1 π‘šπΎ1 NOUN cana-1654 171 36 graph graph NOUN cana-1654 171 37 etc etc X cana-1654 171 38 . . X cana-1654 171 39 to to PART cana-1654 171 40 derive derive VERB cana-1654 171 41 similar similar ADJ cana-1654 171 42 results result NOUN cana-1654 171 43 for for ADP cana-1654 171 44 other other ADJ cana-1654 171 45 graph graph NOUN cana-1654 171 46 families family NOUN cana-1654 171 47 is be AUX cana-1654 171 48 an an DET cana-1654 171 49 open open ADJ cana-1654 171 50 problem problem NOUN cana-1654 171 51 . . PUNCT cana-1654 172 1 refrences refrence VERB cana-1654 173 1 [ [ X cana-1654 173 2 1 1 NUM cana-1654 173 3 ] ] X cana-1654 173 4 bondy bondy PROPN cana-1654 173 5 , , PUNCT cana-1654 173 6 j. j. PROPN cana-1654 173 7 a. a. PROPN cana-1654 173 8 , , PUNCT cana-1654 173 9 and and CCONJ cana-1654 173 10 murty murty NOUN cana-1654 173 11 , , PUNCT cana-1654 173 12 u. u. PROPN cana-1654 173 13 s. s. PROPN cana-1654 173 14 r. r. PROPN cana-1654 173 15 , , PUNCT cana-1654 173 16 ( ( PUNCT cana-1654 173 17 1976 1976 NUM cana-1654 173 18 ) ) PUNCT cana-1654 173 19 , , PUNCT cana-1654 173 20 graph graph NOUN cana-1654 173 21 theory theory NOUN cana-1654 173 22 with with ADP cana-1654 173 23 applications application NOUN cana-1654 173 24 , , PUNCT cana-1654 173 25 2nd 2nd PROPN cana-1654 173 26 edition edition PROPN cana-1654 173 27 , , PUNCT cana-1654 173 28 macmillan macmillan PROPN cana-1654 173 29 , , PUNCT cana-1654 173 30 new new PROPN cana-1654 173 31 york york PROPN cana-1654 173 32 . . PUNCT cana-1654 174 1 [ [ X cana-1654 174 2 2 2 NUM cana-1654 174 3 ] ] PUNCT cana-1654 174 4 gallian gallian NOUN cana-1654 174 5 , , PUNCT cana-1654 174 6 j. j. PROPN cana-1654 174 7 a. a. PROPN cana-1654 174 8 , , PUNCT cana-1654 174 9 ( ( PUNCT cana-1654 174 10 2019 2019 NUM cana-1654 174 11 ) ) PUNCT cana-1654 174 12 , , PUNCT cana-1654 174 13 a a DET cana-1654 174 14 dynamic dynamic ADJ cana-1654 174 15 survey survey NOUN cana-1654 174 16 of of ADP cana-1654 174 17 graph graph NOUN cana-1654 174 18 labeling labeling NOUN cana-1654 174 19 , , PUNCT cana-1654 174 20 the the DET cana-1654 174 21 electronic electronic ADJ cana-1654 174 22 journal journal NOUN cana-1654 174 23 of of ADP cana-1654 174 24 combinatories,#ds6 combinatories,#ds6 PROPN cana-1654 174 25 , , PUNCT cana-1654 174 26 pp pp ADJ cana-1654 174 27 . . PUNCT cana-1654 175 1 1538.h 1538.h X cana-1654 175 2 . . PUNCT cana-1654 176 1 [ [ X cana-1654 176 2 3 3 NUM cana-1654 176 3 ] ] PUNCT cana-1654 176 4 elsonbaty elsonbaty NOUN cana-1654 176 5 , , PUNCT cana-1654 176 6 amr amr NOUN cana-1654 176 7 and and CCONJ cana-1654 176 8 daoud daoud PROPN cana-1654 176 9 , , PUNCT cana-1654 176 10 salama salama PROPN cana-1654 176 11 nagy nagy NOUN cana-1654 176 12 , , PUNCT cana-1654 176 13 edge edge VERB cana-1654 176 14 even even ADV cana-1654 176 15 graceful graceful ADJ cana-1654 176 16 labeling labeling NOUN cana-1654 176 17 of of ADP cana-1654 176 18 some some DET cana-1654 176 19 path path NOUN cana-1654 176 20 and and CCONJ cana-1654 176 21 cycle cycle NOUN cana-1654 176 22 related relate VERB cana-1654 176 23 graphs graph NOUN cana-1654 176 24 , , PUNCT cana-1654 176 25 ars ar VERB cana-1654 176 26 combinatoria combinatoria NOUN cana-1654 176 27 , , PUNCT cana-1654 176 28 130 130 NUM cana-1654 176 29 ( ( PUNCT cana-1654 176 30 2017 2017 NUM cana-1654 176 31 ) ) PUNCT cana-1654 176 32 , , PUNCT cana-1654 176 33 79–96 79–96 NUM cana-1654 176 34 . . PUNCT cana-1654 177 1 [ [ X cana-1654 177 2 4 4 NUM cana-1654 177 3 ] ] PUNCT cana-1654 177 4 hafidhyah hafidhyah PROPN cana-1654 177 5 dwi dwi PROPN cana-1654 177 6 wahyuna wahyuna PROPN cana-1654 177 7 and and CCONJ cana-1654 177 8 diari diari PROPN cana-1654 177 9 indriati indriati NOUN cana-1654 177 10 , , PUNCT cana-1654 177 11 β€œ " PUNCT cana-1654 177 12 on on ADP cana-1654 177 13 the the DET cana-1654 177 14 total total ADJ cana-1654 177 15 edge edge NOUN cana-1654 177 16 irregularity irregularity NOUN cana-1654 177 17 strength strength NOUN cana-1654 177 18 of of ADP cana-1654 177 19 generalized generalized ADJ cana-1654 177 20 butterfly butterfly NOUN cana-1654 177 21 graph graph NOUN cana-1654 177 22 ” " PUNCT cana-1654 177 23 , , PUNCT cana-1654 177 24 journal journal NOUN cana-1654 177 25 of of ADP cana-1654 177 26 physics physics PROPN cana-1654 177 27 : : PUNCT cana-1654 177 28 conf conf PROPN cana-1654 177 29 . . PUNCT cana-1654 178 1 series series PROPN cana-1654 178 2 , , PUNCT cana-1654 178 3 vol vol NOUN cana-1654 178 4 . . PROPN cana-1654 178 5 1008 1008 NUM cana-1654 178 6 , , PUNCT cana-1654 178 7 pp pp ADJ cana-1654 178 8 . . PUNCT cana-1654 179 1 1 1 NUM cana-1654 179 2 - - SYM cana-1654 179 3 6 6 NUM cana-1654 179 4 , , PUNCT cana-1654 179 5 2018 2018 NUM cana-1654 179 6 . . PUNCT cana-1654 180 1 [ [ X cana-1654 180 2 5 5 NUM cana-1654 180 3 ] ] X cana-1654 180 4 rosa rosa PROPN cana-1654 180 5 , , PUNCT cana-1654 180 6 a. a. PROPN cana-1654 180 7 , , PUNCT cana-1654 180 8 ( ( PUNCT cana-1654 180 9 1967 1967 NUM cana-1654 180 10 ) ) PUNCT cana-1654 180 11 , , PUNCT cana-1654 180 12 on on ADP cana-1654 180 13 certain certain ADJ cana-1654 180 14 valuations valuation NOUN cana-1654 180 15 of of ADP cana-1654 180 16 the the DET cana-1654 180 17 vertices vertex NOUN cana-1654 180 18 of of ADP cana-1654 180 19 a a DET cana-1654 180 20 graph graph NOUN cana-1654 180 21 , , PUNCT cana-1654 180 22 theory theory NOUN cana-1654 180 23 of of ADP cana-1654 180 24 graphs graph NOUN cana-1654 180 25 , , PUNCT cana-1654 180 26 internat internat PROPN cana-1654 180 27 . . PUNCT cana-1654 180 28 sympos sympos PROPN cana-1654 180 29 . . PROPN cana-1654 180 30 , , PUNCT cana-1654 180 31 icc icc PROPN cana-1654 180 32 rome rome PROPN cana-1654 180 33 , , PUNCT cana-1654 180 34 paris paris PROPN cana-1654 180 35 dunod dunod PROPN cana-1654 180 36 , , PUNCT cana-1654 180 37 pp pp PROPN cana-1654 180 38 . . PUNCT cana-1654 181 1 339 339 NUM cana-1654 181 2 - - SYM cana-1654 181 3 355 355 NUM cana-1654 181 4 [ [ X cana-1654 181 5 6 6 NUM cana-1654 181 6 ] ] SYM cana-1654 181 7 g.thamizhendhi g.thamizhendhi NOUN cana-1654 181 8 , , PUNCT cana-1654 181 9 and and CCONJ cana-1654 181 10 k.kanakambika k.kanakambika NOUN cana-1654 181 11 , , PUNCT cana-1654 181 12 some some DET cana-1654 181 13 results result NOUN cana-1654 181 14 on on ADP cana-1654 181 15 odd odd ADJ cana-1654 181 16 - - PUNCT cana-1654 181 17 even even ADV cana-1654 181 18 congruence congruence ADJ cana-1654 181 19 labelling labelling NOUN cana-1654 181 20 of of ADP cana-1654 181 21 graphs graph NOUN cana-1654 181 22 , , PUNCT cana-1654 181 23 communications communication NOUN cana-1654 181 24 on on ADP cana-1654 181 25 applied apply VERB cana-1654 181 26 nonlinear nonlinear ADJ cana-1654 181 27 analysis analysis NOUN cana-1654 181 28 , , PUNCT cana-1654 181 29 vol.31 vol.31 ADJ cana-1654 181 30 , , PUNCT cana-1654 181 31 no.1 no.1 NUM cana-1654 181 32 , , PUNCT cana-1654 181 33 141 141 NUM cana-1654 181 34 - - SYM cana-1654 181 35 149 149 NUM cana-1654 181 36 ( ( PUNCT cana-1654 181 37 2024 2024 NUM cana-1654 181 38 ) ) PUNCT cana-1654 181 39 . . PUNCT cana-1654 182 1 [ [ X cana-1654 182 2 7 7 NUM cana-1654 182 3 ] ] X cana-1654 182 4 prajapati prajapati PROPN cana-1654 182 5 , , PUNCT cana-1654 182 6 u. u. PROPN cana-1654 182 7 m. m. PROPN cana-1654 182 8 , , PUNCT cana-1654 182 9 and and CCONJ cana-1654 182 10 patel patel PROPN cana-1654 182 11 , , PUNCT cana-1654 182 12 n. n. PROPN cana-1654 182 13 b. b. PROPN cana-1654 182 14 , , PUNCT cana-1654 182 15 ( ( PUNCT cana-1654 182 16 2022 2022 NUM cana-1654 182 17 ) ) PUNCT cana-1654 182 18 , , PUNCT cana-1654 182 19 edge edge NOUN cana-1654 182 20 product product NOUN cana-1654 182 21 cordial cordial ADJ cana-1654 182 22 cordial cordial ADJ cana-1654 182 23 labeling labeling NOUN cana-1654 182 24 of of ADP cana-1654 182 25 switching switch VERB cana-1654 182 26 operations operation NOUN cana-1654 182 27 on on ADP cana-1654 182 28 some some DET cana-1654 182 29 graphs graph NOUN cana-1654 182 30 , , PUNCT cana-1654 182 31 twms twms PROPN cana-1654 182 32 journal journal PROPN cana-1654 182 33 of of ADP cana-1654 182 34 application application NOUN cana-1654 182 35 and and CCONJ cana-1654 182 36 engineering engineering NOUN cana-1654 182 37 mathematics mathematic NOUN cana-1654 182 38 , , PUNCT cana-1654 182 39 12(1 12(1 NUM cana-1654 182 40 ) ) PUNCT cana-1654 182 41 , , PUNCT cana-1654 182 42 pp pp ADP cana-1654 182 43 . . PUNCT cana-1654 183 1 191 191 NUM cana-1654 183 2 – – SYM cana-1654 183 3 199 199 NUM cana-1654 183 4 . . PUNCT cana-1654 184 1 [ [ X cana-1654 184 2 8 8 NUM cana-1654 184 3 ] ] X cana-1654 184 4 shrimali shrimali VERB cana-1654 184 5 , , PUNCT cana-1654 184 6 n.p n.p PROPN cana-1654 184 7 . . PROPN cana-1654 184 8 and and CCONJ cana-1654 184 9 rathod rathod PROPN cana-1654 184 10 , , PUNCT cana-1654 184 11 a.k a.k PROPN cana-1654 184 12 . . PROPN cana-1654 184 13 , , PUNCT cana-1654 184 14 ( ( PUNCT cana-1654 184 15 2021 2021 NUM cana-1654 184 16 ) ) PUNCT cana-1654 184 17 , , PUNCT cana-1654 184 18 some some PRON cana-1654 184 19 results result VERB cana-1654 184 20 on on ADP cana-1654 184 21 vertex vertex NOUN cana-1654 184 22 edge edge NOUN cana-1654 184 23 neighborhood neighborhood NOUN cana-1654 184 24 prime prime ADJ cana-1654 184 25 labeling labeling NOUN cana-1654 184 26 , , PUNCT cana-1654 184 27 twms twms PROPN cana-1654 184 28 journal journal NOUN cana-1654 184 29 of of ADP cana-1654 184 30 application application NOUN cana-1654 184 31 and and CCONJ cana-1654 184 32 engineering engineering NOUN cana-1654 184 33 mathematics mathematic NOUN cana-1654 184 34 , , PUNCT cana-1654 184 35 11(2 11(2 NUM cana-1654 184 36 ) ) PUNCT cana-1654 184 37 , , PUNCT cana-1654 184 38 pp pp ADP cana-1654 184 39 . . PUNCT cana-1654 185 1 450 450 NUM cana-1654 185 2 501 501 NUM cana-1654 185 3 . . PUNCT cana-1654 186 1 [ [ X cana-1654 186 2 9 9 NUM cana-1654 186 3 ] ] X cana-1654 186 4 meena meena PROPN cana-1654 186 5 , , PUNCT cana-1654 186 6 s s X cana-1654 186 7 and and CCONJ cana-1654 186 8 vaithilingam vaithilingam ADV cana-1654 186 9 , , PUNCT cana-1654 186 10 k k NOUN cana-1654 186 11 , , PUNCT cana-1654 186 12 prime prime ADJ cana-1654 186 13 labeling labeling NOUN cana-1654 186 14 for for ADP cana-1654 186 15 some some DET cana-1654 186 16 fan fan NOUN cana-1654 186 17 related relate VERB cana-1654 186 18 graphs graph NOUN cana-1654 186 19 , , PUNCT cana-1654 186 20 international international ADJ cana-1654 186 21 journal journal NOUN cana-1654 186 22 of of ADP cana-1654 186 23 engineering engineering PROPN cana-1654 186 24 research research NOUN cana-1654 186 25 & & CCONJ cana-1654 186 26 technology technology PROPN cana-1654 186 27 ( ( PUNCT cana-1654 186 28 ijert ijert PROPN cana-1654 186 29 ) ) PUNCT cana-1654 186 30 , , PUNCT cana-1654 186 31 1(9 1(9 NUM cana-1654 186 32 ) ) PUNCT cana-1654 186 33 ( ( PUNCT cana-1654 186 34 2012 2012 NUM cana-1654 186 35 ) ) PUNCT cana-1654 186 36 . . PUNCT cana-1654 187 1 [ [ X cana-1654 187 2 10 10 NUM cana-1654 187 3 ] ] X cana-1654 187 4 sivasakthi sivasakthi NOUN cana-1654 187 5 , , PUNCT cana-1654 187 6 s s PROPN cana-1654 187 7 meena meena PROPN cana-1654 187 8 m m PROPN cana-1654 187 9 renugha renugha PROPN cana-1654 187 10 m m PROPN cana-1654 187 11 , , PUNCT cana-1654 187 12 cordial cordial ADJ cana-1654 187 13 labeling labeling NOUN cana-1654 187 14 for for ADP cana-1654 187 15 different different ADJ cana-1654 187 16 types type NOUN cana-1654 187 17 of of ADP cana-1654 187 18 shell shell ADJ cana-1654 187 19 graph graph NOUN cana-1654 187 20 , , PUNCT cana-1654 187 21 international international ADJ cana-1654 187 22 journal journal NOUN cana-1654 187 23 of of ADP cana-1654 187 24 scientific scientific PROPN cana-1654 187 25 & & CCONJ cana-1654 187 26 engineering engineering PROPN cana-1654 187 27 research research NOUN cana-1654 187 28 , , PUNCT cana-1654 187 29 6 6 NUM cana-1654 187 30 ( ( PUNCT cana-1654 187 31 2015 2015 NUM cana-1654 187 32 ) ) PUNCT cana-1654 187 33 . . PUNCT