id sid tid token lemma pos cana-4038 1 1 untitled untitle VERB cana-4038 1 2 communications communication NOUN cana-4038 1 3 on on ADP cana-4038 1 4 applied apply VERB cana-4038 1 5 nonlinear nonlinear ADJ cana-4038 1 6 analysis analysis NOUN cana-4038 1 7 issn issn NOUN cana-4038 1 8 : : PUNCT cana-4038 1 9 1074 1074 NUM cana-4038 1 10 - - PUNCT cana-4038 1 11 133x 133x NUM cana-4038 1 12 vol vol NOUN cana-4038 1 13 32 32 NUM cana-4038 1 14 no no NOUN cana-4038 1 15 . . PUNCT cana-4038 2 1 9s 9s NUM cana-4038 2 2 ( ( PUNCT cana-4038 2 3 2025 2025 NUM cana-4038 2 4 ) ) PUNCT cana-4038 2 5 901 901 NUM cana-4038 2 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 2 7 sp sp ADP cana-4038 2 8 mean mean NOUN cana-4038 2 9 e e ADJ cana-4038 2 10 - - ADJ cana-4038 2 11 cordial cordial ADJ cana-4038 2 12 labeling labeling NOUN cana-4038 2 13 1 1 NUM cana-4038 2 14 m. m. NOUN cana-4038 2 15 aishwarya aishwarya PROPN cana-4038 2 16 , , PUNCT cana-4038 2 17 2*v 2*v NUM cana-4038 2 18 . . X cana-4038 2 19 maheswari maheswari PROPN cana-4038 2 20 and and CCONJ cana-4038 2 21 3v 3v NUM cana-4038 2 22 . . PUNCT cana-4038 3 1 balaji balaji PROPN cana-4038 3 2 1research 1research NUM cana-4038 3 3 scholar scholar NOUN cana-4038 3 4 , , PUNCT cana-4038 3 5 department department NOUN cana-4038 3 6 of of ADP cana-4038 3 7 mathematics mathematics PROPN cana-4038 3 8 , , PUNCT cana-4038 3 9 vels vels PROPN cana-4038 3 10 institute institute PROPN cana-4038 3 11 of of ADP cana-4038 3 12 science science PROPN cana-4038 3 13 technology technology NOUN cana-4038 3 14 and and CCONJ cana-4038 3 15 advanced advanced ADJ cana-4038 3 16 studies study NOUN cana-4038 3 17 , , PUNCT cana-4038 3 18 chennai chennai PROPN cana-4038 3 19 , , PUNCT cana-4038 3 20 tamil tamil PROPN cana-4038 3 21 nadu nadu PROPN cana-4038 3 22 , , PUNCT cana-4038 3 23 india india PROPN cana-4038 3 24 . . PUNCT cana-4038 3 25 email email NOUN cana-4038 3 26 : : PUNCT cana-4038 4 1 aishwaryamuthazhagu@gmail.com aishwaryamuthazhagu@gmail.com X cana-4038 4 2 2,*research 2,*research NUM cana-4038 4 3 supervisor supervisor NOUN cana-4038 4 4 , , PUNCT cana-4038 4 5 professor professor NOUN cana-4038 4 6 , , PUNCT cana-4038 4 7 department department NOUN cana-4038 4 8 of of ADP cana-4038 4 9 mathematics mathematics PROPN cana-4038 4 10 , , PUNCT cana-4038 4 11 vels vels PROPN cana-4038 4 12 institute institute PROPN cana-4038 4 13 of of ADP cana-4038 4 14 science science PROPN cana-4038 4 15 technology technology NOUN cana-4038 4 16 and and CCONJ cana-4038 4 17 advanced advanced ADJ cana-4038 4 18 studies study NOUN cana-4038 4 19 , , PUNCT cana-4038 4 20 chennai chennai PROPN cana-4038 4 21 , , PUNCT cana-4038 4 22 tamil tamil PROPN cana-4038 4 23 nadu nadu PROPN cana-4038 4 24 , , PUNCT cana-4038 4 25 india india PROPN cana-4038 4 26 . . PUNCT cana-4038 5 1 corresponding correspond VERB cana-4038 5 2 author author NOUN cana-4038 5 3 : : PUNCT cana-4038 5 4 3associate 3associate NUM cana-4038 5 5 professor professor NOUN cana-4038 5 6 , , PUNCT cana-4038 5 7 department department NOUN cana-4038 5 8 of of ADP cana-4038 5 9 mathematics mathematic NOUN cana-4038 5 10 , , PUNCT cana-4038 5 11 sacred sacred ADJ cana-4038 5 12 heart heart NOUN cana-4038 5 13 college college NOUN cana-4038 5 14 , , PUNCT cana-4038 5 15 tirupattur tirupattur PROPN cana-4038 5 16 , , PUNCT cana-4038 5 17 tamil tamil PROPN cana-4038 5 18 nadu nadu PROPN cana-4038 5 19 , , PUNCT cana-4038 5 20 india india PROPN cana-4038 5 21 . . PUNCT cana-4038 5 22 email email NOUN cana-4038 5 23 : : PUNCT cana-4038 6 1 pulibala70@gmail.com pulibala70@gmail.com X cana-4038 6 2 article article NOUN cana-4038 6 3 history history NOUN cana-4038 6 4 : : PUNCT cana-4038 6 5 received receive VERB cana-4038 6 6 : : PUNCT cana-4038 6 7 12 12 NUM cana-4038 6 8 - - SYM cana-4038 6 9 11 11 NUM cana-4038 6 10 - - PUNCT cana-4038 6 11 2024 2024 NUM cana-4038 6 12 revised revise VERB cana-4038 6 13 : : PUNCT cana-4038 6 14 10 10 NUM cana-4038 6 15 - - SYM cana-4038 6 16 12 12 NUM cana-4038 6 17 - - PUNCT cana-4038 6 18 2024 2024 NUM cana-4038 6 19 accepted accept VERB cana-4038 6 20 : : PUNCT cana-4038 6 21 16 16 NUM cana-4038 6 22 - - SYM cana-4038 6 23 01 01 NUM cana-4038 6 24 - - PUNCT cana-4038 6 25 2025 2025 NUM cana-4038 6 26 abstract abstract NOUN cana-4038 6 27 : : PUNCT cana-4038 6 28 assigning assign VERB cana-4038 6 29 an an DET cana-4038 6 30 integer integer NOUN cana-4038 6 31 to to ADP cana-4038 6 32 a a DET cana-4038 6 33 vertices vertex NOUN cana-4038 6 34 or or CCONJ cana-4038 6 35 edges edge NOUN cana-4038 6 36 is be AUX cana-4038 6 37 called call VERB cana-4038 6 38 a a DET cana-4038 6 39 vertex vertex NOUN cana-4038 6 40 or or CCONJ cana-4038 6 41 edge edge VERB cana-4038 6 42 labeling labeling NOUN cana-4038 6 43 respectively respectively ADV cana-4038 6 44 . . PUNCT cana-4038 7 1 suppose suppose VERB cana-4038 7 2 g g PROPN cana-4038 7 3 is be AUX cana-4038 7 4 a a DET cana-4038 7 5 simple simple ADJ cana-4038 7 6 graph graph NOUN cana-4038 7 7 . . PUNCT cana-4038 8 1 consider consider VERB cana-4038 8 2 the the DET cana-4038 8 3 function function NOUN cana-4038 8 4 f f PROPN cana-4038 8 5 for for ADP cana-4038 8 6 the the DET cana-4038 8 7 edge edge NOUN cana-4038 8 8 set set VERB cana-4038 8 9 𝑓 𝑓 PRON cana-4038 8 10 : : PUNCT cana-4038 8 11 𝑅 𝑅 PROPN cana-4038 8 12 β†’ β†’ SYM cana-4038 8 13 { { PUNCT cana-4038 8 14 0 0 NUM cana-4038 8 15 , , PUNCT cana-4038 8 16 1 1 NUM cana-4038 8 17 } } PUNCT cana-4038 8 18 . . PUNCT cana-4038 9 1 for for ADP cana-4038 9 2 each each DET cana-4038 9 3 vertex vertex NOUN cana-4038 9 4 t t PROPN cana-4038 9 5 ∈ ∈ PROPN cana-4038 9 6 𝑇 𝑇 PROPN cana-4038 9 7 , , PUNCT cana-4038 9 8 define define VERB cana-4038 9 9 f(t)=βˆ‘ f(t)=βˆ‘ NOUN cana-4038 9 10 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 9 11 ) ) PUNCT cana-4038 9 12 βˆ– βˆ– X cana-4038 9 13 𝑒𝑑 𝑒𝑑 PROPN cana-4038 9 14 ∈ ∈ PROPN cana-4038 9 15 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 9 16 ) ) PUNCT cana-4038 9 17 ( ( PUNCT cana-4038 9 18 mod2 mod2 NOUN cana-4038 9 19 ) ) PUNCT cana-4038 9 20 . . PUNCT cana-4038 10 1 the the DET cana-4038 10 2 function function NOUN cana-4038 10 3 f f PROPN cana-4038 10 4 is be AUX cana-4038 10 5 known know VERB cana-4038 10 6 as as ADP cana-4038 10 7 an an DET cana-4038 10 8 e e ADJ cana-4038 10 9 - - ADJ cana-4038 10 10 cordial cordial ADJ cana-4038 10 11 labeling labeling NOUN cana-4038 10 12 ( ( PUNCT cana-4038 10 13 ecl ecl NOUN cana-4038 10 14 ) ) PUNCT cana-4038 10 15 of of ADP cana-4038 10 16 g g PROPN cana-4038 10 17 if if SCONJ cana-4038 10 18 |π‘Ÿπ‘“(0 |π‘Ÿπ‘“(0 NUM cana-4038 10 19 ) ) PUNCT cana-4038 11 1 βˆ’ βˆ’ PROPN cana-4038 11 2 π‘Ÿπ‘“(1)| π‘Ÿπ‘“(1)| ADJ cana-4038 11 3 ≀ ≀ NUM cana-4038 11 4 1 1 NUM cana-4038 11 5 , , PUNCT cana-4038 11 6 and and CCONJ cana-4038 11 7 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 11 8 ) ) PUNCT cana-4038 12 1 βˆ’ βˆ’ PROPN cana-4038 12 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 12 3 ≀ ≀ NOUN cana-4038 12 4 1 1 NUM cana-4038 12 5 where where SCONJ cana-4038 12 6 π‘Ÿπ‘“(0 π‘Ÿπ‘“(0 PROPN cana-4038 12 7 ) ) PUNCT cana-4038 12 8 , , PUNCT cana-4038 12 9 π‘Ÿπ‘“(1)and π‘Ÿπ‘“(1)and PROPN cana-4038 12 10 𝑑𝑓(0 𝑑𝑓(0 PROPN cana-4038 12 11 ) ) PUNCT cana-4038 12 12 , , PUNCT cana-4038 12 13 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 12 14 ) ) PUNCT cana-4038 12 15 are be AUX cana-4038 12 16 the the DET cana-4038 12 17 number number NOUN cana-4038 12 18 of of ADP cana-4038 12 19 edges edge NOUN cana-4038 12 20 and and CCONJ cana-4038 12 21 vertices vertex NOUN cana-4038 12 22 labeled label VERB cana-4038 12 23 with with ADP cana-4038 12 24 0 0 NUM cana-4038 12 25 and and CCONJ cana-4038 12 26 labeled label VERB cana-4038 12 27 by by ADP cana-4038 12 28 1 1 NUM cana-4038 12 29 respectively respectively ADV cana-4038 12 30 . . PUNCT cana-4038 13 1 a a DET cana-4038 13 2 graph graph NOUN cana-4038 13 3 that that PRON cana-4038 13 4 admits admit VERB cana-4038 13 5 e e NOUN cana-4038 13 6 - - NOUN cana-4038 13 7 cl cl NOUN cana-4038 13 8 is be AUX cana-4038 13 9 said say VERB cana-4038 13 10 to to PART cana-4038 13 11 be be AUX cana-4038 13 12 e e ADJ cana-4038 13 13 - - ADJ cana-4038 13 14 cordial cordial ADJ cana-4038 13 15 graphs graph NOUN cana-4038 13 16 ( ( PUNCT cana-4038 13 17 e e NOUN cana-4038 13 18 - - NOUN cana-4038 13 19 cg cg NOUN cana-4038 13 20 ) ) PUNCT cana-4038 13 21 . . PUNCT cana-4038 14 1 based base VERB cana-4038 14 2 on on ADP cana-4038 14 3 the the DET cana-4038 14 4 above above ADJ cana-4038 14 5 definition definition NOUN cana-4038 14 6 we we PRON cana-4038 14 7 propose propose VERB cana-4038 14 8 a a DET cana-4038 14 9 novel novel ADJ cana-4038 14 10 labeling labeling NOUN cana-4038 14 11 known know VERB cana-4038 14 12 as as ADP cana-4038 14 13 sp sp ADP cana-4038 14 14 mean mean NOUN cana-4038 14 15 e e ADJ cana-4038 14 16 - - ADJ cana-4038 14 17 cordial cordial ADJ cana-4038 14 18 labeling labeling NOUN cana-4038 14 19 ( ( PUNCT cana-4038 14 20 e e NOUN cana-4038 14 21 - - NOUN cana-4038 14 22 cl cl NOUN cana-4038 14 23 ) ) PUNCT cana-4038 14 24 . . PUNCT cana-4038 15 1 in in ADP cana-4038 15 2 this this DET cana-4038 15 3 paper paper NOUN cana-4038 15 4 , , PUNCT cana-4038 15 5 we we PRON cana-4038 15 6 study study VERB cana-4038 15 7 sp sp ADP cana-4038 15 8 mean mean NOUN cana-4038 15 9 e e NOUN cana-4038 15 10 - - NOUN cana-4038 15 11 cl cl NOUN cana-4038 15 12 of of ADP cana-4038 15 13 several several ADJ cana-4038 15 14 families family NOUN cana-4038 15 15 of of ADP cana-4038 15 16 graphs graph NOUN cana-4038 15 17 such such ADJ cana-4038 15 18 as as ADP cana-4038 15 19 complete complete ADJ cana-4038 15 20 bipartite bipartite NOUN cana-4038 15 21 graphs graph NOUN cana-4038 15 22 , , PUNCT cana-4038 15 23 complete complete ADJ cana-4038 15 24 graphs graph NOUN cana-4038 15 25 , , PUNCT cana-4038 15 26 wheels wheel NOUN cana-4038 15 27 , , PUNCT cana-4038 15 28 etc etc X cana-4038 15 29 . . X cana-4038 16 1 keywords keyword NOUN cana-4038 16 2 : : PUNCT cana-4038 16 3 mean mean VERB cana-4038 16 4 e e ADJ cana-4038 16 5 - - ADJ cana-4038 16 6 cordial cordial ADJ cana-4038 16 7 labeling labeling NOUN cana-4038 16 8 , , PUNCT cana-4038 16 9 wheel wheel NOUN cana-4038 16 10 , , PUNCT cana-4038 16 11 complete complete ADJ cana-4038 16 12 graphs graph NOUN cana-4038 16 13 1 1 NUM cana-4038 16 14 . . PUNCT cana-4038 17 1 introduction introduction NOUN cana-4038 17 2 in in ADP cana-4038 17 3 this this DET cana-4038 17 4 research research NOUN cana-4038 17 5 , , PUNCT cana-4038 17 6 a a DET cana-4038 17 7 graph graph NOUN cana-4038 17 8 is be AUX cana-4038 17 9 defined define VERB cana-4038 17 10 as as ADP cana-4038 17 11 a a DET cana-4038 17 12 simple simple ADJ cana-4038 17 13 graph graph NOUN cana-4038 17 14 g. g. NOUN cana-4038 17 15 for for ADP cana-4038 17 16 graph graph NOUN cana-4038 17 17 theory theory NOUN cana-4038 17 18 definition definition NOUN cana-4038 17 19 and and CCONJ cana-4038 17 20 results result NOUN cana-4038 17 21 we we PRON cana-4038 17 22 refer refer VERB cana-4038 17 23 to to ADP cana-4038 17 24 f. f. PROPN cana-4038 17 25 harary harary PROPN cana-4038 18 1 [ [ X cana-4038 18 2 1,2 1,2 NUM cana-4038 18 3 ] ] PUNCT cana-4038 18 4 . . PUNCT cana-4038 19 1 rosa rosa PROPN cana-4038 19 2 [ [ X cana-4038 19 3 3 3 NUM cana-4038 19 4 ] ] PUNCT cana-4038 19 5 introduce introduce VERB cana-4038 19 6 a a DET cana-4038 19 7 new new ADJ cana-4038 19 8 idea idea NOUN cana-4038 19 9 called call VERB cana-4038 19 10 labeling label VERB cana-4038 19 11 the the DET cana-4038 19 12 graph graph NOUN cana-4038 19 13 . . PUNCT cana-4038 20 1 assigning assign VERB cana-4038 20 2 an an DET cana-4038 20 3 integer integer NOUN cana-4038 20 4 to to ADP cana-4038 20 5 a a DET cana-4038 20 6 vertices vertex NOUN cana-4038 20 7 or or CCONJ cana-4038 20 8 edges edge NOUN cana-4038 20 9 is be AUX cana-4038 20 10 called call VERB cana-4038 20 11 a a DET cana-4038 20 12 vertex vertex NOUN cana-4038 20 13 or or CCONJ cana-4038 20 14 edge edge VERB cana-4038 20 15 labeling labeling NOUN cana-4038 20 16 respectively respectively ADV cana-4038 20 17 . . PUNCT cana-4038 21 1 several several ADJ cana-4038 21 2 labeling labeling NOUN cana-4038 21 3 introduced introduce VERB cana-4038 21 4 by by ADP cana-4038 21 5 various various ADJ cana-4038 21 6 authors author NOUN cana-4038 21 7 [ [ X cana-4038 21 8 4,5,6,7,8,9 4,5,6,7,8,9 NUM cana-4038 21 9 ] ] PUNCT cana-4038 21 10 . . PUNCT cana-4038 22 1 here here ADV cana-4038 22 2 we we PRON cana-4038 22 3 discuss discuss VERB cana-4038 22 4 with with ADP cana-4038 22 5 edge edge NOUN cana-4038 22 6 labeling labeling NOUN cana-4038 22 7 . . PUNCT cana-4038 23 1 if if SCONJ cana-4038 23 2 a a DET cana-4038 23 3 mapping mapping NOUN cana-4038 23 4 𝑓 𝑓 NOUN cana-4038 23 5 : : PUNCT cana-4038 23 6 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 23 7 ) ) PUNCT cana-4038 23 8 β†’ β†’ SYM cana-4038 23 9 { { PUNCT cana-4038 23 10 0,1,2 0,1,2 ADJ cana-4038 23 11 … … PUNCT cana-4038 23 12 … … PUNCT cana-4038 23 13 , , PUNCT cana-4038 23 14 π‘ž π‘ž X cana-4038 23 15 } } PUNCT cana-4038 23 16 exists exist VERB cana-4038 23 17 , , PUNCT cana-4038 23 18 a a DET cana-4038 23 19 graph graph NOUN cana-4038 23 20 g g NOUN cana-4038 23 21 is be AUX cana-4038 23 22 said say VERB cana-4038 23 23 to to PART cana-4038 23 24 have have VERB cana-4038 23 25 elegant elegant ADJ cana-4038 23 26 edges edge NOUN cana-4038 23 27 such such ADJ cana-4038 23 28 that that SCONJ cana-4038 23 29 the the DET cana-4038 23 30 induced induced ADJ cana-4038 23 31 mapping mapping NOUN cana-4038 23 32 π‘“βˆ— π‘“βˆ— NOUN cana-4038 23 33 : : PUNCT cana-4038 23 34 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 23 35 ) ) PUNCT cana-4038 23 36 β†’ β†’ SYM cana-4038 23 37 { { PUNCT cana-4038 23 38 0 0 NUM cana-4038 23 39 , , PUNCT cana-4038 23 40 1 1 NUM cana-4038 23 41 , , PUNCT cana-4038 23 42 2 2 NUM cana-4038 23 43 , , PUNCT cana-4038 23 44 … … PUNCT cana-4038 23 45 . . PUNCT cana-4038 23 46 . . PUNCT cana-4038 24 1 , , PUNCT cana-4038 24 2 𝑝 𝑝 ADV cana-4038 24 3 βˆ’ βˆ’ NOUN cana-4038 24 4 1 1 NUM cana-4038 24 5 } } PUNCT cana-4038 24 6 given give VERB cana-4038 24 7 by by ADP cana-4038 24 8 𝑓(𝑒 𝑓(𝑒 PROPN cana-4038 24 9 ) ) PUNCT cana-4038 24 10 ≀ ≀ ADV cana-4038 24 11 βˆ‘ βˆ‘ PUNCT cana-4038 24 12 𝑓(𝑒𝑑)π‘šπ‘œπ‘‘|𝑑| 𝑓(𝑒𝑑)π‘šπ‘œπ‘‘|𝑑| PROPN cana-4038 24 13 , , PUNCT cana-4038 24 14 𝑒𝑑 𝑒𝑑 VERB cana-4038 24 15 βˆˆπ‘…(𝐺 βˆˆπ‘…(𝐺 NOUN cana-4038 24 16 ) ) PUNCT cana-4038 24 17 . . PUNCT cana-4038 25 1 [ [ X cana-4038 25 2 10,11,12,13,14,15,16 10,11,12,13,14,15,16 NUM cana-4038 25 3 ] ] X cana-4038 25 4 . . PUNCT cana-4038 26 1 the the DET cana-4038 26 2 label label NOUN cana-4038 26 3 of of ADP cana-4038 26 4 vertex vertex NOUN cana-4038 26 5 v v NOUN cana-4038 26 6 under under ADP cana-4038 26 7 f f PROPN cana-4038 26 8 is be AUX cana-4038 26 9 said say VERB cana-4038 26 10 to to PART cana-4038 26 11 be be AUX cana-4038 26 12 f(v f(v NOUN cana-4038 26 13 ) ) PUNCT cana-4038 26 14 . . PUNCT cana-4038 27 1 a a DET cana-4038 27 2 mapping mapping NOUN cana-4038 27 3 𝑓 𝑓 NOUN cana-4038 27 4 : : PUNCT cana-4038 27 5 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 27 6 ) ) PUNCT cana-4038 27 7 β†’{0 β†’{0 NUM cana-4038 27 8 , , PUNCT cana-4038 27 9 1 1 NUM cana-4038 27 10 } } PUNCT cana-4038 27 11 is be AUX cana-4038 27 12 known know VERB cana-4038 27 13 as as ADP cana-4038 27 14 binary binary ADJ cana-4038 27 15 vertex vertex NOUN cana-4038 27 16 labeling labeling NOUN cana-4038 27 17 of of ADP cana-4038 27 18 g. g. PROPN cana-4038 27 19 cordial cordial ADJ cana-4038 27 20 labeling labeling NOUN cana-4038 27 21 is be AUX cana-4038 27 22 the the DET cana-4038 27 23 binary binary ADJ cana-4038 27 24 vertex vertex NOUN cana-4038 27 25 labeling labeling NOUN cana-4038 27 26 of of ADP cana-4038 27 27 a a DET cana-4038 27 28 graph graph NOUN cana-4038 27 29 g g NOUN cana-4038 27 30 if if SCONJ cana-4038 27 31 |π‘Ÿπ‘“(0 |π‘Ÿπ‘“(0 NUM cana-4038 27 32 ) ) PUNCT cana-4038 28 1 βˆ’ βˆ’ PROPN cana-4038 29 1 π‘Ÿπ‘“(1)| π‘Ÿπ‘“(1)| ADJ cana-4038 29 2 ≀ ≀ NOUN cana-4038 29 3 1 1 NUM cana-4038 29 4 and and CCONJ cana-4038 29 5 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 29 6 ) ) PUNCT cana-4038 30 1 βˆ’ βˆ’ PROPN cana-4038 30 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 30 3 ≀ ≀ ADJ cana-4038 30 4 1 1 NUM cana-4038 30 5 . . PUNCT cana-4038 31 1 a a DET cana-4038 31 2 graph graph NOUN cana-4038 31 3 satisfies satisfy VERB cana-4038 31 4 cordial cordial ADJ cana-4038 31 5 labeling labeling NOUN cana-4038 31 6 is be AUX cana-4038 31 7 defined define VERB cana-4038 31 8 as as ADP cana-4038 31 9 cordial cordial ADJ cana-4038 31 10 graph graph NOUN cana-4038 31 11 , , PUNCT cana-4038 31 12 where where SCONJ cana-4038 31 13 π‘Ÿπ‘“(0 π‘Ÿπ‘“(0 PROPN cana-4038 31 14 ) ) PUNCT cana-4038 31 15 , , PUNCT cana-4038 31 16 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 31 17 ) ) PUNCT cana-4038 31 18 and and CCONJ cana-4038 31 19 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 31 20 ) ) PUNCT cana-4038 31 21 , , PUNCT cana-4038 31 22 𝑑𝑓(1)are 𝑑𝑓(1)are VERB cana-4038 31 23 the the DET cana-4038 31 24 number number NOUN cana-4038 31 25 of of ADP cana-4038 31 26 edges edge NOUN cana-4038 31 27 and and CCONJ cana-4038 31 28 vertices vertex NOUN cana-4038 31 29 labeled label VERB cana-4038 31 30 by by ADP cana-4038 31 31 0 0 NUM cana-4038 31 32 and and CCONJ cana-4038 31 33 1 1 NUM cana-4038 31 34 respectively respectively ADV cana-4038 31 35 . . PUNCT cana-4038 32 1 the the DET cana-4038 32 2 cordial cordial ADJ cana-4038 32 3 labeling labeling NOUN cana-4038 32 4 introduced introduce VERB cana-4038 32 5 in in ADP cana-4038 32 6 1987 1987 NUM cana-4038 32 7 by by ADP cana-4038 32 8 cahit cahit ADJ cana-4038 32 9 [ [ X cana-4038 32 10 17 17 NUM cana-4038 32 11 ] ] SYM cana-4038 32 12 . . PUNCT cana-4038 33 1 2 2 X cana-4038 33 2 . . X cana-4038 33 3 sp sp ADP cana-4038 33 4 mean mean VERB cana-4038 33 5 e e ADJ cana-4038 33 6 - - ADJ cana-4038 33 7 cordial cordial ADJ cana-4038 33 8 labeling labeling NOUN cana-4038 33 9 definition definition NOUN cana-4038 33 10 2.1 2.1 NUM cana-4038 33 11 let let VERB cana-4038 33 12 g g NOUN cana-4038 33 13 be be AUX cana-4038 33 14 a a DET cana-4038 33 15 simple simple ADJ cana-4038 33 16 graph graph NOUN cana-4038 33 17 . . PUNCT cana-4038 34 1 let let VERB cana-4038 34 2 f f PRON cana-4038 34 3 be be AUX cana-4038 34 4 a a DET cana-4038 34 5 function function NOUN cana-4038 34 6 𝑓:𝑅 𝑓:𝑅 NOUN cana-4038 34 7 β†’ β†’ SYM cana-4038 34 8 { { PUNCT cana-4038 34 9 1,2 1,2 NUM cana-4038 34 10 } } PUNCT cana-4038 34 11 and and CCONJ cana-4038 34 12 an an DET cana-4038 34 13 induced induced ADJ cana-4038 34 14 function function NOUN cana-4038 34 15 π‘“βˆ— π‘“βˆ— NOUN cana-4038 34 16 : : PUNCT cana-4038 34 17 𝑇 𝑇 PROPN cana-4038 34 18 β†’ β†’ SYM cana-4038 34 19 { { PUNCT cana-4038 34 20 0,1 0,1 NUM cana-4038 34 21 } } PUNCT cana-4038 34 22 . . PUNCT cana-4038 35 1 we we PRON cana-4038 35 2 associate associate VERB cana-4038 35 3 two two NUM cana-4038 35 4 integers integer NOUN cana-4038 35 5 . . PUNCT cana-4038 36 1 𝑆 𝑆 PROPN cana-4038 36 2 = = SYM cana-4038 36 3 βˆ‘ βˆ‘ PUNCT cana-4038 36 4 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 36 5 ) ) PUNCT cana-4038 36 6 βˆ– βˆ– X cana-4038 36 7 𝑒𝑑 𝑒𝑑 PROPN cana-4038 36 8 ∈ ∈ PROPN cana-4038 36 9 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 36 10 ) ) PUNCT cana-4038 36 11 and and CCONJ cana-4038 36 12 𝑃 𝑃 PROPN cana-4038 36 13 = = SYM cana-4038 36 14 ∏ ∏ NUM cana-4038 36 15 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 36 16 ) ) PUNCT cana-4038 36 17 . . PUNCT cana-4038 37 1 for for SCONJ cana-4038 37 2 each each DET cana-4038 37 3 maheswari.sbs@vistas.ac.in maheswari.sbs@vistas.ac.in PROPN cana-4038 37 4 mailto:aishwaryamuthazhagu@gmail.com mailto:aishwaryamuthazhagu@gmail.com X cana-4038 37 5 mailto:sasikala.sbs@velsuniv.ac.in mailto:sasikala.sbs@velsuniv.ac.in PROPN cana-4038 37 6 mailto:pulibala70@gmail.com mailto:pulibala70@gmail.com PROPN cana-4038 37 7 communications communication NOUN cana-4038 37 8 on on ADP cana-4038 37 9 applied apply VERB cana-4038 37 10 nonlinear nonlinear ADJ cana-4038 37 11 analysis analysis NOUN cana-4038 37 12 issn issn NOUN cana-4038 37 13 : : PUNCT cana-4038 37 14 1074 1074 NUM cana-4038 37 15 - - PUNCT cana-4038 37 16 133x 133x NUM cana-4038 37 17 vol vol NOUN cana-4038 37 18 32 32 NUM cana-4038 37 19 no no NOUN cana-4038 37 20 . . PUNCT cana-4038 38 1 9s 9s NUM cana-4038 38 2 ( ( PUNCT cana-4038 38 3 2025 2025 NUM cana-4038 38 4 ) ) PUNCT cana-4038 38 5 902 902 NUM cana-4038 39 1 https://internationalpubls.com https://internationalpubls.com X cana-4038 39 2 vertex vertex NOUN cana-4038 39 3 u u PRON cana-4038 39 4 assign assign VERB cana-4038 39 5 the the DET cana-4038 39 6 label label NOUN cana-4038 39 7 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 39 8 βŒ‹ βŒ‹ NOUN cana-4038 39 9 ( ( PUNCT cana-4038 39 10 mod mod PROPN cana-4038 39 11 2 2 NUM cana-4038 39 12 ) ) PUNCT cana-4038 39 13 , , PUNCT cana-4038 39 14 then then ADV cana-4038 39 15 f f PROPN cana-4038 39 16 is be AUX cana-4038 39 17 defined define VERB cana-4038 39 18 as as ADP cana-4038 39 19 sp sp ADP cana-4038 39 20 mean mean NOUN cana-4038 39 21 e e ADJ cana-4038 39 22 - - ADJ cana-4038 39 23 cordial cordial ADJ cana-4038 39 24 labeling labeling NOUN cana-4038 39 25 if if SCONJ cana-4038 39 26 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 39 27 ) ) PUNCT cana-4038 39 28 βˆ’π‘Ÿπ‘“(2)| βˆ’π‘Ÿπ‘“(2)| VERB cana-4038 39 29 ≀ ≀ ADV cana-4038 39 30 1 1 NUM cana-4038 39 31 and and CCONJ cana-4038 39 32 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 39 33 ) ) PUNCT cana-4038 40 1 βˆ’ βˆ’ PROPN cana-4038 40 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 40 3 ≀ ≀ NOUN cana-4038 40 4 1 1 NUM cana-4038 40 5 where where SCONJ cana-4038 40 6 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 40 7 ) ) PUNCT cana-4038 40 8 and and CCONJ cana-4038 40 9 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 40 10 ) ) PUNCT cana-4038 40 11 is be AUX cana-4038 40 12 the the DET cana-4038 40 13 number number NOUN cana-4038 40 14 of of ADP cana-4038 40 15 vertices vertex NOUN cana-4038 40 16 labeled label VERB cana-4038 40 17 with with ADP cana-4038 40 18 0 0 NUM cana-4038 40 19 and and CCONJ cana-4038 40 20 labeled label VERB cana-4038 40 21 by by ADP cana-4038 40 22 1 1 NUM cana-4038 40 23 and and CCONJ cana-4038 40 24 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 40 25 ) ) PUNCT cana-4038 40 26 and and CCONJ cana-4038 40 27 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 40 28 ) ) PUNCT cana-4038 40 29 is be AUX cana-4038 40 30 the the DET cana-4038 40 31 number number NOUN cana-4038 40 32 of of ADP cana-4038 40 33 edges edge NOUN cana-4038 40 34 labeled label VERB cana-4038 40 35 with with ADP cana-4038 40 36 1 1 NUM cana-4038 40 37 and and CCONJ cana-4038 40 38 labeled label VERB cana-4038 40 39 by by ADP cana-4038 40 40 2 2 NUM cana-4038 40 41 respectively respectively ADV cana-4038 40 42 . . PUNCT cana-4038 41 1 a a DET cana-4038 41 2 graph graph NOUN cana-4038 41 3 which which PRON cana-4038 41 4 admits admit VERB cana-4038 41 5 sp sp ADP cana-4038 41 6 mean mean NOUN cana-4038 41 7 e e NOUN cana-4038 41 8 - - NOUN cana-4038 41 9 cl cl NOUN cana-4038 41 10 is be AUX cana-4038 41 11 known know VERB cana-4038 41 12 as as ADP cana-4038 41 13 sp sp ADP cana-4038 41 14 mean mean NOUN cana-4038 41 15 e e NOUN cana-4038 41 16 - - NOUN cana-4038 41 17 cg cg INTJ cana-4038 41 18 ( ( PUNCT cana-4038 41 19 figure figure NOUN cana-4038 41 20 1 1 NUM cana-4038 41 21 ) ) PUNCT cana-4038 41 22 . . PUNCT cana-4038 42 1 example example NOUN cana-4038 42 2 2.2 2.2 NUM cana-4038 42 3 figure figure NOUN cana-4038 42 4 1 1 NUM cana-4038 42 5 . . PUNCT cana-4038 43 1 sp sp ADP cana-4038 43 2 mean mean VERB cana-4038 43 3 e e NOUN cana-4038 43 4 - - NOUN cana-4038 43 5 cg cg NOUN cana-4038 43 6 for for ADP cana-4038 43 7 vertex vertex NOUN cana-4038 43 8 𝑒1 𝑒1 NOUN cana-4038 43 9 , , PUNCT cana-4038 43 10 then then ADV cana-4038 43 11 𝑆 𝑆 PROPN cana-4038 43 12 = = SYM cana-4038 43 13 1 1 NUM cana-4038 43 14 and and CCONJ cana-4038 43 15 𝑃 𝑃 NOUN cana-4038 43 16 = = SYM cana-4038 43 17 1 1 NUM cana-4038 43 18 then then ADV cana-4038 43 19 π‘“βˆ—(𝑒1 π‘“βˆ—(𝑒1 PROPN cana-4038 43 20 ) ) PUNCT cana-4038 43 21 = = SYM cana-4038 43 22 1 1 NUM cana-4038 43 23 for for ADP cana-4038 43 24 vertex vertex NOUN cana-4038 43 25 𝑒2 𝑒2 PROPN cana-4038 43 26 , , PUNCT cana-4038 43 27 then then ADV cana-4038 43 28 𝑆 𝑆 PROPN cana-4038 43 29 = = SYM cana-4038 43 30 4 4 NUM cana-4038 43 31 and and CCONJ cana-4038 43 32 𝑃 𝑃 NOUN cana-4038 43 33 = = SYM cana-4038 43 34 2 2 NUM cana-4038 43 35 then then ADV cana-4038 43 36 π‘“βˆ—(𝑒2 π‘“βˆ—(𝑒2 PROPN cana-4038 43 37 ) ) PUNCT cana-4038 43 38 = = SYM cana-4038 44 1 1 1 NUM cana-4038 44 2 for for ADP cana-4038 44 3 vertex vertex NOUN cana-4038 44 4 𝑒3 𝑒3 NOUN cana-4038 44 5 , , PUNCT cana-4038 44 6 then then ADV cana-4038 44 7 𝑆 𝑆 PROPN cana-4038 44 8 = = SYM cana-4038 44 9 4 4 NUM cana-4038 44 10 and and CCONJ cana-4038 44 11 𝑃 𝑃 NOUN cana-4038 44 12 = = SYM cana-4038 44 13 2 2 NUM cana-4038 44 14 then then ADV cana-4038 44 15 π‘“βˆ—(𝑒3 π‘“βˆ—(𝑒3 NUM cana-4038 44 16 ) ) PUNCT cana-4038 44 17 = = SYM cana-4038 44 18 0 0 NUM cana-4038 45 1 for for ADP cana-4038 45 2 vertex vertex NOUN cana-4038 45 3 𝑒4 𝑒4 NOUN cana-4038 45 4 , , PUNCT cana-4038 45 5 then then ADV cana-4038 45 6 𝑆 𝑆 PROPN cana-4038 45 7 = = SYM cana-4038 45 8 4 4 NUM cana-4038 45 9 and and CCONJ cana-4038 45 10 𝑃 𝑃 NOUN cana-4038 45 11 = = SYM cana-4038 45 12 2 2 NUM cana-4038 45 13 then then ADV cana-4038 45 14 π‘“βˆ—(𝑒4 π‘“βˆ—(𝑒4 PROPN cana-4038 45 15 ) ) PUNCT cana-4038 45 16 = = SYM cana-4038 45 17 1 1 NUM cana-4038 45 18 for for ADP cana-4038 45 19 vertex vertex NOUN cana-4038 45 20 𝑒5 𝑒5 NOUN cana-4038 45 21 , , PUNCT cana-4038 45 22 then then ADV cana-4038 45 23 𝑆 𝑆 PROPN cana-4038 45 24 = = SYM cana-4038 45 25 5 5 NUM cana-4038 45 26 and and CCONJ cana-4038 45 27 𝑃 𝑃 NOUN cana-4038 45 28 = = NOUN cana-4038 45 29 4 4 NUM cana-4038 45 30 then then ADV cana-4038 45 31 π‘“βˆ—(𝑒5 π‘“βˆ—(𝑒5 PROPN cana-4038 45 32 ) ) PUNCT cana-4038 45 33 = = SYM cana-4038 45 34 0 0 NUM cana-4038 45 35 for for ADP cana-4038 45 36 vertex vertex NOUN cana-4038 45 37 𝑒6 𝑒6 NOUN cana-4038 45 38 , , PUNCT cana-4038 45 39 then then ADV cana-4038 45 40 𝑆 𝑆 PROPN cana-4038 45 41 = = SYM cana-4038 45 42 2 2 NUM cana-4038 45 43 and and CCONJ cana-4038 45 44 𝑃 𝑃 NOUN cana-4038 45 45 = = SYM cana-4038 45 46 2 2 NUM cana-4038 45 47 then then ADV cana-4038 45 48 π‘“βˆ—(𝑒6 π‘“βˆ—(𝑒6 ADJ cana-4038 45 49 ) ) PUNCT cana-4038 45 50 = = SYM cana-4038 45 51 0 0 NUM cana-4038 46 1 for for ADP cana-4038 46 2 vertex vertex NOUN cana-4038 46 3 𝑒7 𝑒7 PROPN cana-4038 46 4 , , PUNCT cana-4038 46 5 then then ADV cana-4038 46 6 𝑆 𝑆 PROPN cana-4038 46 7 = = SYM cana-4038 46 8 2 2 NUM cana-4038 46 9 and and CCONJ cana-4038 46 10 𝑃 𝑃 NOUN cana-4038 46 11 = = SYM cana-4038 46 12 2 2 NUM cana-4038 46 13 then then ADV cana-4038 46 14 π‘“βˆ—(𝑒7 π‘“βˆ—(𝑒7 PROPN cana-4038 46 15 ) ) PUNCT cana-4038 46 16 = = SYM cana-4038 46 17 0 0 PUNCT cana-4038 46 18 hence hence ADV cana-4038 46 19 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 46 20 ) ) PUNCT cana-4038 46 21 = = SYM cana-4038 46 22 4 4 NUM cana-4038 46 23 , , PUNCT cana-4038 46 24 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 46 25 ) ) PUNCT cana-4038 46 26 = = SYM cana-4038 46 27 3 3 NUM cana-4038 46 28 this this PRON cana-4038 46 29 implies imply VERB cana-4038 46 30 |𝑑𝑓(0 |𝑑𝑓(0 PROPN cana-4038 46 31 ) ) PUNCT cana-4038 47 1 βˆ’ βˆ’ PROPN cana-4038 47 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 47 3 ≀ ≀ NOUN cana-4038 47 4 1 1 NUM cana-4038 47 5 hence hence ADV cana-4038 47 6 the the DET cana-4038 47 7 above above ADJ cana-4038 47 8 graph graph NOUN cana-4038 47 9 is be AUX cana-4038 47 10 a a DET cana-4038 47 11 sp sp NOUN cana-4038 47 12 mean mean NOUN cana-4038 47 13 e e NOUN cana-4038 47 14 - - NOUN cana-4038 47 15 cg cg NOUN cana-4038 47 16 . . PUNCT cana-4038 48 1 theorem theorem VERB cana-4038 48 2 2.3 2.3 NUM cana-4038 48 3 for for ADP cana-4038 48 4 any any DET cana-4038 48 5 star star NOUN cana-4038 48 6 ( ( PUNCT cana-4038 48 7 𝐾1,𝑛 𝐾1,𝑛 PROPN cana-4038 48 8 ) ) PUNCT cana-4038 48 9 is be AUX cana-4038 48 10 sp sp ADP cana-4038 48 11 mean mean NOUN cana-4038 48 12 e e ADJ cana-4038 48 13 - - ADJ cana-4038 48 14 cordial cordial ADJ cana-4038 48 15 graph graph NOUN cana-4038 48 16 if if SCONJ cana-4038 48 17 n n CCONJ cana-4038 48 18 even even ADV cana-4038 48 19 . . PUNCT cana-4038 49 1 proof proof NOUN cana-4038 49 2 : : PUNCT cana-4038 49 3 let let VERB cana-4038 49 4 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PRON cana-4038 49 5 ) ) PUNCT cana-4038 49 6 = = PRON cana-4038 49 7 { { PUNCT cana-4038 49 8 𝑒 𝑒 X cana-4038 49 9 , , PUNCT cana-4038 49 10 𝑒𝑖 𝑒𝑖 INTJ cana-4038 49 11 / / SYM cana-4038 49 12 𝑖 𝑖 SYM cana-4038 49 13 = = SYM cana-4038 49 14 1,2 1,2 NUM cana-4038 49 15 , , PUNCT cana-4038 49 16 … … PUNCT cana-4038 49 17 . . PUNCT cana-4038 50 1 𝑛 𝑛 X cana-4038 50 2 } } PUNCT cana-4038 50 3 be be VERB cana-4038 50 4 the the DET cana-4038 50 5 vertices vertex NOUN cana-4038 50 6 and and CCONJ cana-4038 50 7 𝑅(𝐾1,𝑛 𝑅(𝐾1,𝑛 ADJ cana-4038 50 8 ) ) PUNCT cana-4038 51 1 = = PRON cana-4038 51 2 { { PUNCT cana-4038 51 3 𝑒𝑒𝑖 𝑒𝑒𝑖 PROPN cana-4038 51 4 / / SYM cana-4038 51 5 𝑖 𝑖 SYM cana-4038 51 6 = = SYM cana-4038 51 7 1,2 1,2 NUM cana-4038 51 8 , , PUNCT cana-4038 51 9 … … PUNCT cana-4038 51 10 . . PUNCT cana-4038 52 1 𝑛 𝑛 X cana-4038 52 2 } } PUNCT cana-4038 52 3 be be VERB cana-4038 52 4 the the DET cana-4038 52 5 edges edge NOUN cana-4038 52 6 . . PUNCT cana-4038 53 1 then then ADV cana-4038 53 2 | | ADV cana-4038 53 3 t(𝐾1,𝑛)| t(𝐾1,𝑛)| NUM cana-4038 53 4 = = SYM cana-4038 53 5 𝑛 𝑛 NOUN cana-4038 53 6 + + NUM cana-4038 53 7 1 1 NUM cana-4038 53 8 and and CCONJ cana-4038 53 9 | | ADV cana-4038 53 10 e(𝐾1,𝑛)| e(𝐾1,𝑛)| ADV cana-4038 53 11 = = NOUN cana-4038 53 12 𝑛 𝑛 PART cana-4038 53 13 define define VERB cana-4038 53 14 the the DET cana-4038 53 15 function function NOUN cana-4038 53 16 𝑓 𝑓 NOUN cana-4038 53 17 : : PUNCT cana-4038 53 18 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 53 19 ) ) PUNCT cana-4038 53 20 β†’ β†’ SYM cana-4038 53 21 { { PUNCT cana-4038 53 22 1,2 1,2 NUM cana-4038 53 23 } } PUNCT cana-4038 53 24 as as ADP cana-4038 53 25 𝑓(𝑒𝑒𝑖 𝑓(𝑒𝑒𝑖 PROPN cana-4038 53 26 ) ) PUNCT cana-4038 53 27 = = PUNCT cana-4038 53 28 { { PUNCT cana-4038 53 29 2 2 NUM cana-4038 53 30 𝑖𝑓 𝑖𝑓 SYM cana-4038 53 31 𝑖 𝑖 PUNCT cana-4038 53 32 𝑖𝑠 𝑖𝑠 PROPN cana-4038 53 33 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 53 34 𝑖𝑓 𝑖𝑓 ADP cana-4038 53 35 𝑖 𝑖 PUNCT cana-4038 53 36 𝑖𝑠 𝑖𝑠 NOUN cana-4038 54 1 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 55 1 find find VERB cana-4038 55 2 𝑆 𝑆 PROPN cana-4038 55 3 = = SYM cana-4038 55 4 βˆ‘ βˆ‘ PUNCT cana-4038 55 5 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 55 6 ) ) PUNCT cana-4038 55 7 βˆ– βˆ– X cana-4038 55 8 𝑒𝑑 𝑒𝑑 PROPN cana-4038 55 9 ∈ ∈ PROPN cana-4038 55 10 𝐸(𝐺 𝐸(𝐺 PROPN cana-4038 55 11 ) ) PUNCT cana-4038 55 12 and and CCONJ cana-4038 55 13 𝑃 𝑃 PROPN cana-4038 55 14 = = SYM cana-4038 55 15 ∏ ∏ PROPN cana-4038 55 16 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 55 17 ) ) PUNCT cana-4038 55 18 for for ADP cana-4038 55 19 each each DET cana-4038 55 20 vertex vertex NOUN cana-4038 55 21 in in ADP cana-4038 55 22 t(𝐾1,𝑛 t(𝐾1,𝑛 PROPN cana-4038 55 23 ) ) PUNCT cana-4038 55 24 define define VERB cana-4038 55 25 π‘“βˆ— π‘“βˆ— NOUN cana-4038 55 26 : : PUNCT cana-4038 55 27 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 55 28 ) ) PUNCT cana-4038 55 29 β†’ β†’ SYM cana-4038 55 30 { { PUNCT cana-4038 55 31 0,1 0,1 NOUN cana-4038 55 32 } } PUNCT cana-4038 55 33 defined define VERB cana-4038 55 34 by by ADP cana-4038 55 35 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 55 36 ) ) PUNCT cana-4038 55 37 = = X cana-4038 55 38 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 VERB cana-4038 55 39 βŒ‹(mod βŒ‹(mod NOUN cana-4038 55 40 2 2 NUM cana-4038 55 41 ) ) PUNCT cana-4038 55 42 . . PUNCT cana-4038 56 1 communications communication NOUN cana-4038 56 2 on on ADP cana-4038 56 3 applied apply VERB cana-4038 56 4 nonlinear nonlinear ADJ cana-4038 56 5 analysis analysis NOUN cana-4038 56 6 issn issn NOUN cana-4038 56 7 : : PUNCT cana-4038 56 8 1074 1074 NUM cana-4038 56 9 - - PUNCT cana-4038 56 10 133x 133x NUM cana-4038 56 11 vol vol NOUN cana-4038 56 12 32 32 NUM cana-4038 56 13 no no NOUN cana-4038 56 14 . . PUNCT cana-4038 57 1 9s 9s NUM cana-4038 57 2 ( ( PUNCT cana-4038 57 3 2025 2025 NUM cana-4038 57 4 ) ) PUNCT cana-4038 57 5 903 903 NUM cana-4038 57 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 57 7 then then ADV cana-4038 57 8 for for ADP cana-4038 57 9 𝑛 𝑛 DET cana-4038 57 10 ≑ ≑ PROPN cana-4038 57 11 0(π‘šπ‘œπ‘‘ 0(π‘šπ‘œπ‘‘ NOUN cana-4038 57 12 4 4 X cana-4038 57 13 ) ) PUNCT cana-4038 57 14 𝑑𝑓(0 𝑑𝑓(0 PROPN cana-4038 57 15 ) ) PUNCT cana-4038 57 16 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 57 17 ) ) PUNCT cana-4038 57 18 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 57 19 ) ) PUNCT cana-4038 57 20 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 57 21 ) ) PUNCT cana-4038 57 22 𝑛≑ 𝑛≑ VERB cana-4038 57 23 0(π‘šπ‘œπ‘‘ 0(π‘šπ‘œπ‘‘ ADJ cana-4038 57 24 4 4 NUM cana-4038 57 25 ) ) PUNCT cana-4038 57 26 𝑛2 𝑛2 NOUN cana-4038 57 27 𝑛2 𝑛2 NOUN cana-4038 57 28 + + SYM cana-4038 57 29 1 1 NUM cana-4038 57 30 𝑛2 𝑛2 NOUN cana-4038 57 31 𝑛2 𝑛2 NOUN cana-4038 57 32 𝑛≑ 𝑛≑ PART cana-4038 57 33 0(π‘šπ‘œπ‘‘ 0(π‘šπ‘œπ‘‘ ADJ cana-4038 57 34 4 4 NUM cana-4038 57 35 ) ) PUNCT cana-4038 57 36 𝑛2 𝑛2 NOUN cana-4038 57 37 + + SYM cana-4038 57 38 1 1 NUM cana-4038 57 39 𝑛2 𝑛2 NOUN cana-4038 57 40 𝑛2 𝑛2 NOUN cana-4038 57 41 𝑛2 𝑛2 NOUN cana-4038 57 42 the the DET cana-4038 57 43 above above ADJ cana-4038 57 44 labeling labeling NOUN cana-4038 57 45 satisfies satisfie NOUN cana-4038 57 46 , , PUNCT cana-4038 57 47 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 57 48 ) ) PUNCT cana-4038 57 49 βˆ’ βˆ’ PROPN cana-4038 58 1 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 58 2 ≀ ≀ NOUN cana-4038 58 3 1 1 NUM cana-4038 58 4 and and CCONJ cana-4038 58 5 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 58 6 ) ) PUNCT cana-4038 59 1 βˆ’ βˆ’ NOUN cana-4038 59 2 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 59 3 ≀ ≀ NUM cana-4038 59 4 1 1 NUM cana-4038 59 5 . . PUNCT cana-4038 60 1 the the DET cana-4038 60 2 star star NOUN cana-4038 60 3 graph graph NOUN cana-4038 60 4 admits admit VERB cana-4038 60 5 sp sp ADP cana-4038 60 6 mean mean NOUN cana-4038 60 7 e e NOUN cana-4038 60 8 - - NOUN cana-4038 60 9 cl cl NOUN cana-4038 60 10 . . PUNCT cana-4038 61 1 hence hence ADV cana-4038 61 2 star star NOUN cana-4038 61 3 graph graph NOUN cana-4038 61 4 is be AUX cana-4038 61 5 a a DET cana-4038 61 6 sp sp NOUN cana-4038 61 7 mean mean NOUN cana-4038 61 8 e e NOUN cana-4038 61 9 - - NOUN cana-4038 61 10 cg cg NOUN cana-4038 61 11 if if SCONJ cana-4038 61 12 n n NOUN cana-4038 61 13 is be AUX cana-4038 61 14 even even ADV cana-4038 61 15 . . PUNCT cana-4038 62 1 theorem theorem VERB cana-4038 62 2 2.4 2.4 NUM cana-4038 62 3 for for ADP cana-4038 62 4 any any DET cana-4038 62 5 path path NOUN cana-4038 62 6 graph graph NOUN cana-4038 62 7 𝑃𝑛 𝑃𝑛 PROPN cana-4038 62 8 is be AUX cana-4038 62 9 sp sp ADP cana-4038 62 10 mean mean NOUN cana-4038 62 11 e e NOUN cana-4038 62 12 - - NOUN cana-4038 62 13 cg cg NOUN cana-4038 62 14 if if SCONJ cana-4038 62 15 n n NOUN cana-4038 62 16 is be AUX cana-4038 62 17 odd odd ADJ cana-4038 62 18 . . PUNCT cana-4038 63 1 proof proof NOUN cana-4038 63 2 : : PUNCT cana-4038 63 3 let let VERB cana-4038 63 4 𝑇(𝑃𝑛 𝑇(𝑃𝑛 NOUN cana-4038 63 5 ) ) PUNCT cana-4038 64 1 = = PRON cana-4038 64 2 { { PUNCT cana-4038 64 3 𝑒1 𝑒1 PROPN cana-4038 64 4 , , PUNCT cana-4038 64 5 𝑒2 𝑒2 PROPN cana-4038 64 6 , , PUNCT cana-4038 64 7 … … PUNCT cana-4038 64 8 , , PUNCT cana-4038 64 9 𝑒𝑛 𝑒𝑛 NOUN cana-4038 64 10 } } PUNCT cana-4038 64 11 and and CCONJ cana-4038 64 12 𝑅(𝑃𝑛 𝑅(𝑃𝑛 NUM cana-4038 64 13 ) ) PUNCT cana-4038 64 14 = = PRON cana-4038 64 15 { { PUNCT cana-4038 64 16 𝑒1𝑒2 𝑒1𝑒2 NOUN cana-4038 64 17 , , PUNCT cana-4038 64 18 𝑒2𝑒3 𝑒2𝑒3 NOUN cana-4038 64 19 , , PUNCT cana-4038 64 20 𝑒3𝑒4 𝑒3𝑒4 PROPN cana-4038 64 21 , , PUNCT cana-4038 64 22 … … PUNCT cana-4038 64 23 , , PUNCT cana-4038 64 24 π‘’π‘›βˆ’1𝑒𝑛 π‘’π‘›βˆ’1𝑒𝑛 NOUN cana-4038 64 25 } } PUNCT cana-4038 64 26 be be VERB cana-4038 64 27 the the DET cana-4038 64 28 vertices vertex NOUN cana-4038 64 29 and and CCONJ cana-4038 64 30 edges edge NOUN cana-4038 64 31 respectively respectively ADV cana-4038 64 32 . . PUNCT cana-4038 65 1 then then ADV cana-4038 65 2 | | ADV cana-4038 65 3 t(𝑃𝑛)| t(𝑃𝑛)| ADV cana-4038 65 4 = = PUNCT cana-4038 65 5 𝑛 𝑛 PROPN cana-4038 65 6 and and CCONJ cana-4038 65 7 | | ADV cana-4038 65 8 r(𝐾1,𝑛)| r(𝐾1,𝑛)| PROPN cana-4038 66 1 = = NOUN cana-4038 66 2 𝑛 𝑛 DET cana-4038 66 3 βˆ’ βˆ’ NOUN cana-4038 66 4 1 1 NUM cana-4038 66 5 define define VERB cana-4038 66 6 the the DET cana-4038 66 7 labeling labeling NOUN cana-4038 66 8 function function NOUN cana-4038 66 9 𝑓 𝑓 NOUN cana-4038 66 10 : : PUNCT cana-4038 66 11 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 66 12 ) ) PUNCT cana-4038 66 13 β†’ β†’ SYM cana-4038 66 14 { { PUNCT cana-4038 66 15 1,2 1,2 NUM cana-4038 66 16 } } PUNCT cana-4038 66 17 as as SCONJ cana-4038 66 18 follows follow VERB cana-4038 66 19 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 NOUN cana-4038 66 20 ) ) PUNCT cana-4038 66 21 = = NOUN cana-4038 66 22 { { PUNCT cana-4038 66 23 1 1 NUM cana-4038 66 24 𝑖𝑓 𝑖𝑓 ADP cana-4038 66 25 1 1 NUM cana-4038 66 26 ≀ ≀ NOUN cana-4038 66 27 𝑖 𝑖 SYM cana-4038 66 28 ≀ ≀ NUM cana-4038 66 29 𝑛 𝑛 PRON cana-4038 66 30 βˆ’ βˆ’ PROPN cana-4038 66 31 122 122 NUM cana-4038 66 32 𝑖𝑓 𝑖𝑓 NUM cana-4038 66 33 𝑛 𝑛 DET cana-4038 66 34 βˆ’ βˆ’ PROPN cana-4038 66 35 12 12 NUM cana-4038 66 36 ≀ ≀ PROPN cana-4038 66 37 𝑖 𝑖 SYM cana-4038 66 38 ≀ ≀ NUM cana-4038 66 39 𝑛 𝑛 PRON cana-4038 66 40 βˆ’ βˆ’ NOUN cana-4038 66 41 1 1 NUM cana-4038 66 42 find find VERB cana-4038 66 43 𝑆 𝑆 PROPN cana-4038 66 44 = = SYM cana-4038 66 45 βˆ‘ βˆ‘ PUNCT cana-4038 66 46 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 66 47 ) ) PUNCT cana-4038 66 48 βˆ– βˆ– X cana-4038 66 49 𝑒𝑑 𝑒𝑑 PROPN cana-4038 66 50 ∈ ∈ PROPN cana-4038 66 51 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 66 52 ) ) PUNCT cana-4038 66 53 and and CCONJ cana-4038 66 54 𝑃 𝑃 PROPN cana-4038 66 55 = = SYM cana-4038 66 56 ∏ ∏ PROPN cana-4038 66 57 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 66 58 ) ) PUNCT cana-4038 66 59 for for ADP cana-4038 66 60 each each DET cana-4038 66 61 vertex vertex NOUN cana-4038 66 62 in in ADP cana-4038 66 63 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 66 64 ) ) PUNCT cana-4038 66 65 defineπ‘“βˆ— defineπ‘“βˆ— VERB cana-4038 66 66 : : PUNCT cana-4038 66 67 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 66 68 ) ) PUNCT cana-4038 66 69 β†’ β†’ SYM cana-4038 66 70 { { PUNCT cana-4038 66 71 0,1 0,1 NOUN cana-4038 66 72 } } PUNCT cana-4038 66 73 defined define VERB cana-4038 66 74 by by ADP cana-4038 66 75 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 66 76 ) ) PUNCT cana-4038 66 77 = = NOUN cana-4038 66 78 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 66 79 βŒ‹ βŒ‹ NOUN cana-4038 66 80 ( ( PUNCT cana-4038 66 81 mod mod PROPN cana-4038 66 82 2 2 NUM cana-4038 66 83 ) ) PUNCT cana-4038 66 84 . . PUNCT cana-4038 67 1 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 67 2 ) ) PUNCT cana-4038 67 3 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 67 4 ) ) PUNCT cana-4038 67 5 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 67 6 ) ) PUNCT cana-4038 67 7 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 67 8 ) ) PUNCT cana-4038 67 9 𝑛 𝑛 PRON cana-4038 67 10 + + SYM cana-4038 67 11 12 12 NUM cana-4038 67 12 𝑛2 𝑛2 NOUN cana-4038 67 13 𝑛 𝑛 PROPN cana-4038 67 14 βˆ’ βˆ’ PROPN cana-4038 67 15 12 12 NUM cana-4038 67 16 𝑛 𝑛 PRON cana-4038 67 17 βˆ’ βˆ’ PROPN cana-4038 67 18 12 12 NUM cana-4038 67 19 the the DET cana-4038 67 20 above above ADJ cana-4038 67 21 labeling labeling NOUN cana-4038 67 22 satisfies satisfie NOUN cana-4038 67 23 , , PUNCT cana-4038 67 24 and and CCONJ cana-4038 67 25 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 67 26 ) ) PUNCT cana-4038 68 1 βˆ’ βˆ’ PROPN cana-4038 68 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 68 3 ≀ ≀ NOUN cana-4038 68 4 1 1 NUM cana-4038 68 5 and and CCONJ cana-4038 68 6 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 68 7 ) ) PUNCT cana-4038 69 1 βˆ’ βˆ’ NOUN cana-4038 69 2 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 69 3 ≀ ≀ NUM cana-4038 69 4 1 1 NUM cana-4038 69 5 . . PUNCT cana-4038 70 1 the the DET cana-4038 70 2 path path NOUN cana-4038 70 3 graph graph NOUN cana-4038 70 4 admits admit VERB cana-4038 70 5 sp sp ADP cana-4038 70 6 mean mean NOUN cana-4038 70 7 e e NOUN cana-4038 70 8 - - NOUN cana-4038 70 9 cl cl NOUN cana-4038 70 10 . . PUNCT cana-4038 71 1 hence hence ADV cana-4038 71 2 path path NOUN cana-4038 71 3 graph graph NOUN cana-4038 71 4 is be AUX cana-4038 71 5 a a DET cana-4038 71 6 sp sp NOUN cana-4038 71 7 mean mean NOUN cana-4038 71 8 e e NOUN cana-4038 71 9 - - NOUN cana-4038 71 10 cg cg NOUN cana-4038 71 11 . . PUNCT cana-4038 72 1 theorem theorem VERB cana-4038 72 2 2.5 2.5 NUM cana-4038 72 3 for for ADP cana-4038 72 4 any any DET cana-4038 72 5 cycle cycle NOUN cana-4038 72 6 graph graph NOUN cana-4038 72 7 𝐢𝑛 𝐢𝑛 PROPN cana-4038 72 8 is be AUX cana-4038 72 9 sp sp ADP cana-4038 72 10 mean mean NOUN cana-4038 72 11 e e NOUN cana-4038 72 12 - - NOUN cana-4038 72 13 cg cg NOUN cana-4038 72 14 if if SCONJ cana-4038 72 15 n n NOUN cana-4038 72 16 is be AUX cana-4038 72 17 odd odd ADJ cana-4038 72 18 . . PUNCT cana-4038 73 1 proof proof NOUN cana-4038 73 2 : : PUNCT cana-4038 73 3 let let VERB cana-4038 73 4 𝑇(𝐢𝑛 𝑇(𝐢𝑛 NUM cana-4038 73 5 ) ) PUNCT cana-4038 74 1 = = PRON cana-4038 74 2 { { PUNCT cana-4038 74 3 𝑒1 𝑒1 PROPN cana-4038 74 4 , , PUNCT cana-4038 74 5 𝑒2 𝑒2 PROPN cana-4038 74 6 , , PUNCT cana-4038 74 7 … … PUNCT cana-4038 74 8 , , PUNCT cana-4038 74 9 𝑒𝑛 𝑒𝑛 NOUN cana-4038 74 10 } } PUNCT cana-4038 74 11 and and CCONJ cana-4038 74 12 𝑅(𝐢𝑛 𝑅(𝐢𝑛 NUM cana-4038 74 13 ) ) PUNCT cana-4038 74 14 = = PRON cana-4038 74 15 { { PUNCT cana-4038 74 16 𝑒1𝑒2 𝑒1𝑒2 NOUN cana-4038 74 17 , , PUNCT cana-4038 74 18 𝑒2𝑒3 𝑒2𝑒3 NOUN cana-4038 74 19 , , PUNCT cana-4038 74 20 𝑒3𝑒4 𝑒3𝑒4 PROPN cana-4038 74 21 , , PUNCT cana-4038 74 22 … … PUNCT cana-4038 74 23 , , PUNCT cana-4038 74 24 π‘’π‘›βˆ’1𝑒𝑛 π‘’π‘›βˆ’1𝑒𝑛 NOUN cana-4038 74 25 , , PUNCT cana-4038 74 26 𝑒𝑛𝑒𝑛1 𝑒𝑛𝑒𝑛1 VERB cana-4038 74 27 } } PUNCT cana-4038 74 28 be be AUX cana-4038 74 29 the the DET cana-4038 74 30 vertices vertex NOUN cana-4038 74 31 and and CCONJ cana-4038 74 32 edges edge NOUN cana-4038 74 33 respectively respectively ADV cana-4038 74 34 . . PUNCT cana-4038 75 1 then then ADV cana-4038 75 2 | | ADV cana-4038 75 3 t(𝑃𝑛)| t(𝑃𝑛)| ADV cana-4038 75 4 = = PUNCT cana-4038 75 5 𝑛 𝑛 PROPN cana-4038 75 6 and and CCONJ cana-4038 75 7 | | ADV cana-4038 75 8 r(𝐾1,𝑛)| r(𝐾1,𝑛)| PROPN cana-4038 76 1 = = NOUN cana-4038 76 2 𝑛 𝑛 DET cana-4038 76 3 βˆ’ βˆ’ NUM cana-4038 76 4 1 1 NUM cana-4038 76 5 communications communication NOUN cana-4038 76 6 on on ADP cana-4038 76 7 applied apply VERB cana-4038 76 8 nonlinear nonlinear ADJ cana-4038 76 9 analysis analysis NOUN cana-4038 76 10 issn issn NOUN cana-4038 76 11 : : PUNCT cana-4038 76 12 1074 1074 NUM cana-4038 76 13 - - PUNCT cana-4038 76 14 133x 133x NUM cana-4038 76 15 vol vol NOUN cana-4038 76 16 32 32 NUM cana-4038 76 17 no no NOUN cana-4038 76 18 . . PUNCT cana-4038 77 1 9s 9s NUM cana-4038 77 2 ( ( PUNCT cana-4038 77 3 2025 2025 NUM cana-4038 77 4 ) ) PUNCT cana-4038 77 5 904 904 NUM cana-4038 77 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 77 7 define define VERB cana-4038 77 8 the the DET cana-4038 77 9 labeling labeling NOUN cana-4038 77 10 function function NOUN cana-4038 77 11 𝑓 𝑓 NOUN cana-4038 77 12 : : PUNCT cana-4038 77 13 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 77 14 ) ) PUNCT cana-4038 77 15 β†’ β†’ SYM cana-4038 77 16 { { PUNCT cana-4038 77 17 1,2 1,2 NUM cana-4038 77 18 } } PUNCT cana-4038 77 19 as as SCONJ cana-4038 77 20 follows follow VERB cana-4038 77 21 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 NOUN cana-4038 77 22 ) ) PUNCT cana-4038 77 23 = = NOUN cana-4038 77 24 { { PUNCT cana-4038 77 25 1 1 NUM cana-4038 77 26 𝑖𝑓 𝑖𝑓 ADP cana-4038 77 27 1 1 NUM cana-4038 77 28 ≀ ≀ NOUN cana-4038 77 29 𝑖 𝑖 SYM cana-4038 77 30 ≀ ≀ NOUN cana-4038 78 1 𝑛 𝑛 ADP cana-4038 78 2 + + SYM cana-4038 78 3 122 122 NUM cana-4038 78 4 𝑖𝑓 𝑖𝑓 NOUN cana-4038 78 5 𝑛 𝑛 PRON cana-4038 78 6 + + NOUN cana-4038 78 7 12 12 NUM cana-4038 78 8 ≀ ≀ NUM cana-4038 78 9 𝑖 𝑖 SYM cana-4038 78 10 ≀ ≀ NUM cana-4038 78 11 𝑛 𝑛 PRON cana-4038 78 12 βˆ’ βˆ’ NOUN cana-4038 78 13 1 1 NUM cana-4038 78 14 find find VERB cana-4038 78 15 𝑆 𝑆 PROPN cana-4038 78 16 = = SYM cana-4038 78 17 βˆ‘ βˆ‘ PUNCT cana-4038 78 18 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 78 19 ) ) PUNCT cana-4038 78 20 βˆ– βˆ– X cana-4038 78 21 𝑒𝑑 𝑒𝑑 PROPN cana-4038 78 22 ∈ ∈ PROPN cana-4038 78 23 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 78 24 ) ) PUNCT cana-4038 78 25 and and CCONJ cana-4038 78 26 𝑃 𝑃 PROPN cana-4038 78 27 = = SYM cana-4038 78 28 ∏ ∏ PROPN cana-4038 78 29 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 78 30 ) ) PUNCT cana-4038 78 31 for for ADP cana-4038 78 32 each each DET cana-4038 78 33 vertex vertex NOUN cana-4038 78 34 in in ADP cana-4038 78 35 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 78 36 ) ) PUNCT cana-4038 78 37 defineπ‘“βˆ— defineπ‘“βˆ— VERB cana-4038 78 38 : : PUNCT cana-4038 78 39 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 78 40 ) ) PUNCT cana-4038 78 41 β†’ β†’ SYM cana-4038 78 42 { { PUNCT cana-4038 78 43 0,1 0,1 NOUN cana-4038 78 44 } } PUNCT cana-4038 78 45 defined define VERB cana-4038 78 46 byπ‘“βˆ—(𝑒 byπ‘“βˆ—(𝑒 NOUN cana-4038 78 47 ) ) PUNCT cana-4038 78 48 = = NOUN cana-4038 78 49 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 78 50 βŒ‹ βŒ‹ NOUN cana-4038 78 51 ( ( PUNCT cana-4038 78 52 mod mod PROPN cana-4038 78 53 2 2 NUM cana-4038 78 54 ) ) PUNCT cana-4038 78 55 . . PUNCT cana-4038 79 1 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 79 2 ) ) PUNCT cana-4038 79 3 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 79 4 ) ) PUNCT cana-4038 79 5 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 79 6 ) ) PUNCT cana-4038 79 7 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 79 8 ) ) PUNCT cana-4038 79 9 𝑛 𝑛 PRON cana-4038 80 1 + + NUM cana-4038 80 2 12 12 NUM cana-4038 80 3 𝑛 𝑛 DET cana-4038 80 4 βˆ’ βˆ’ PROPN cana-4038 80 5 12 12 NUM cana-4038 80 6 𝑛 𝑛 PROPN cana-4038 80 7 + + NUM cana-4038 80 8 12 12 NUM cana-4038 80 9 𝑛 𝑛 PRON cana-4038 80 10 βˆ’ βˆ’ PROPN cana-4038 80 11 12 12 NUM cana-4038 81 1 the the DET cana-4038 81 2 above above ADP cana-4038 81 3 labeling labeling NOUN cana-4038 81 4 satisfies,|𝑑𝑓(0 satisfies,|𝑑𝑓(0 PROPN cana-4038 81 5 ) ) PUNCT cana-4038 81 6 βˆ’ βˆ’ PROPN cana-4038 82 1 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 82 2 ≀ ≀ NUM cana-4038 82 3 1 1 NUM cana-4038 82 4 and and CCONJ cana-4038 82 5 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 82 6 ) ) PUNCT cana-4038 82 7 βˆ’ βˆ’ NOUN cana-4038 82 8 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 82 9 ≀ ≀ NUM cana-4038 82 10 1 1 NUM cana-4038 82 11 . . PUNCT cana-4038 83 1 the the DET cana-4038 83 2 cycle cycle NOUN cana-4038 83 3 graph graph NOUN cana-4038 83 4 admits admit VERB cana-4038 83 5 sp sp ADP cana-4038 83 6 mean mean NOUN cana-4038 83 7 e e NOUN cana-4038 83 8 - - NOUN cana-4038 83 9 cl cl NOUN cana-4038 83 10 . . PUNCT cana-4038 84 1 hence hence ADV cana-4038 84 2 cycle cycle NOUN cana-4038 84 3 graph graph NOUN cana-4038 84 4 is be AUX cana-4038 84 5 a a DET cana-4038 84 6 sp sp NOUN cana-4038 84 7 mean mean NOUN cana-4038 84 8 e e ADJ cana-4038 84 9 - - ADJ cana-4038 84 10 cordial cordial ADJ cana-4038 84 11 graph graph NOUN cana-4038 84 12 . . PUNCT cana-4038 85 1 note note VERB cana-4038 85 2 2.6 2.6 NUM cana-4038 85 3 𝐾3 𝐾3 NOUN cana-4038 85 4 , , PUNCT cana-4038 85 5 𝐾4 𝐾4 NOUN cana-4038 85 6 are be AUX cana-4038 85 7 the the DET cana-4038 85 8 only only ADJ cana-4038 85 9 complete complete ADJ cana-4038 85 10 graph graph NOUN cana-4038 85 11 in in ADP cana-4038 85 12 sp sp ADP cana-4038 85 13 mean mean NOUN cana-4038 85 14 e e ADJ cana-4038 85 15 - - ADJ cana-4038 85 16 cordial cordial ADJ cana-4038 85 17 graph graph NOUN cana-4038 85 18 . . PUNCT cana-4038 86 1 theorem theorem VERB cana-4038 86 2 2.7 2.7 NUM cana-4038 86 3 for for ADP cana-4038 86 4 any any DET cana-4038 86 5 wheel wheel NOUN cana-4038 86 6 graph graph NOUN cana-4038 86 7 π‘Šπ‘›is π‘Šπ‘›is PROPN cana-4038 86 8 a a DET cana-4038 86 9 sp sp NOUN cana-4038 86 10 mean mean NOUN cana-4038 86 11 e e ADJ cana-4038 86 12 - - ADJ cana-4038 86 13 cordial cordial ADJ cana-4038 86 14 graph graph NOUN cana-4038 86 15 if if SCONJ cana-4038 86 16 n n PRON cana-4038 86 17 is be AUX cana-4038 86 18 even even ADV cana-4038 86 19 . . PUNCT cana-4038 87 1 proof proof NOUN cana-4038 87 2 : : PUNCT cana-4038 87 3 let let VERB cana-4038 87 4 𝑇(π‘Šπ‘› 𝑇(π‘Šπ‘› NOUN cana-4038 87 5 ) ) PUNCT cana-4038 87 6 = = PRON cana-4038 87 7 { { PUNCT cana-4038 87 8 𝑒 𝑒 PROPN cana-4038 87 9 , , PUNCT cana-4038 87 10 𝑒𝑖 𝑒𝑖 DET cana-4038 87 11 𝑖 𝑖 NOUN cana-4038 87 12 = = SYM cana-4038 87 13 1,2,3 1,2,3 NUM cana-4038 87 14 … … PUNCT cana-4038 87 15 𝑛 𝑛 NOUN cana-4038 87 16 } } PUNCT cana-4038 87 17 and and CCONJ cana-4038 87 18 𝑅(π‘Šπ‘› 𝑅(π‘Šπ‘› NOUN cana-4038 87 19 ) ) PUNCT cana-4038 87 20 = = PRON cana-4038 87 21 { { PUNCT cana-4038 87 22 𝑒𝑒𝑖 𝑒𝑒𝑖 PROPN cana-4038 87 23 , , PUNCT cana-4038 87 24 𝑒1𝑒2 𝑒1𝑒2 NOUN cana-4038 87 25 , , PUNCT cana-4038 87 26 𝑒2𝑒3 𝑒2𝑒3 NOUN cana-4038 87 27 , , PUNCT cana-4038 87 28 𝑒3𝑒4 𝑒3𝑒4 PROPN cana-4038 87 29 , , PUNCT cana-4038 87 30 … … PUNCT cana-4038 87 31 , , PUNCT cana-4038 87 32 𝑒𝑛𝑒1 𝑒𝑛𝑒1 PROPN cana-4038 87 33 } } PUNCT cana-4038 87 34 be be VERB cana-4038 87 35 the the DET cana-4038 87 36 vertices vertex NOUN cana-4038 87 37 and and CCONJ cana-4038 87 38 edges edge NOUN cana-4038 87 39 respectively respectively ADV cana-4038 87 40 , , PUNCT cana-4038 87 41 u u NOUN cana-4038 87 42 is be AUX cana-4038 87 43 apex apex ADJ cana-4038 87 44 vertex vertex NOUN cana-4038 87 45 and and CCONJ cana-4038 87 46 𝑒1 𝑒1 NOUN cana-4038 87 47 , , PUNCT cana-4038 87 48 𝑒2 𝑒2 PROPN cana-4038 87 49 , , PUNCT cana-4038 87 50 𝑒3 𝑒3 PROPN cana-4038 87 51 … … PUNCT cana-4038 87 52 , , PUNCT cana-4038 87 53 𝑒𝑛 𝑒𝑛 NOUN cana-4038 87 54 be be AUX cana-4038 87 55 the the DET cana-4038 87 56 vertices vertex NOUN cana-4038 87 57 of of ADP cana-4038 87 58 cycle cycle NOUN cana-4038 87 59 𝐢𝑛. 𝐢𝑛. NOUN cana-4038 87 60 then then ADV cana-4038 87 61 | | ADV cana-4038 87 62 t(π‘Šπ‘›)| t(π‘Šπ‘›)| VERB cana-4038 87 63 = = SYM cana-4038 87 64 𝑛 𝑛 NOUN cana-4038 88 1 + + SYM cana-4038 88 2 1 1 NUM cana-4038 88 3 and and CCONJ cana-4038 88 4 | | ADV cana-4038 88 5 r(π‘Šπ‘›)| r(π‘Šπ‘›)| ADV cana-4038 88 6 = = SYM cana-4038 88 7 𝑛 𝑛 PART cana-4038 88 8 define define VERB cana-4038 88 9 the the DET cana-4038 88 10 labeling labeling NOUN cana-4038 88 11 𝑓 𝑓 NOUN cana-4038 88 12 : : PUNCT cana-4038 88 13 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 88 14 ) ) PUNCT cana-4038 88 15 β†’ β†’ SYM cana-4038 88 16 { { PUNCT cana-4038 88 17 1,2 1,2 NUM cana-4038 88 18 } } PUNCT cana-4038 88 19 as as ADP cana-4038 88 20 below below ADP cana-4038 88 21 𝑓(𝑒𝑒𝑖 𝑓(𝑒𝑒𝑖 PROPN cana-4038 88 22 ) ) PUNCT cana-4038 88 23 = = SYM cana-4038 88 24 2 2 NUM cana-4038 88 25 , , PUNCT cana-4038 88 26 𝑖𝑓 𝑖𝑓 ADP cana-4038 88 27 𝑖 𝑖 PRON cana-4038 88 28 𝑖𝑠 𝑖𝑠 PROPN cana-4038 88 29 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-4038 88 30 𝑓(𝑒𝑒𝑖 𝑓(𝑒𝑒𝑖 PROPN cana-4038 88 31 ) ) PUNCT cana-4038 88 32 = = SYM cana-4038 88 33 1 1 NUM cana-4038 88 34 , , PUNCT cana-4038 88 35 𝑖𝑓 𝑖𝑓 ADP cana-4038 88 36 𝑖 𝑖 PUNCT cana-4038 88 37 𝑖𝑠 𝑖𝑠 NOUN cana-4038 88 38 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 88 39 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 X cana-4038 88 40 ) ) PUNCT cana-4038 88 41 = = SYM cana-4038 88 42 2 2 NUM cana-4038 88 43 , , PUNCT cana-4038 88 44 𝑖𝑓 𝑖𝑓 ADP cana-4038 88 45 𝑖 𝑖 PRON cana-4038 88 46 𝑖𝑠 𝑖𝑠 ADV cana-4038 88 47 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-4038 88 48 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 PROPN cana-4038 88 49 ) ) PUNCT cana-4038 88 50 = = SYM cana-4038 88 51 1 1 NUM cana-4038 88 52 , , PUNCT cana-4038 88 53 𝑖𝑓 𝑖𝑓 ADP cana-4038 88 54 𝑖 𝑖 PUNCT cana-4038 88 55 𝑖𝑠 𝑖𝑠 NOUN cana-4038 88 56 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 88 57 find find VERB cana-4038 88 58 𝑆 𝑆 PROPN cana-4038 88 59 = = SYM cana-4038 88 60 βˆ‘ βˆ‘ PUNCT cana-4038 88 61 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 88 62 ) ) PUNCT cana-4038 88 63 βˆ– βˆ– X cana-4038 88 64 𝑒𝑑 𝑒𝑑 PROPN cana-4038 88 65 ∈ ∈ PROPN cana-4038 88 66 𝐸(𝐺 𝐸(𝐺 PROPN cana-4038 88 67 ) ) PUNCT cana-4038 88 68 and and CCONJ cana-4038 88 69 𝑃 𝑃 PROPN cana-4038 88 70 = = SYM cana-4038 88 71 ∏ ∏ PROPN cana-4038 88 72 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 88 73 ) ) PUNCT cana-4038 88 74 for for ADP cana-4038 88 75 each each DET cana-4038 88 76 vertex vertex NOUN cana-4038 88 77 in in ADP cana-4038 88 78 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 88 79 ) ) PUNCT cana-4038 88 80 define define VERB cana-4038 88 81 π‘“βˆ— π‘“βˆ— NOUN cana-4038 88 82 : : PUNCT cana-4038 88 83 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 88 84 ) ) PUNCT cana-4038 88 85 β†’ β†’ SYM cana-4038 88 86 { { PUNCT cana-4038 88 87 0,1 0,1 NOUN cana-4038 88 88 } } PUNCT cana-4038 88 89 defined define VERB cana-4038 88 90 by by ADP cana-4038 88 91 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 88 92 ) ) PUNCT cana-4038 88 93 = = NOUN cana-4038 88 94 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 88 95 βŒ‹ βŒ‹ NOUN cana-4038 88 96 ( ( PUNCT cana-4038 88 97 mod mod PROPN cana-4038 88 98 2 2 NUM cana-4038 88 99 ) ) PUNCT cana-4038 88 100 . . PUNCT cana-4038 89 1 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 89 2 ) ) PUNCT cana-4038 89 3 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 89 4 ) ) PUNCT cana-4038 89 5 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 89 6 ) ) PUNCT cana-4038 89 7 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 89 8 ) ) PUNCT cana-4038 89 9 𝑛2 𝑛2 NOUN cana-4038 89 10 𝑛2 𝑛2 NOUN cana-4038 89 11 𝑛 𝑛 ADP cana-4038 89 12 𝑛 𝑛 VERB cana-4038 89 13 the the DET cana-4038 89 14 above above ADJ cana-4038 89 15 labeling labeling NOUN cana-4038 89 16 satisfies satisfie NOUN cana-4038 89 17 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 89 18 ) ) PUNCT cana-4038 89 19 βˆ’ βˆ’ NOUN cana-4038 89 20 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 89 21 ≀ ≀ NUM cana-4038 89 22 1 1 NUM cana-4038 89 23 and and CCONJ cana-4038 89 24 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 89 25 ) ) PUNCT cana-4038 90 1 βˆ’ βˆ’ PROPN cana-4038 90 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 90 3 ≀ ≀ ADJ cana-4038 90 4 1 1 NUM cana-4038 90 5 . . PUNCT cana-4038 91 1 communications communication NOUN cana-4038 91 2 on on ADP cana-4038 91 3 applied apply VERB cana-4038 91 4 nonlinear nonlinear ADJ cana-4038 91 5 analysis analysis NOUN cana-4038 91 6 issn issn NOUN cana-4038 91 7 : : PUNCT cana-4038 91 8 1074 1074 NUM cana-4038 91 9 - - PUNCT cana-4038 91 10 133x 133x NUM cana-4038 91 11 vol vol NOUN cana-4038 91 12 32 32 NUM cana-4038 91 13 no no NOUN cana-4038 91 14 . . PUNCT cana-4038 92 1 9s 9s NUM cana-4038 92 2 ( ( PUNCT cana-4038 92 3 2025 2025 NUM cana-4038 92 4 ) ) PUNCT cana-4038 92 5 905 905 NUM cana-4038 92 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 92 7 hence hence ADV cana-4038 92 8 wheel wheel NOUN cana-4038 92 9 graph graph NOUN cana-4038 93 1 π‘Šπ‘› π‘Šπ‘› PROPN cana-4038 93 2 admits admit VERB cana-4038 93 3 sp sp ADP cana-4038 93 4 mean mean NOUN cana-4038 93 5 e e ADJ cana-4038 93 6 - - ADJ cana-4038 93 7 cordial cordial ADJ cana-4038 93 8 graph graph NOUN cana-4038 93 9 labeling labeling NOUN cana-4038 93 10 . . PUNCT cana-4038 94 1 theorem theorem VERB cana-4038 94 2 2.8 2.8 NUM cana-4038 94 3 for for ADP cana-4038 94 4 any any DET cana-4038 94 5 bistar bistar NOUN cana-4038 94 6 𝐡𝑛,𝑛 𝐡𝑛,𝑛 PROPN cana-4038 94 7 is be AUX cana-4038 94 8 sp sp ADP cana-4038 94 9 mean mean NOUN cana-4038 94 10 e e NOUN cana-4038 94 11 - - NOUN cana-4038 94 12 cg cg NOUN cana-4038 94 13 if if SCONJ cana-4038 94 14 n n NOUN cana-4038 94 15 is be AUX cana-4038 94 16 even even ADV cana-4038 94 17 . . PUNCT cana-4038 95 1 proof proof NOUN cana-4038 95 2 : : PUNCT cana-4038 95 3 let let VERB cana-4038 95 4 𝑇(𝐡𝑛,𝑛 𝑇(𝐡𝑛,𝑛 NOUN cana-4038 95 5 ) ) PUNCT cana-4038 95 6 = = PRON cana-4038 95 7 { { PUNCT cana-4038 95 8 𝑒 𝑒 X cana-4038 95 9 , , PUNCT cana-4038 95 10 𝑑 𝑑 NOUN cana-4038 95 11 , , PUNCT cana-4038 95 12 𝑒1 𝑒1 NOUN cana-4038 95 13 , , PUNCT cana-4038 95 14 𝑒2 𝑒2 PROPN cana-4038 95 15 , , PUNCT cana-4038 95 16 𝑒3 𝑒3 PROPN cana-4038 95 17 … … PUNCT cana-4038 95 18 . . PUNCT cana-4038 95 19 . . PUNCT cana-4038 96 1 𝑒𝑛 𝑒𝑛 X cana-4038 96 2 , , PUNCT cana-4038 96 3 𝑑1 𝑑1 PROPN cana-4038 96 4 , , PUNCT cana-4038 96 5 𝑑2 𝑑2 PROPN cana-4038 96 6 , , PUNCT cana-4038 96 7 𝑑3 𝑑3 PROPN cana-4038 96 8 … … PUNCT cana-4038 96 9 . . PUNCT cana-4038 97 1 , , PUNCT cana-4038 97 2 𝑑𝑛 𝑑𝑛 NOUN cana-4038 97 3 } } PUNCT cana-4038 97 4 be be VERB cana-4038 97 5 the the DET cana-4038 97 6 vertices vertex NOUN cana-4038 97 7 and and CCONJ cana-4038 97 8 𝑅(𝐡𝑛,𝑛 𝑅(𝐡𝑛,𝑛 PUNCT cana-4038 97 9 ) ) PUNCT cana-4038 98 1 = = PRON cana-4038 98 2 { { PUNCT cana-4038 98 3 𝑒𝑑 𝑒𝑑 NOUN cana-4038 98 4 , , PUNCT cana-4038 98 5 𝑒𝑒1 𝑒𝑒1 PROPN cana-4038 98 6 , , PUNCT cana-4038 98 7 𝑒𝑒2 𝑒𝑒2 NOUN cana-4038 98 8 , , PUNCT cana-4038 98 9 𝑒𝑒3 𝑒𝑒3 NOUN cana-4038 98 10 … … PUNCT cana-4038 98 11 . . PUNCT cana-4038 98 12 . . PUNCT cana-4038 99 1 𝑒𝑒𝑛 𝑒𝑒𝑛 ADV cana-4038 99 2 , , PUNCT cana-4038 99 3 𝑑𝑑1 𝑑𝑑1 PROPN cana-4038 99 4 , , PUNCT cana-4038 99 5 𝑑𝑑2 𝑑𝑑2 PROPN cana-4038 99 6 , , PUNCT cana-4038 99 7 𝑑𝑑3 𝑑𝑑3 PROPN cana-4038 99 8 … … PUNCT cana-4038 99 9 . . PUNCT cana-4038 100 1 , , PUNCT cana-4038 100 2 𝑑𝑑𝑛 𝑑𝑑𝑛 AUX cana-4038 100 3 } } PUNCT cana-4038 100 4 be be AUX cana-4038 100 5 the the DET cana-4038 100 6 edges edge NOUN cana-4038 100 7 . . PUNCT cana-4038 101 1 then then ADV cana-4038 101 2 | | ADV cana-4038 101 3 t(𝐡𝑛,𝑛)| t(𝐡𝑛,𝑛)| ADJ cana-4038 101 4 = = SYM cana-4038 101 5 2𝑛 2𝑛 NOUN cana-4038 102 1 + + CCONJ cana-4038 102 2 2 2 NUM cana-4038 102 3 and and CCONJ cana-4038 102 4 | | ADV cana-4038 102 5 r(π‘Šπ‘›)| r(π‘Šπ‘›)| ADV cana-4038 102 6 = = SYM cana-4038 102 7 2𝑛 2𝑛 PROPN cana-4038 103 1 + + CCONJ cana-4038 103 2 1 1 NUM cana-4038 103 3 to to PART cana-4038 103 4 define define VERB cana-4038 103 5 the the DET cana-4038 103 6 labeling labeling NOUN cana-4038 103 7 function function NOUN cana-4038 103 8 𝑓 𝑓 NOUN cana-4038 103 9 : : PUNCT cana-4038 103 10 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 103 11 ) ) PUNCT cana-4038 103 12 β†’ β†’ SYM cana-4038 103 13 { { PUNCT cana-4038 103 14 1,2 1,2 NUM cana-4038 103 15 } } PUNCT cana-4038 103 16 as as SCONJ cana-4038 103 17 follows follow VERB cana-4038 103 18 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 103 19 ) ) PUNCT cana-4038 103 20 = = SYM cana-4038 103 21 1 1 NUM cana-4038 103 22 𝑓(𝑒𝑑𝑖 𝑓(𝑒𝑑𝑖 PROPN cana-4038 103 23 ) ) PUNCT cana-4038 103 24 = = SYM cana-4038 103 25 1 1 NUM cana-4038 103 26 𝑓(𝑑𝑑𝑖 𝑓(𝑑𝑑𝑖 NOUN cana-4038 103 27 ) ) PUNCT cana-4038 103 28 = = SYM cana-4038 103 29 2 2 NUM cana-4038 103 30 find find VERB cana-4038 103 31 𝑆 𝑆 PROPN cana-4038 103 32 = = SYM cana-4038 103 33 βˆ‘ βˆ‘ PUNCT cana-4038 103 34 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 103 35 ) ) PUNCT cana-4038 103 36 βˆ– βˆ– X cana-4038 103 37 𝑒𝑑 𝑒𝑑 PROPN cana-4038 103 38 ∈ ∈ PROPN cana-4038 103 39 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 103 40 ) ) PUNCT cana-4038 103 41 and and CCONJ cana-4038 103 42 𝑃 𝑃 PROPN cana-4038 103 43 = = SYM cana-4038 103 44 ∏ ∏ PROPN cana-4038 103 45 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 103 46 ) ) PUNCT cana-4038 103 47 for for ADP cana-4038 103 48 each each DET cana-4038 103 49 vertex vertex NOUN cana-4038 103 50 in in ADP cana-4038 103 51 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 103 52 ) ) PUNCT cana-4038 103 53 defineπ‘“βˆ— defineπ‘“βˆ— VERB cana-4038 103 54 : : PUNCT cana-4038 103 55 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 103 56 ) ) PUNCT cana-4038 103 57 β†’ β†’ SYM cana-4038 103 58 { { PUNCT cana-4038 103 59 0,1 0,1 NOUN cana-4038 103 60 } } PUNCT cana-4038 103 61 defined define VERB cana-4038 103 62 by by ADP cana-4038 103 63 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 103 64 ) ) PUNCT cana-4038 103 65 = = NOUN cana-4038 103 66 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 103 67 βŒ‹ βŒ‹ NOUN cana-4038 103 68 ( ( PUNCT cana-4038 103 69 mod mod PROPN cana-4038 103 70 2 2 NUM cana-4038 103 71 ) ) PUNCT cana-4038 103 72 . . PUNCT cana-4038 104 1 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 104 2 ) ) PUNCT cana-4038 104 3 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 104 4 ) ) PUNCT cana-4038 104 5 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 104 6 ) ) PUNCT cana-4038 104 7 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 104 8 ) ) PUNCT cana-4038 104 9 𝑛 𝑛 PRON cana-4038 105 1 + + NOUN cana-4038 105 2 1 1 NUM cana-4038 105 3 𝑛 𝑛 VERB cana-4038 105 4 + + NUM cana-4038 105 5 1 1 NUM cana-4038 105 6 𝑛 𝑛 DET cana-4038 105 7 𝑛 𝑛 PROPN cana-4038 105 8 + + CCONJ cana-4038 105 9 1 1 NUM cana-4038 105 10 the the DET cana-4038 105 11 above above ADJ cana-4038 105 12 labeling labeling NOUN cana-4038 105 13 satisfies satisfie NOUN cana-4038 105 14 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 105 15 ) ) PUNCT cana-4038 106 1 βˆ’ βˆ’ PROPN cana-4038 106 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 106 3 ≀ ≀ NOUN cana-4038 106 4 1 1 NUM cana-4038 106 5 and and CCONJ cana-4038 106 6 |π‘Ÿπ‘“(𝑖 |π‘Ÿπ‘“(𝑖 NUM cana-4038 106 7 ) ) PUNCT cana-4038 107 1 βˆ’ βˆ’ NOUN cana-4038 107 2 π‘Ÿπ‘“(𝑗)| π‘Ÿπ‘“(𝑗)| PUNCT cana-4038 107 3 ≀ ≀ ADV cana-4038 107 4 1 1 NUM cana-4038 107 5 . . PUNCT cana-4038 108 1 the the DET cana-4038 108 2 bistar bistar PROPN cana-4038 108 3 𝐡𝑛,𝑛admits 𝐡𝑛,𝑛admit NOUN cana-4038 108 4 sp sp ADP cana-4038 108 5 mean mean NOUN cana-4038 108 6 e e NOUN cana-4038 108 7 - - NOUN cana-4038 108 8 cl cl NOUN cana-4038 108 9 . . PUNCT cana-4038 109 1 therefore therefore ADV cana-4038 109 2 , , PUNCT cana-4038 109 3 bistar bistar PROPN cana-4038 109 4 𝐡𝑛,𝑛is 𝐡𝑛,𝑛is VERB cana-4038 109 5 a a DET cana-4038 109 6 sp sp NOUN cana-4038 109 7 mean mean NOUN cana-4038 109 8 e e NOUN cana-4038 109 9 - - NOUN cana-4038 109 10 cg cg NOUN cana-4038 109 11 . . PUNCT cana-4038 110 1 theorem theorem VERB cana-4038 110 2 2.9 2.9 NUM cana-4038 110 3 combo combo NOUN cana-4038 110 4 graph graph NOUN cana-4038 110 5 𝑃𝑛 𝑃𝑛 PROPN cana-4038 110 6 ⊚ ⊚ NOUN cana-4038 110 7 𝐾2 𝐾2 NOUN cana-4038 110 8 is be AUX cana-4038 110 9 a a DET cana-4038 110 10 sp sp NOUN cana-4038 110 11 mean mean NOUN cana-4038 110 12 e e NOUN cana-4038 110 13 - - NOUN cana-4038 110 14 cg cg NOUN cana-4038 110 15 . . PUNCT cana-4038 111 1 proof proof NOUN cana-4038 111 2 : : PUNCT cana-4038 111 3 let let VERB cana-4038 111 4 𝑇(𝑃𝑛 𝑇(𝑃𝑛 PROPN cana-4038 111 5 ⊚ ⊚ VERB cana-4038 111 6 𝐾2 𝐾2 NOUN cana-4038 111 7 ) ) PUNCT cana-4038 111 8 = = PRON cana-4038 111 9 { { PUNCT cana-4038 111 10 𝑒𝑖 𝑒𝑖 INTJ cana-4038 111 11 , , PUNCT cana-4038 111 12 𝑒𝑖′1 𝑒𝑖′1 NOUN cana-4038 111 13 ≀ ≀ NOUN cana-4038 111 14 𝑖 𝑖 SYM cana-4038 111 15 ≀ ≀ NUM cana-4038 111 16 𝑛 𝑛 PRON cana-4038 111 17 } } PUNCT cana-4038 111 18 be be VERB cana-4038 111 19 the the DET cana-4038 111 20 vertices vertex NOUN cana-4038 111 21 and and CCONJ cana-4038 111 22 𝑅(𝑃𝑛 𝑅(𝑃𝑛 ADJ cana-4038 111 23 ⊚ ⊚ NOUN cana-4038 111 24 𝐾2 𝐾2 NOUN cana-4038 111 25 ) ) PUNCT cana-4038 111 26 = = PRON cana-4038 111 27 { { PUNCT cana-4038 111 28 𝑒𝑖𝑒𝑖+1 𝑒𝑖𝑒𝑖+1 NOUN cana-4038 111 29 , , PUNCT cana-4038 111 30 𝑒𝑒𝑖′ 𝑒𝑒𝑖′ NOUN cana-4038 111 31 , , PUNCT cana-4038 111 32 1 1 NUM cana-4038 111 33 ≀ ≀ NUM cana-4038 111 34 𝑖 𝑖 SYM cana-4038 111 35 ≀ ≀ NUM cana-4038 111 36 𝑛 𝑛 PRON cana-4038 111 37 } } PUNCT cana-4038 111 38 be be VERB cana-4038 111 39 the the DET cana-4038 111 40 edges edge NOUN cana-4038 111 41 . . PUNCT cana-4038 112 1 then then ADV cana-4038 112 2 | | ADV cana-4038 112 3 t(𝑃𝑛 t(𝑃𝑛 NOUN cana-4038 112 4 ⊚ ⊚ INTJ cana-4038 112 5 𝐾2)| 𝐾2)| PROPN cana-4038 112 6 = = SYM cana-4038 112 7 2𝑛 2𝑛 PROPN cana-4038 112 8 and and CCONJ cana-4038 112 9 | | ADV cana-4038 112 10 r(𝑃𝑛 r(𝑃𝑛 PROPN cana-4038 112 11 ⊚ ⊚ NOUN cana-4038 112 12 𝐾2)| 𝐾2)| PUNCT cana-4038 112 13 = = SYM cana-4038 112 14 2𝑛 2𝑛 PROPN cana-4038 112 15 βˆ’ βˆ’ NOUN cana-4038 112 16 1 1 NUM cana-4038 112 17 define define VERB cana-4038 112 18 the the DET cana-4038 112 19 function function NOUN cana-4038 112 20 𝑓 𝑓 NOUN cana-4038 112 21 : : PUNCT cana-4038 112 22 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 112 23 ) ) PUNCT cana-4038 112 24 β†’ β†’ SYM cana-4038 112 25 { { PUNCT cana-4038 112 26 1,2 1,2 NUM cana-4038 112 27 } } PUNCT cana-4038 112 28 as as ADP cana-4038 112 29 below below ADP cana-4038 112 30 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 NOUN cana-4038 112 31 ) ) PUNCT cana-4038 112 32 = = NOUN cana-4038 112 33 { { PUNCT cana-4038 112 34 2 2 NUM cana-4038 112 35 𝑖𝑓 𝑖𝑓 SYM cana-4038 112 36 𝑖 𝑖 PUNCT cana-4038 112 37 𝑖𝑠 𝑖𝑠 PROPN cana-4038 112 38 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 112 39 𝑖𝑓 𝑖𝑓 ADP cana-4038 112 40 𝑖 𝑖 PUNCT cana-4038 112 41 𝑖𝑠 𝑖𝑠 NOUN cana-4038 112 42 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 112 43 𝑓(𝑒𝑖𝑒𝑖′ 𝑓(𝑒𝑖𝑒𝑖′ NOUN cana-4038 112 44 ) ) PUNCT cana-4038 113 1 = = PUNCT cana-4038 113 2 { { PUNCT cana-4038 113 3 2 2 NUM cana-4038 113 4 𝑖𝑓 𝑖𝑓 SYM cana-4038 113 5 𝑖 𝑖 PUNCT cana-4038 113 6 𝑖𝑠 𝑖𝑠 PROPN cana-4038 113 7 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 113 8 𝑖𝑓 𝑖𝑓 ADP cana-4038 113 9 𝑖 𝑖 PUNCT cana-4038 113 10 𝑖𝑠 𝑖𝑠 NOUN cana-4038 113 11 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 113 12 find find VERB cana-4038 113 13 𝑆 𝑆 PROPN cana-4038 113 14 = = SYM cana-4038 113 15 βˆ‘ βˆ‘ PUNCT cana-4038 113 16 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 113 17 ) ) PUNCT cana-4038 113 18 βˆ– βˆ– X cana-4038 113 19 𝑒𝑑 𝑒𝑑 PROPN cana-4038 113 20 ∈ ∈ PROPN cana-4038 113 21 𝐸(𝐺 𝐸(𝐺 PROPN cana-4038 113 22 ) ) PUNCT cana-4038 113 23 and and CCONJ cana-4038 113 24 𝑃 𝑃 PROPN cana-4038 113 25 = = SYM cana-4038 113 26 ∏ ∏ PROPN cana-4038 113 27 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 113 28 ) ) PUNCT cana-4038 113 29 for for ADP cana-4038 113 30 each each DET cana-4038 113 31 vertex vertex NOUN cana-4038 113 32 in in ADP cana-4038 113 33 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 113 34 ) ) PUNCT cana-4038 113 35 define define VERB cana-4038 113 36 π‘“βˆ— π‘“βˆ— NOUN cana-4038 113 37 : : PUNCT cana-4038 113 38 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 113 39 ) ) PUNCT cana-4038 113 40 β†’ β†’ SYM cana-4038 113 41 { { PUNCT cana-4038 113 42 0,1 0,1 NOUN cana-4038 113 43 } } PUNCT cana-4038 113 44 defined define VERB cana-4038 113 45 by by ADP cana-4038 113 46 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 113 47 ) ) PUNCT cana-4038 113 48 = = NOUN cana-4038 113 49 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 113 50 βŒ‹ βŒ‹ NOUN cana-4038 113 51 ( ( PUNCT cana-4038 113 52 mod mod PROPN cana-4038 113 53 2 2 NUM cana-4038 113 54 ) ) PUNCT cana-4038 113 55 . . PUNCT cana-4038 114 1 communications communication NOUN cana-4038 114 2 on on ADP cana-4038 114 3 applied apply VERB cana-4038 114 4 nonlinear nonlinear ADJ cana-4038 114 5 analysis analysis NOUN cana-4038 114 6 issn issn NOUN cana-4038 114 7 : : PUNCT cana-4038 114 8 1074 1074 NUM cana-4038 114 9 - - PUNCT cana-4038 114 10 133x 133x NUM cana-4038 114 11 vol vol NOUN cana-4038 114 12 32 32 NUM cana-4038 114 13 no no NOUN cana-4038 114 14 . . PUNCT cana-4038 115 1 9s 9s NUM cana-4038 115 2 ( ( PUNCT cana-4038 115 3 2025 2025 NUM cana-4038 115 4 ) ) PUNCT cana-4038 115 5 906 906 NUM cana-4038 115 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 115 7 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 115 8 ) ) PUNCT cana-4038 115 9 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 115 10 ) ) PUNCT cana-4038 115 11 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 115 12 ) ) PUNCT cana-4038 115 13 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 115 14 ) ) PUNCT cana-4038 115 15 𝑛 𝑛 DET cana-4038 115 16 π‘›βˆ’ π‘›βˆ’ PROPN cana-4038 115 17 1 1 NUM cana-4038 115 18 𝑛 𝑛 ADP cana-4038 115 19 𝑛 𝑛 NOUN cana-4038 115 20 the the DET cana-4038 115 21 above above ADJ cana-4038 115 22 labeling labeling NOUN cana-4038 115 23 satisfies satisfie NOUN cana-4038 115 24 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 115 25 ) ) PUNCT cana-4038 115 26 βˆ’ βˆ’ NOUN cana-4038 115 27 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 115 28 ≀ ≀ NUM cana-4038 115 29 1 1 NUM cana-4038 115 30 and and CCONJ cana-4038 115 31 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 115 32 ) ) PUNCT cana-4038 116 1 βˆ’ βˆ’ PROPN cana-4038 116 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 116 3 ≀ ≀ ADJ cana-4038 116 4 1 1 NUM cana-4038 116 5 . . PUNCT cana-4038 117 1 combo combo NOUN cana-4038 117 2 graph graph NOUN cana-4038 117 3 𝑃𝑛 𝑃𝑛 PROPN cana-4038 117 4 ⊚ ⊚ ADJ cana-4038 117 5 𝐾2 𝐾2 NOUN cana-4038 117 6 admits admit VERB cana-4038 117 7 sp sp ADP cana-4038 117 8 mean mean NOUN cana-4038 117 9 e e ADJ cana-4038 117 10 - - ADJ cana-4038 117 11 cordial cordial ADJ cana-4038 117 12 labeling labeling NOUN cana-4038 117 13 . . PUNCT cana-4038 118 1 hence hence ADV cana-4038 118 2 combo combo ADJ cana-4038 118 3 graph graph NOUN cana-4038 118 4 𝑃𝑛 𝑃𝑛 PROPN cana-4038 118 5 ⊚ ⊚ NOUN cana-4038 118 6 𝐾2is 𝐾2is PRON cana-4038 118 7 a a DET cana-4038 118 8 sp sp NOUN cana-4038 118 9 mean mean NOUN cana-4038 118 10 e e ADJ cana-4038 118 11 - - ADJ cana-4038 118 12 cordial cordial ADJ cana-4038 118 13 graph graph NOUN cana-4038 118 14 . . PUNCT cana-4038 119 1 theorem theorem VERB cana-4038 119 2 2.10 2.10 NUM cana-4038 119 3 crown crown NOUN cana-4038 119 4 graph graph NOUN cana-4038 119 5 𝐢𝑛 𝐢𝑛 PROPN cana-4038 119 6 ⊚ ⊚ NOUN cana-4038 119 7 𝐾2 𝐾2 NOUN cana-4038 119 8 is be AUX cana-4038 119 9 a a DET cana-4038 119 10 sp sp NOUN cana-4038 119 11 mean mean NOUN cana-4038 119 12 e e NOUN cana-4038 119 13 - - NOUN cana-4038 119 14 cg cg NOUN cana-4038 119 15 . . PUNCT cana-4038 120 1 proof proof NOUN cana-4038 120 2 : : PUNCT cana-4038 120 3 let let VERB cana-4038 120 4 𝑇(𝐢𝑛 𝑇(𝐢𝑛 PROPN cana-4038 120 5 ⊚ ⊚ PRON cana-4038 120 6 𝐾2 𝐾2 NOUN cana-4038 120 7 ) ) PUNCT cana-4038 120 8 = = PRON cana-4038 120 9 { { PUNCT cana-4038 120 10 𝑒1 𝑒1 PROPN cana-4038 120 11 , , PUNCT cana-4038 120 12 𝑒2 𝑒2 PROPN cana-4038 120 13 , , PUNCT cana-4038 120 14 𝑒3 𝑒3 PROPN cana-4038 120 15 … … PUNCT cana-4038 120 16 , , PUNCT cana-4038 120 17 𝑒𝑛 𝑒𝑛 NOUN cana-4038 120 18 , , PUNCT cana-4038 120 19 𝑒1 𝑒1 NOUN cana-4038 120 20 , , PUNCT cana-4038 120 21 , , PUNCT cana-4038 120 22 𝑒2 𝑒2 PROPN cana-4038 120 23 , , PUNCT cana-4038 120 24 , , PUNCT cana-4038 120 25 𝑒3 𝑒3 PROPN cana-4038 120 26 , , PUNCT cana-4038 120 27 … … PUNCT cana-4038 120 28 , , PUNCT cana-4038 120 29 𝑒𝑛 𝑒𝑛 NOUN cana-4038 120 30 , , PUNCT cana-4038 120 31 } } PUNCT cana-4038 120 32 be be AUX cana-4038 120 33 the the DET cana-4038 120 34 vertices vertex NOUN cana-4038 120 35 and and CCONJ cana-4038 120 36 𝑅(𝐢𝑛 𝑅(𝐢𝑛 ADP cana-4038 120 37 ⊚ ⊚ NUM cana-4038 120 38 𝐾2 𝐾2 NOUN cana-4038 120 39 ) ) PUNCT cana-4038 121 1 = = NOUN cana-4038 121 2 { { PUNCT cana-4038 121 3 𝑒1𝑒2 𝑒1𝑒2 NOUN cana-4038 121 4 , , PUNCT cana-4038 121 5 𝑒2𝑒3 𝑒2𝑒3 NOUN cana-4038 121 6 , , PUNCT cana-4038 121 7 𝑒3𝑒4 𝑒3𝑒4 PROPN cana-4038 121 8 , , PUNCT cana-4038 121 9 … … PUNCT cana-4038 121 10 , , PUNCT cana-4038 121 11 𝑒𝑛𝑒1 𝑒𝑛𝑒1 PROPN cana-4038 121 12 , , PUNCT cana-4038 121 13 𝑒1𝑒1 𝑒1𝑒1 NOUN cana-4038 121 14 , , PUNCT cana-4038 121 15 , , PUNCT cana-4038 121 16 𝑒2𝑒2 𝑒2𝑒2 X cana-4038 121 17 , , PUNCT cana-4038 121 18 , , PUNCT cana-4038 121 19 𝑒3𝑒3 𝑒3𝑒3 ADP cana-4038 121 20 , , PUNCT cana-4038 121 21 … … PUNCT cana-4038 121 22 , , PUNCT cana-4038 121 23 𝑒𝑛𝑒𝑛 𝑒𝑛𝑒𝑛 NOUN cana-4038 121 24 , , PUNCT cana-4038 121 25 } } PUNCT cana-4038 121 26 be be AUX cana-4038 121 27 the the DET cana-4038 121 28 edges edge NOUN cana-4038 121 29 . . PUNCT cana-4038 122 1 then then ADV cana-4038 122 2 | | ADV cana-4038 122 3 t(𝐢𝑛 t(𝐢𝑛 VERB cana-4038 122 4 ⊚ ⊚ NOUN cana-4038 122 5 𝐾2)| 𝐾2)| X cana-4038 122 6 = = SYM cana-4038 122 7 2𝑛 2𝑛 PROPN cana-4038 122 8 and and CCONJ cana-4038 122 9 | | ADV cana-4038 122 10 r(𝐢𝑛 r(𝐢𝑛 VERB cana-4038 122 11 ⊚ ⊚ NOUN cana-4038 122 12 𝐾2)| 𝐾2)| SYM cana-4038 123 1 = = SYM cana-4038 124 1 2𝑛 2𝑛 PROPN cana-4038 124 2 define define VERB cana-4038 124 3 the the DET cana-4038 124 4 labeling labeling NOUN cana-4038 124 5 function function NOUN cana-4038 124 6 𝑓 𝑓 NOUN cana-4038 124 7 : : PUNCT cana-4038 124 8 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 124 9 ) ) PUNCT cana-4038 124 10 β†’ β†’ SYM cana-4038 124 11 { { PUNCT cana-4038 124 12 1,2 1,2 NUM cana-4038 124 13 } } PUNCT cana-4038 124 14 as as ADP cana-4038 124 15 below below ADV cana-4038 124 16 for for ADP cana-4038 124 17 1≀ 1≀ NUM cana-4038 124 18 𝑖 𝑖 SYM cana-4038 124 19 ≀ ≀ NOUN cana-4038 124 20 𝑛 𝑛 DET cana-4038 124 21 βˆ’ βˆ’ NUM cana-4038 124 22 1 1 NUM cana-4038 124 23 if if SCONJ cana-4038 124 24 n n NOUN cana-4038 124 25 is be AUX cana-4038 124 26 odd odd ADJ cana-4038 124 27 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 NOUN cana-4038 124 28 ) ) PUNCT cana-4038 124 29 = = NOUN cana-4038 124 30 { { PUNCT cana-4038 124 31 2 2 NUM cana-4038 124 32 𝑖𝑓 𝑖𝑓 SYM cana-4038 124 33 𝑖 𝑖 PUNCT cana-4038 124 34 𝑖𝑠 𝑖𝑠 PROPN cana-4038 124 35 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 124 36 𝑖𝑓 𝑖𝑓 ADP cana-4038 124 37 𝑖 𝑖 PUNCT cana-4038 124 38 𝑖𝑠 𝑖𝑠 NOUN cana-4038 124 39 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 124 40 𝑓(𝑒𝑖𝑒𝑖′ 𝑓(𝑒𝑖𝑒𝑖′ NOUN cana-4038 124 41 ) ) PUNCT cana-4038 124 42 = = PUNCT cana-4038 124 43 { { PUNCT cana-4038 125 1 1 1 NUM cana-4038 125 2 𝑖𝑓 𝑖𝑓 SYM cana-4038 125 3 𝑖 𝑖 PUNCT cana-4038 125 4 𝑖𝑠 𝑖𝑠 INTJ cana-4038 126 1 𝑒𝑣𝑒𝑛2 𝑒𝑣𝑒𝑛2 PROPN cana-4038 127 1 𝑖𝑓 𝑖𝑓 CCONJ cana-4038 127 2 𝑖 𝑖 PRON cana-4038 127 3 𝑖𝑠 𝑖𝑠 ADV cana-4038 127 4 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 128 1 if if SCONJ cana-4038 128 2 n n PRON cana-4038 128 3 is be AUX cana-4038 128 4 even even ADV cana-4038 128 5 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 NOUN cana-4038 128 6 ) ) PUNCT cana-4038 128 7 = = SYM cana-4038 128 8 { { PUNCT cana-4038 128 9 2 2 NUM cana-4038 128 10 𝑖𝑓 𝑖𝑓 SYM cana-4038 128 11 𝑖 𝑖 PUNCT cana-4038 128 12 𝑖𝑠 𝑖𝑠 PROPN cana-4038 128 13 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 128 14 𝑖𝑓 𝑖𝑓 ADP cana-4038 128 15 𝑖 𝑖 PUNCT cana-4038 128 16 𝑖𝑠 𝑖𝑠 NOUN cana-4038 128 17 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 128 18 𝑓(𝑒𝑖𝑒𝑖′ 𝑓(𝑒𝑖𝑒𝑖′ NOUN cana-4038 128 19 ) ) PUNCT cana-4038 129 1 = = PUNCT cana-4038 129 2 { { PUNCT cana-4038 129 3 2 2 NUM cana-4038 129 4 𝑖𝑓 𝑖𝑓 SYM cana-4038 129 5 𝑖 𝑖 PUNCT cana-4038 129 6 𝑖𝑠 𝑖𝑠 PROPN cana-4038 129 7 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 129 8 𝑖𝑓 𝑖𝑓 ADP cana-4038 129 9 𝑖 𝑖 PUNCT cana-4038 129 10 𝑖𝑠 𝑖𝑠 NOUN cana-4038 129 11 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 129 12 find find VERB cana-4038 129 13 𝑃 𝑃 NOUN cana-4038 129 14 = = SYM cana-4038 129 15 ∏ ∏ NUM cana-4038 129 16 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 129 17 ) ) PUNCT cana-4038 129 18 and and CCONJ cana-4038 129 19 𝑆 𝑆 PROPN cana-4038 129 20 = = SYM cana-4038 129 21 βˆ‘ βˆ‘ PUNCT cana-4038 129 22 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 129 23 ) ) PUNCT cana-4038 129 24 βˆ– βˆ– X cana-4038 129 25 𝑒𝑑 𝑒𝑑 PROPN cana-4038 129 26 ∈ ∈ PROPN cana-4038 129 27 𝐸(𝐺 𝐸(𝐺 PROPN cana-4038 129 28 ) ) PUNCT cana-4038 129 29 for for ADP cana-4038 129 30 each each DET cana-4038 129 31 vertex vertex NOUN cana-4038 129 32 in in ADP cana-4038 129 33 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 129 34 ) ) PUNCT cana-4038 129 35 defineπ‘“βˆ— defineπ‘“βˆ— VERB cana-4038 129 36 : : PUNCT cana-4038 129 37 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 129 38 ) ) PUNCT cana-4038 129 39 β†’ β†’ SYM cana-4038 129 40 { { PUNCT cana-4038 129 41 0,1 0,1 NOUN cana-4038 129 42 } } PUNCT cana-4038 129 43 by by ADP cana-4038 129 44 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 129 45 ) ) PUNCT cana-4038 129 46 = = NOUN cana-4038 129 47 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 129 48 βŒ‹ βŒ‹ NOUN cana-4038 129 49 ( ( PUNCT cana-4038 129 50 mod mod PROPN cana-4038 129 51 2 2 NUM cana-4038 129 52 ) ) PUNCT cana-4038 129 53 . . PUNCT cana-4038 130 1 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 130 2 ) ) PUNCT cana-4038 130 3 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 130 4 ) ) PUNCT cana-4038 130 5 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 130 6 ) ) PUNCT cana-4038 130 7 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 130 8 ) ) PUNCT cana-4038 130 9 𝑛 𝑛 PROPN cana-4038 130 10 is be AUX cana-4038 130 11 even even ADV cana-4038 130 12 𝑛 𝑛 DET cana-4038 130 13 𝑛 𝑛 PROPN cana-4038 130 14 𝑛 𝑛 ADP cana-4038 130 15 𝑛 𝑛 PRON cana-4038 130 16 𝑛 𝑛 PROPN cana-4038 130 17 is be AUX cana-4038 130 18 odd odd ADJ cana-4038 130 19 𝑛 𝑛 PRON cana-4038 130 20 𝑛 𝑛 DET cana-4038 130 21 βˆ’ βˆ’ PROPN cana-4038 130 22 1 1 NUM cana-4038 130 23 𝑛 𝑛 PROPN cana-4038 130 24 𝑛 𝑛 PROPN cana-4038 130 25 communications communication NOUN cana-4038 130 26 on on ADP cana-4038 130 27 applied apply VERB cana-4038 130 28 nonlinear nonlinear ADJ cana-4038 130 29 analysis analysis NOUN cana-4038 130 30 issn issn NOUN cana-4038 130 31 : : PUNCT cana-4038 130 32 1074 1074 NUM cana-4038 130 33 - - PUNCT cana-4038 130 34 133x 133x NUM cana-4038 130 35 vol vol NOUN cana-4038 130 36 32 32 NUM cana-4038 130 37 no no NOUN cana-4038 130 38 . . PUNCT cana-4038 131 1 9s 9s NUM cana-4038 131 2 ( ( PUNCT cana-4038 131 3 2025 2025 NUM cana-4038 131 4 ) ) PUNCT cana-4038 131 5 907 907 NUM cana-4038 131 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 131 7 the the DET cana-4038 131 8 above above ADJ cana-4038 131 9 labeling labeling NOUN cana-4038 131 10 satisfies satisfie NOUN cana-4038 131 11 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 131 12 ) ) PUNCT cana-4038 131 13 βˆ’ βˆ’ PROPN cana-4038 131 14 𝑑𝑓(2)| 𝑑𝑓(2)| PROPN cana-4038 131 15 ≀ ≀ NUM cana-4038 131 16 1 1 NUM cana-4038 131 17 and and CCONJ cana-4038 131 18 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 131 19 ) ) PUNCT cana-4038 132 1 βˆ’ βˆ’ PROPN cana-4038 132 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 132 3 ≀ ≀ ADJ cana-4038 132 4 1 1 NUM cana-4038 132 5 . . PUNCT cana-4038 133 1 the the DET cana-4038 133 2 crown crown NOUN cana-4038 133 3 graph graph NOUN cana-4038 133 4 𝐢𝑛 𝐢𝑛 NOUN cana-4038 133 5 ⊚ ⊚ NOUN cana-4038 133 6 𝐾2admits 𝐾2admit NOUN cana-4038 133 7 sp sp ADP cana-4038 133 8 mean mean NOUN cana-4038 133 9 e e NOUN cana-4038 133 10 - - NOUN cana-4038 133 11 cl cl NOUN cana-4038 133 12 . . PUNCT cana-4038 134 1 hence hence ADV cana-4038 134 2 crown crown NOUN cana-4038 134 3 graph graph NOUN cana-4038 134 4 𝐢𝑛 𝐢𝑛 PROPN cana-4038 134 5 ⊚ ⊚ NOUN cana-4038 134 6 𝐾2is 𝐾2is VERB cana-4038 134 7 a a DET cana-4038 134 8 sp sp NOUN cana-4038 134 9 mean mean NOUN cana-4038 134 10 e e ADJ cana-4038 134 11 - - ADJ cana-4038 134 12 cordial cordial ADJ cana-4038 134 13 graph graph NOUN cana-4038 134 14 . . PUNCT cana-4038 135 1 theorem theorem VERB cana-4038 135 2 2.11 2.11 NUM cana-4038 135 3 triangular triangular NOUN cana-4038 135 4 snake snake NOUN cana-4038 135 5 𝑛𝐢3 𝑛𝐢3 NUM cana-4038 135 6 is be AUX cana-4038 135 7 a a DET cana-4038 135 8 sp sp NOUN cana-4038 135 9 mean mean NOUN cana-4038 135 10 e e NOUN cana-4038 135 11 - - NOUN cana-4038 135 12 cg cg NOUN cana-4038 135 13 . . PUNCT cana-4038 136 1 proof proof NOUN cana-4038 136 2 : : PUNCT cana-4038 136 3 let let VERB cana-4038 136 4 the the DET cana-4038 136 5 path path NOUN cana-4038 137 1 𝑃𝑛 𝑃𝑛 PROPN cana-4038 137 2 having have VERB cana-4038 137 3 the the DET cana-4038 137 4 edges edge NOUN cana-4038 137 5 π‘Ÿ1 π‘Ÿ1 NOUN cana-4038 137 6 , , PUNCT cana-4038 137 7 π‘Ÿ2 π‘Ÿ2 NOUN cana-4038 137 8 , , PUNCT cana-4038 137 9 π‘Ÿ3 π‘Ÿ3 NOUN cana-4038 137 10 … … PUNCT cana-4038 137 11 … … PUNCT cana-4038 137 12 . . PUNCT cana-4038 138 1 . . PUNCT cana-4038 139 1 , , PUNCT cana-4038 139 2 π‘Ÿπ‘›βˆ’1 π‘Ÿπ‘›βˆ’1 NOUN cana-4038 139 3 and and CCONJ cana-4038 139 4 vertices vertice VERB cana-4038 139 5 𝑑1 𝑑1 NOUN cana-4038 139 6 , , PUNCT cana-4038 139 7 𝑑2 𝑑2 PROPN cana-4038 139 8 , , PUNCT cana-4038 139 9 𝑑3 𝑑3 PROPN cana-4038 139 10 , , PUNCT cana-4038 139 11 … … PUNCT cana-4038 139 12 , , PUNCT cana-4038 139 13 𝑑𝑛. 𝑑𝑛. NOUN cana-4038 139 14 to to PART cana-4038 139 15 construct construct VERB cana-4038 139 16 triangular triangular NOUN cana-4038 139 17 snake snake NOUN cana-4038 139 18 𝑛𝐢3 𝑛𝐢3 NUM cana-4038 139 19 from from ADP cana-4038 139 20 path path NOUN cana-4038 139 21 𝑃𝑛 𝑃𝑛 PROPN cana-4038 139 22 join join VERB cana-4038 139 23 𝑒𝑖 𝑒𝑖 PRON cana-4038 139 24 and and CCONJ cana-4038 139 25 𝑒𝑖+1 𝑒𝑖+1 NUM cana-4038 139 26 to to ADP cana-4038 139 27 a a DET cana-4038 139 28 new new ADJ cana-4038 139 29 edge edge NOUN cana-4038 139 30 𝑑𝑖 𝑑𝑖 NOUN cana-4038 139 31 by by ADP cana-4038 139 32 edges edge NOUN cana-4038 139 33 𝑒𝑖𝑑𝑖 𝑒𝑖𝑑𝑖 NOUN cana-4038 139 34 and and CCONJ cana-4038 139 35 𝑒𝑖+1𝑑𝑖 𝑒𝑖+1𝑑𝑖 PROPN cana-4038 139 36 , , PUNCT cana-4038 139 37 for for ADP cana-4038 139 38 i i PROPN cana-4038 139 39 = = NOUN cana-4038 139 40 1,2,3 1,2,3 NUM cana-4038 139 41 … … PUNCT cana-4038 139 42 … … PUNCT cana-4038 139 43 .. .. PUNCT cana-4038 139 44 , , PUNCT cana-4038 139 45 𝑛 𝑛 DET cana-4038 139 46 βˆ’ βˆ’ NOUN cana-4038 139 47 1 1 NUM cana-4038 140 1 then then ADV cana-4038 140 2 | | ADV cana-4038 140 3 t(𝑛𝐢3)| t(𝑛𝐢3)| PRON cana-4038 141 1 = = SYM cana-4038 141 2 2𝑛 2𝑛 PROPN cana-4038 142 1 βˆ’ βˆ’ NOUN cana-4038 142 2 1 1 NUM cana-4038 142 3 and and CCONJ cana-4038 142 4 | | ADV cana-4038 142 5 r(𝑛𝐢3)| r(𝑛𝐢3)| NOUN cana-4038 142 6 = = SYM cana-4038 142 7 3𝑛 3𝑛 NUM cana-4038 142 8 βˆ’ βˆ’ NOUN cana-4038 142 9 3 3 NUM cana-4038 142 10 define define VERB cana-4038 142 11 the the DET cana-4038 142 12 labeling labeling NOUN cana-4038 142 13 function function NOUN cana-4038 142 14 𝑓 𝑓 NOUN cana-4038 142 15 : : PUNCT cana-4038 142 16 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 142 17 ) ) PUNCT cana-4038 142 18 β†’ β†’ SYM cana-4038 142 19 { { PUNCT cana-4038 142 20 1,2 1,2 NUM cana-4038 142 21 } } PUNCT cana-4038 142 22 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 NOUN cana-4038 142 23 ) ) PUNCT cana-4038 142 24 = = NOUN cana-4038 142 25 { { PUNCT cana-4038 143 1 2 2 NUM cana-4038 143 2 𝑖𝑓 𝑖𝑓 SYM cana-4038 143 3 𝑖 𝑖 PRON cana-4038 143 4 𝑖𝑠 𝑖𝑠 NOUN cana-4038 143 5 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-4038 144 1 1 1 NUM cana-4038 144 2 𝑖𝑓 𝑖𝑓 SYM cana-4038 144 3 𝑖 𝑖 PUNCT cana-4038 144 4 𝑖𝑠 𝑖𝑠 NOUN cana-4038 144 5 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 144 6 𝑓(𝑒𝑖𝑒𝑖′ 𝑓(𝑒𝑖𝑒𝑖′ NOUN cana-4038 144 7 ) ) PUNCT cana-4038 144 8 = = SYM cana-4038 144 9 1 1 NUM cana-4038 144 10 𝑓(𝑒𝑖+1𝑒𝑖′ 𝑓(𝑒𝑖+1𝑒𝑖′ VERB cana-4038 144 11 ) ) PUNCT cana-4038 144 12 = = SYM cana-4038 144 13 2 2 NUM cana-4038 144 14 find find VERB cana-4038 144 15 𝑆 𝑆 PROPN cana-4038 144 16 = = SYM cana-4038 144 17 βˆ‘ βˆ‘ PUNCT cana-4038 144 18 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 144 19 ) ) PUNCT cana-4038 144 20 βˆ– βˆ– X cana-4038 144 21 𝑒𝑑 𝑒𝑑 PROPN cana-4038 144 22 ∈ ∈ PROPN cana-4038 144 23 𝐸(𝐺 𝐸(𝐺 PROPN cana-4038 144 24 ) ) PUNCT cana-4038 144 25 and and CCONJ cana-4038 144 26 𝑃 𝑃 PROPN cana-4038 144 27 = = SYM cana-4038 144 28 ∏ ∏ NUM cana-4038 144 29 𝑓(𝑒𝑣 𝑓(𝑒𝑣 PROPN cana-4038 144 30 ) ) PUNCT cana-4038 144 31 for for ADP cana-4038 144 32 each each DET cana-4038 144 33 vertex vertex NOUN cana-4038 144 34 in in ADP cana-4038 144 35 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 144 36 ) ) PUNCT cana-4038 144 37 defineπ‘“βˆ— defineπ‘“βˆ— VERB cana-4038 144 38 : : PUNCT cana-4038 144 39 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 144 40 ) ) PUNCT cana-4038 144 41 β†’ β†’ SYM cana-4038 144 42 { { PUNCT cana-4038 144 43 0,1 0,1 NOUN cana-4038 144 44 } } PUNCT cana-4038 144 45 defined define VERB cana-4038 144 46 by by ADP cana-4038 144 47 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 144 48 ) ) PUNCT cana-4038 144 49 = = NOUN cana-4038 144 50 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 144 51 βŒ‹ βŒ‹ NOUN cana-4038 144 52 ( ( PUNCT cana-4038 144 53 mod mod PROPN cana-4038 144 54 2 2 NUM cana-4038 144 55 ) ) PUNCT cana-4038 144 56 . . PUNCT cana-4038 145 1 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 145 2 ) ) PUNCT cana-4038 145 3 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 145 4 ) ) PUNCT cana-4038 145 5 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 145 6 ) ) PUNCT cana-4038 145 7 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 145 8 ) ) PUNCT cana-4038 145 9 𝑛 𝑛 PRON cana-4038 145 10 is be AUX cana-4038 145 11 even even ADV cana-4038 145 12 𝑛 𝑛 DET cana-4038 145 13 𝑛 𝑛 NOUN cana-4038 145 14 βˆ’ βˆ’ NUM cana-4038 145 15 1 1 NUM cana-4038 145 16 𝑛 𝑛 PROPN cana-4038 145 17 + + NOUN cana-4038 145 18 1 1 NUM cana-4038 145 19 𝑛 𝑛 PRON cana-4038 145 20 + + NOUN cana-4038 145 21 1 1 NUM cana-4038 145 22 𝑛 𝑛 NOUN cana-4038 145 23 is be AUX cana-4038 145 24 odd odd ADJ cana-4038 145 25 𝑛 𝑛 PRON cana-4038 145 26 𝑛 𝑛 PRON cana-4038 145 27 βˆ’ βˆ’ PROPN cana-4038 145 28 1 1 NUM cana-4038 145 29 𝑛 𝑛 PROPN cana-4038 145 30 + + NOUN cana-4038 145 31 1 1 NUM cana-4038 145 32 𝑛 𝑛 NOUN cana-4038 145 33 + + NUM cana-4038 145 34 1 1 NUM cana-4038 145 35 the the DET cana-4038 145 36 above above ADJ cana-4038 145 37 labeling labeling NOUN cana-4038 145 38 satisfies satisfie NOUN cana-4038 145 39 β€œ " PUNCT cana-4038 145 40 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 145 41 ) ) PUNCT cana-4038 145 42 βˆ’ βˆ’ NOUN cana-4038 145 43 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 145 44 ≀ ≀ NUM cana-4038 145 45 1 1 NUM cana-4038 145 46 and and CCONJ cana-4038 145 47 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 145 48 ) ) PUNCT cana-4038 146 1 βˆ’ βˆ’ PROPN cana-4038 146 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 146 3 ≀ ≀ ADJ cana-4038 146 4 1 1 NUM cana-4038 146 5 . . PUNCT cana-4038 147 1 the the DET cana-4038 147 2 triangular triangular NOUN cana-4038 147 3 snake snake NOUN cana-4038 147 4 𝑛𝐢3 𝑛𝐢3 PROPN cana-4038 147 5 admits admit NOUN cana-4038 147 6 sp sp ADP cana-4038 147 7 mean mean NOUN cana-4038 147 8 e e ADJ cana-4038 147 9 - - ADJ cana-4038 147 10 cordial cordial ADJ cana-4038 147 11 labeling labeling NOUN cana-4038 147 12 hence hence ADV cana-4038 147 13 triangular triangular NOUN cana-4038 147 14 snake snake NOUN cana-4038 147 15 𝑛𝐢3is 𝑛𝐢3i NOUN cana-4038 147 16 a a DET cana-4038 147 17 sp sp NOUN cana-4038 147 18 mean mean NOUN cana-4038 147 19 e e ADJ cana-4038 147 20 - - ADJ cana-4038 147 21 cordial cordial ADJ cana-4038 147 22 graph graph NOUN cana-4038 147 23 . . PUNCT cana-4038 147 24 theorem theorem VERB cana-4038 147 25 2.12 2.12 NUM cana-4038 147 26 fan fan NOUN cana-4038 147 27 graph graph NOUN cana-4038 147 28 𝐹1,𝑛 𝐹1,𝑛 PUNCT cana-4038 147 29 is be AUX cana-4038 147 30 a a DET cana-4038 147 31 sp sp NOUN cana-4038 147 32 mean mean NOUN cana-4038 147 33 e e NOUN cana-4038 147 34 - - NOUN cana-4038 147 35 cg cg NOUN cana-4038 147 36 if if SCONJ cana-4038 147 37 n n NOUN cana-4038 147 38 is be AUX cana-4038 147 39 odd odd ADJ cana-4038 147 40 . . PUNCT cana-4038 148 1 proof proof NOUN cana-4038 148 2 : : PUNCT cana-4038 148 3 let let VERB cana-4038 148 4 𝑇(𝐹1,𝑛 𝑇(𝐹1,𝑛 VERB cana-4038 148 5 ) ) PUNCT cana-4038 148 6 = = PRON cana-4038 148 7 { { PUNCT cana-4038 148 8 𝑒 𝑒 PROPN cana-4038 148 9 , , PUNCT cana-4038 148 10 𝑒1 𝑒1 NOUN cana-4038 148 11 , , PUNCT cana-4038 148 12 𝑒2 𝑒2 PROPN cana-4038 148 13 , , PUNCT cana-4038 148 14 𝑒3 𝑒3 PROPN cana-4038 148 15 … … PUNCT cana-4038 148 16 , , PUNCT cana-4038 148 17 𝑒𝑛 𝑒𝑛 NOUN cana-4038 148 18 , , PUNCT cana-4038 148 19 } } PUNCT cana-4038 148 20 be be AUX cana-4038 148 21 the the DET cana-4038 148 22 vertices vertex NOUN cana-4038 148 23 and and CCONJ cana-4038 148 24 𝑅(𝐹1,𝑛 𝑅(𝐹1,𝑛 X cana-4038 148 25 ) ) PUNCT cana-4038 149 1 = = NOUN cana-4038 149 2 { { PUNCT cana-4038 149 3 𝑒1𝑒2 𝑒1𝑒2 NOUN cana-4038 149 4 , , PUNCT cana-4038 149 5 𝑒2𝑒3 𝑒2𝑒3 NOUN cana-4038 149 6 , , PUNCT cana-4038 149 7 𝑒3𝑒4 𝑒3𝑒4 PROPN cana-4038 149 8 , , PUNCT cana-4038 149 9 … … PUNCT cana-4038 149 10 , , PUNCT cana-4038 149 11 π‘’π‘›βˆ’1𝑒𝑛 π‘’π‘›βˆ’1𝑒𝑛 NOUN cana-4038 149 12 , , PUNCT cana-4038 149 13 𝑒𝑒1 𝑒𝑒1 NOUN cana-4038 149 14 , , PUNCT cana-4038 149 15 𝑒𝑒2𝑒𝑒3 𝑒𝑒2𝑒𝑒3 NOUN cana-4038 149 16 , , PUNCT cana-4038 149 17 … … PUNCT cana-4038 149 18 , , PUNCT cana-4038 149 19 𝑒𝑒𝑛 𝑒𝑒𝑛 AUX cana-4038 149 20 } } PUNCT cana-4038 149 21 be be AUX cana-4038 149 22 the the DET cana-4038 149 23 edges edge NOUN cana-4038 149 24 . . PUNCT cana-4038 150 1 here here ADV cana-4038 150 2 u u NOUN cana-4038 150 3 is be AUX cana-4038 150 4 apex apex ADJ cana-4038 150 5 vertex vertex NOUN cana-4038 150 6 and and CCONJ cana-4038 150 7 𝑒1 𝑒1 NOUN cana-4038 150 8 , , PUNCT cana-4038 150 9 𝑒2 𝑒2 PROPN cana-4038 150 10 , , PUNCT cana-4038 150 11 𝑒3 𝑒3 PROPN cana-4038 150 12 … … PUNCT cana-4038 150 13 , , PUNCT cana-4038 150 14 𝑒𝑛 𝑒𝑛 NOUN cana-4038 150 15 be be AUX cana-4038 150 16 the the DET cana-4038 150 17 vertices vertex NOUN cana-4038 150 18 of of ADP cana-4038 150 19 path path NOUN cana-4038 150 20 . . PUNCT cana-4038 151 1 then| then| PROPN cana-4038 151 2 r(𝐹1,𝑛)| r(𝐹1,𝑛)| PROPN cana-4038 152 1 = = PROPN cana-4038 152 2 2𝑛 2𝑛 PROPN cana-4038 152 3 βˆ’ βˆ’ NOUN cana-4038 152 4 1 1 NUM cana-4038 152 5 and and CCONJ cana-4038 152 6 | | ADV cana-4038 152 7 t(𝐹1,𝑛)| t(𝐹1,𝑛)| NUM cana-4038 152 8 = = SYM cana-4038 152 9 𝑛 𝑛 PROPN cana-4038 152 10 + + NOUN cana-4038 152 11 1 1 X cana-4038 152 12 . . X cana-4038 152 13 define define VERB cana-4038 152 14 the the DET cana-4038 152 15 function function NOUN cana-4038 152 16 𝑓 𝑓 NOUN cana-4038 152 17 : : PUNCT cana-4038 152 18 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 152 19 ) ) PUNCT cana-4038 152 20 β†’ β†’ SYM cana-4038 152 21 { { PUNCT cana-4038 152 22 1,2 1,2 NUM cana-4038 152 23 } } PUNCT cana-4038 152 24 as as ADP cana-4038 152 25 below below ADP cana-4038 152 26 𝑓(𝑒𝑒𝑖 𝑓(𝑒𝑒𝑖 PROPN cana-4038 152 27 ) ) PUNCT cana-4038 152 28 = = PRON cana-4038 152 29 { { PUNCT cana-4038 152 30 2 2 NUM cana-4038 152 31 𝑖𝑓 𝑖𝑓 SYM cana-4038 152 32 𝑖 𝑖 PUNCT cana-4038 152 33 𝑖𝑠 𝑖𝑠 PROPN cana-4038 152 34 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 152 35 𝑖𝑓 𝑖𝑓 ADP cana-4038 152 36 𝑖 𝑖 PUNCT cana-4038 152 37 𝑖𝑠 𝑖𝑠 NOUN cana-4038 152 38 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 152 39 communications communication NOUN cana-4038 152 40 on on ADP cana-4038 152 41 applied apply VERB cana-4038 152 42 nonlinear nonlinear ADJ cana-4038 152 43 analysis analysis NOUN cana-4038 152 44 issn issn NOUN cana-4038 152 45 : : PUNCT cana-4038 152 46 1074 1074 NUM cana-4038 152 47 - - PUNCT cana-4038 152 48 133x 133x NUM cana-4038 152 49 vol vol NOUN cana-4038 152 50 32 32 NUM cana-4038 153 1 no no NOUN cana-4038 153 2 . . PUNCT cana-4038 154 1 9s 9s NUM cana-4038 154 2 ( ( PUNCT cana-4038 154 3 2025 2025 NUM cana-4038 154 4 ) ) PUNCT cana-4038 154 5 908 908 NUM cana-4038 154 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 154 7 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 PROPN cana-4038 154 8 ) ) PUNCT cana-4038 154 9 = = PRON cana-4038 154 10 { { PUNCT cana-4038 154 11 1 1 NUM cana-4038 154 12 𝑖𝑓 𝑖𝑓 SYM cana-4038 154 13 𝑖 𝑖 PUNCT cana-4038 154 14 𝑖𝑠 𝑖𝑠 INTJ cana-4038 155 1 𝑒𝑣𝑒𝑛2 𝑒𝑣𝑒𝑛2 PROPN cana-4038 156 1 𝑖𝑓 𝑖𝑓 ADV cana-4038 156 2 𝑖 𝑖 PUNCT cana-4038 156 3 𝑖𝑠 𝑖𝑠 NOUN cana-4038 156 4 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 156 5 find find VERB cana-4038 156 6 𝑃 𝑃 NOUN cana-4038 156 7 = = SYM cana-4038 156 8 ∏ ∏ NUM cana-4038 156 9 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 156 10 ) ) PUNCT cana-4038 156 11 and and CCONJ cana-4038 156 12 𝑆 𝑆 PROPN cana-4038 156 13 = = SYM cana-4038 156 14 βˆ‘ βˆ‘ PUNCT cana-4038 156 15 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 156 16 ) ) PUNCT cana-4038 156 17 /𝑒𝑑 /𝑒𝑑 PUNCT cana-4038 157 1 ∈ ∈ PROPN cana-4038 157 2 𝐸(𝐺 𝐸(𝐺 PROPN cana-4038 157 3 ) ) PUNCT cana-4038 157 4 for for ADP cana-4038 157 5 each each DET cana-4038 157 6 vertex vertex NOUN cana-4038 157 7 in in ADP cana-4038 157 8 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 157 9 ) ) PUNCT cana-4038 157 10 defineπ‘“βˆ— defineπ‘“βˆ— VERB cana-4038 157 11 : : PUNCT cana-4038 157 12 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 157 13 ) ) PUNCT cana-4038 157 14 β†’ β†’ SYM cana-4038 157 15 { { PUNCT cana-4038 157 16 0,1 0,1 NOUN cana-4038 157 17 } } PUNCT cana-4038 157 18 by by ADP cana-4038 157 19 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 157 20 ) ) PUNCT cana-4038 158 1 = = NOUN cana-4038 158 2 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 158 3 βŒ‹ βŒ‹ NOUN cana-4038 158 4 ( ( PUNCT cana-4038 158 5 mod mod PROPN cana-4038 158 6 2 2 NUM cana-4038 158 7 ) ) PUNCT cana-4038 158 8 . . PUNCT cana-4038 159 1 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 159 2 ) ) PUNCT cana-4038 159 3 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 159 4 ) ) PUNCT cana-4038 159 5 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 159 6 ) ) PUNCT cana-4038 159 7 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 159 8 ) ) PUNCT cana-4038 159 9 𝑛 𝑛 PRON cana-4038 160 1 + + NUM cana-4038 160 2 12 12 NUM cana-4038 160 3 𝑛 𝑛 NOUN cana-4038 161 1 + + NUM cana-4038 161 2 12 12 NUM cana-4038 161 3 𝑛 𝑛 DET cana-4038 161 4 𝑛 𝑛 PRON cana-4038 161 5 βˆ’ βˆ’ PROPN cana-4038 161 6 1 1 NUM cana-4038 161 7 the the DET cana-4038 161 8 above above ADJ cana-4038 161 9 labeling labeling NOUN cana-4038 161 10 satisfies satisfie NOUN cana-4038 161 11 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 161 12 ) ) PUNCT cana-4038 162 1 βˆ’ βˆ’ PROPN cana-4038 162 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 162 3 ≀ ≀ NOUN cana-4038 162 4 1 1 NUM cana-4038 162 5 and and CCONJ cana-4038 162 6 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 162 7 ) ) PUNCT cana-4038 163 1 βˆ’ βˆ’ NOUN cana-4038 163 2 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 163 3 ≀ ≀ NUM cana-4038 163 4 1 1 NUM cana-4038 163 5 . . PUNCT cana-4038 164 1 the the DET cana-4038 164 2 fan fan NOUN cana-4038 164 3 graph graph NOUN cana-4038 164 4 𝐹1,𝑛 𝐹1,𝑛 PUNCT cana-4038 164 5 admits admit VERB cana-4038 164 6 sp sp ADP cana-4038 164 7 mean mean NOUN cana-4038 164 8 e e ADJ cana-4038 164 9 - - ADJ cana-4038 164 10 cordial cordial ADJ cana-4038 164 11 labeling labeling NOUN cana-4038 164 12 . . PUNCT cana-4038 165 1 hence hence ADV cana-4038 165 2 fan fan NOUN cana-4038 165 3 graph graph NOUN cana-4038 165 4 𝐹1,𝑛 𝐹1,𝑛 PUNCT cana-4038 165 5 is be AUX cana-4038 165 6 a a DET cana-4038 165 7 sp sp NOUN cana-4038 165 8 mean mean NOUN cana-4038 165 9 e e NOUN cana-4038 165 10 - - NOUN cana-4038 165 11 cg cg NOUN cana-4038 165 12 . . PUNCT cana-4038 166 1 theorem theorem VERB cana-4038 166 2 2.13 2.13 NUM cana-4038 166 3 semi semi ADJ cana-4038 166 4 point point NOUN cana-4038 166 5 total total ADJ cana-4038 166 6 graph graph NOUN cana-4038 166 7 of of ADP cana-4038 166 8 path path NOUN cana-4038 166 9 𝑇2(𝑃𝑛)is 𝑇2(𝑃𝑛)is VERB cana-4038 166 10 a a DET cana-4038 166 11 sp sp NOUN cana-4038 166 12 mean mean NOUN cana-4038 166 13 e e NOUN cana-4038 166 14 - - NOUN cana-4038 166 15 cg cg NOUN cana-4038 166 16 . . PUNCT cana-4038 167 1 proof proof NOUN cana-4038 167 2 : : PUNCT cana-4038 167 3 let let VERB cana-4038 167 4 the the DET cana-4038 167 5 path path NOUN cana-4038 168 1 𝑃𝑛 𝑃𝑛 PROPN cana-4038 168 2 having have VERB cana-4038 168 3 the the DET cana-4038 168 4 edges edge NOUN cana-4038 168 5 π‘Ÿ1 π‘Ÿ1 NOUN cana-4038 168 6 , , PUNCT cana-4038 168 7 π‘Ÿ2 π‘Ÿ2 NOUN cana-4038 168 8 , , PUNCT cana-4038 168 9 π‘Ÿ3 π‘Ÿ3 NOUN cana-4038 168 10 … … PUNCT cana-4038 168 11 … … PUNCT cana-4038 168 12 . . PUNCT cana-4038 169 1 . . PUNCT cana-4038 170 1 , , PUNCT cana-4038 170 2 π‘Ÿπ‘›βˆ’1 π‘Ÿπ‘›βˆ’1 NOUN cana-4038 170 3 and and CCONJ cana-4038 170 4 the the DET cana-4038 170 5 vertices vertex NOUN cana-4038 170 6 𝑒1 𝑒1 NOUN cana-4038 170 7 , , PUNCT cana-4038 170 8 𝑒2 𝑒2 PROPN cana-4038 170 9 , , PUNCT cana-4038 170 10 𝑒3 𝑒3 PROPN cana-4038 170 11 , , PUNCT cana-4038 170 12 … … PUNCT cana-4038 170 13 𝑒𝑛. 𝑒𝑛. NOUN cana-4038 170 14 to to PART cana-4038 170 15 construct construct VERB cana-4038 170 16 semi semi ADJ cana-4038 170 17 point point NOUN cana-4038 170 18 total total ADJ cana-4038 170 19 graph graph NOUN cana-4038 170 20 of of ADP cana-4038 170 21 path path NOUN cana-4038 170 22 𝑇2(𝑃𝑛)from 𝑇2(𝑃𝑛)from ADP cana-4038 170 23 path path NOUN cana-4038 170 24 𝑃𝑛 𝑃𝑛 PROPN cana-4038 170 25 join join VERB cana-4038 170 26 𝑒𝑖 𝑒𝑖 PRON cana-4038 170 27 and and CCONJ cana-4038 170 28 𝑒𝑖+1 𝑒𝑖+1 NUM cana-4038 170 29 to to ADP cana-4038 170 30 a a DET cana-4038 170 31 new new ADJ cana-4038 170 32 edge edge NOUN cana-4038 170 33 𝑑𝑖 𝑑𝑖 NOUN cana-4038 170 34 by by ADP cana-4038 170 35 edges edge NOUN cana-4038 170 36 𝑒𝑖𝑑𝑖 𝑒𝑖𝑑𝑖 NOUN cana-4038 170 37 and and CCONJ cana-4038 170 38 𝑒𝑖+1𝑑𝑖 𝑒𝑖+1𝑑𝑖 PROPN cana-4038 170 39 , , PUNCT cana-4038 170 40 for for ADP cana-4038 170 41 i i PROPN cana-4038 170 42 = = NOUN cana-4038 170 43 1,2,3 1,2,3 NUM cana-4038 170 44 … … PUNCT cana-4038 170 45 … … PUNCT cana-4038 170 46 .. .. PUNCT cana-4038 170 47 , , PUNCT cana-4038 170 48 𝑛 𝑛 DET cana-4038 170 49 βˆ’ βˆ’ NOUN cana-4038 170 50 1 1 NUM cana-4038 170 51 . . PUNCT cana-4038 171 1 then then ADV cana-4038 171 2 | | ADV cana-4038 171 3 t(𝑇2(𝑃𝑛))| t(𝑇2(𝑃𝑛))| PUNCT cana-4038 172 1 = = PUNCT cana-4038 172 2 2𝑛 2𝑛 PROPN cana-4038 172 3 βˆ’ βˆ’ NOUN cana-4038 172 4 1 1 NUM cana-4038 172 5 and and CCONJ cana-4038 172 6 | | ADV cana-4038 172 7 r(𝑇2(𝑃𝑛))| r(𝑇2(𝑃𝑛))| NOUN cana-4038 172 8 = = SYM cana-4038 172 9 3𝑛 3𝑛 NUM cana-4038 172 10 βˆ’ βˆ’ NOUN cana-4038 172 11 3 3 NUM cana-4038 172 12 define define VERB cana-4038 172 13 the the DET cana-4038 172 14 labeling labeling NOUN cana-4038 172 15 function function NOUN cana-4038 172 16 𝑓 𝑓 NOUN cana-4038 172 17 : : PUNCT cana-4038 172 18 𝑅(𝐺 𝑅(𝐺 PROPN cana-4038 172 19 ) ) PUNCT cana-4038 172 20 β†’ β†’ SYM cana-4038 172 21 { { PUNCT cana-4038 172 22 1,2 1,2 NUM cana-4038 172 23 } } PUNCT cana-4038 172 24 𝑓(𝑒𝑖𝑒𝑖+1 𝑓(𝑒𝑖𝑒𝑖+1 NOUN cana-4038 172 25 ) ) PUNCT cana-4038 172 26 = = NOUN cana-4038 172 27 { { PUNCT cana-4038 172 28 2 2 NUM cana-4038 172 29 𝑖𝑓 𝑖𝑓 SYM cana-4038 172 30 𝑖 𝑖 PUNCT cana-4038 172 31 𝑖𝑠 𝑖𝑠 PROPN cana-4038 172 32 𝑒𝑣𝑒𝑛1 𝑒𝑣𝑒𝑛1 NOUN cana-4038 172 33 𝑖𝑓 𝑖𝑓 ADP cana-4038 172 34 𝑖 𝑖 PUNCT cana-4038 172 35 𝑖𝑠 𝑖𝑠 NOUN cana-4038 172 36 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-4038 172 37 𝑓(𝑒𝑖𝑒𝑖′ 𝑓(𝑒𝑖𝑒𝑖′ NOUN cana-4038 172 38 ) ) PUNCT cana-4038 172 39 = = SYM cana-4038 172 40 1 1 NUM cana-4038 172 41 𝑓(𝑒𝑖+1𝑒𝑖′ 𝑓(𝑒𝑖+1𝑒𝑖′ VERB cana-4038 172 42 ) ) PUNCT cana-4038 172 43 = = SYM cana-4038 172 44 2 2 NUM cana-4038 172 45 find find VERB cana-4038 172 46 𝑆 𝑆 PROPN cana-4038 172 47 = = SYM cana-4038 172 48 βˆ‘ βˆ‘ PUNCT cana-4038 172 49 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 172 50 ) ) PUNCT cana-4038 172 51 βˆ– βˆ– X cana-4038 172 52 𝑒𝑑 𝑒𝑑 PROPN cana-4038 172 53 ∈ ∈ PROPN cana-4038 172 54 𝐸(𝐺 𝐸(𝐺 PROPN cana-4038 172 55 ) ) PUNCT cana-4038 172 56 and and CCONJ cana-4038 172 57 𝑃 𝑃 PROPN cana-4038 172 58 = = SYM cana-4038 172 59 ∏ ∏ PROPN cana-4038 172 60 𝑓(𝑒𝑑 𝑓(𝑒𝑑 NUM cana-4038 172 61 ) ) PUNCT cana-4038 172 62 for for ADP cana-4038 172 63 each each DET cana-4038 172 64 vertex vertex NOUN cana-4038 172 65 in in ADP cana-4038 172 66 𝑇(𝐾1,𝑛 𝑇(𝐾1,𝑛 PROPN cana-4038 172 67 ) ) PUNCT cana-4038 172 68 define define VERB cana-4038 172 69 π‘“βˆ— π‘“βˆ— NOUN cana-4038 172 70 : : PUNCT cana-4038 172 71 𝑇(𝐺 𝑇(𝐺 NUM cana-4038 172 72 ) ) PUNCT cana-4038 172 73 β†’ β†’ SYM cana-4038 172 74 { { PUNCT cana-4038 172 75 0,1 0,1 NOUN cana-4038 172 76 } } PUNCT cana-4038 172 77 defined define VERB cana-4038 172 78 by by ADP cana-4038 172 79 π‘“βˆ—(𝑒 π‘“βˆ—(𝑒 PROPN cana-4038 172 80 ) ) PUNCT cana-4038 172 81 = = NOUN cana-4038 172 82 βŒŠπ‘†+𝑃2 βŒŠπ‘†+𝑃2 NOUN cana-4038 172 83 βŒ‹ βŒ‹ NOUN cana-4038 172 84 ( ( PUNCT cana-4038 172 85 mod mod PROPN cana-4038 172 86 2 2 NUM cana-4038 172 87 ) ) PUNCT cana-4038 172 88 . . PUNCT cana-4038 173 1 π‘Ÿπ‘“(1 π‘Ÿπ‘“(1 NOUN cana-4038 173 2 ) ) PUNCT cana-4038 173 3 π‘Ÿπ‘“(2 π‘Ÿπ‘“(2 NOUN cana-4038 173 4 ) ) PUNCT cana-4038 173 5 𝑑𝑓(0 𝑑𝑓(0 NOUN cana-4038 173 6 ) ) PUNCT cana-4038 173 7 𝑑𝑓(1 𝑑𝑓(1 NOUN cana-4038 173 8 ) ) PUNCT cana-4038 173 9 𝑛 𝑛 PROPN cana-4038 173 10 is be AUX cana-4038 173 11 even even ADV cana-4038 173 12 𝑛 𝑛 DET cana-4038 173 13 + + NUM cana-4038 173 14 1 1 NUM cana-4038 173 15 𝑛 𝑛 DET cana-4038 173 16 + + NUM cana-4038 173 17 1 1 NUM cana-4038 173 18 𝑛 𝑛 DET cana-4038 173 19 𝑛 𝑛 PRON cana-4038 173 20 βˆ’ βˆ’ NUM cana-4038 173 21 1 1 NUM cana-4038 173 22 𝑛 𝑛 PROPN cana-4038 173 23 is be AUX cana-4038 173 24 odd odd ADJ cana-4038 173 25 𝑛 𝑛 PRON cana-4038 173 26 + + PROPN cana-4038 173 27 1 1 NUM cana-4038 173 28 𝑛 𝑛 PRON cana-4038 173 29 + + NUM cana-4038 173 30 1 1 NUM cana-4038 173 31 𝑛 𝑛 DET cana-4038 173 32 𝑛 𝑛 DET cana-4038 173 33 βˆ’ βˆ’ PROPN cana-4038 173 34 1 1 NUM cana-4038 173 35 the the DET cana-4038 173 36 above above ADJ cana-4038 173 37 labeling labeling NOUN cana-4038 173 38 satisfies satisfie NOUN cana-4038 173 39 |π‘Ÿπ‘“(1 |π‘Ÿπ‘“(1 NOUN cana-4038 173 40 ) ) PUNCT cana-4038 174 1 βˆ’ βˆ’ NOUN cana-4038 174 2 π‘Ÿπ‘“(2)| π‘Ÿπ‘“(2)| NOUN cana-4038 174 3 ≀ ≀ NUM cana-4038 174 4 1 1 NUM cana-4038 174 5 and and CCONJ cana-4038 174 6 |𝑑𝑓(0 |𝑑𝑓(0 NUM cana-4038 174 7 ) ) PUNCT cana-4038 175 1 βˆ’ βˆ’ PROPN cana-4038 175 2 𝑑𝑓(1)| 𝑑𝑓(1)| NOUN cana-4038 175 3 ≀ ≀ ADJ cana-4038 175 4 1 1 NUM cana-4038 175 5 . . PUNCT cana-4038 175 6 ” " PUNCT cana-4038 176 1 the the DET cana-4038 176 2 semi semi ADJ cana-4038 176 3 point point NOUN cana-4038 176 4 total total ADJ cana-4038 176 5 graph graph NOUN cana-4038 176 6 of of ADP cana-4038 176 7 path path NOUN cana-4038 176 8 𝑇2(𝑃𝑛)admits 𝑇2(𝑃𝑛)admit NOUN cana-4038 176 9 sp sp ADP cana-4038 176 10 mean mean NOUN cana-4038 176 11 e e NOUN cana-4038 176 12 - - NOUN cana-4038 176 13 cl cl NOUN cana-4038 176 14 . . PUNCT cana-4038 177 1 communications communication NOUN cana-4038 177 2 on on ADP cana-4038 177 3 applied apply VERB cana-4038 177 4 nonlinear nonlinear ADJ cana-4038 177 5 analysis analysis NOUN cana-4038 177 6 issn issn NOUN cana-4038 177 7 : : PUNCT cana-4038 177 8 1074 1074 NUM cana-4038 177 9 - - PUNCT cana-4038 177 10 133x 133x NUM cana-4038 177 11 vol vol NOUN cana-4038 177 12 32 32 NUM cana-4038 177 13 no no NOUN cana-4038 177 14 . . PUNCT cana-4038 178 1 9s 9s NUM cana-4038 178 2 ( ( PUNCT cana-4038 178 3 2025 2025 NUM cana-4038 178 4 ) ) PUNCT cana-4038 178 5 909 909 NUM cana-4038 178 6 https://internationalpubls.com https://internationalpubls.com X cana-4038 178 7 hence hence ADV cana-4038 178 8 semi semi ADV cana-4038 178 9 point point VERB cana-4038 178 10 total total ADJ cana-4038 178 11 graph graph NOUN cana-4038 178 12 of of ADP cana-4038 178 13 path path NOUN cana-4038 178 14 𝑇2(𝑃𝑛)is 𝑇2(𝑃𝑛)is VERB cana-4038 178 15 a a DET cana-4038 178 16 sp sp NOUN cana-4038 178 17 mean mean NOUN cana-4038 178 18 e e NOUN cana-4038 178 19 - - NOUN cana-4038 178 20 cg cg NOUN cana-4038 178 21 . . NOUN cana-4038 179 1 3 3 X cana-4038 179 2 . . X cana-4038 179 3 conclusion conclusion NOUN cana-4038 179 4 in in ADP cana-4038 179 5 this this DET cana-4038 179 6 research research NOUN cana-4038 179 7 , , PUNCT cana-4038 179 8 we we PRON cana-4038 179 9 define define VERB cana-4038 179 10 a a DET cana-4038 179 11 new new ADJ cana-4038 179 12 labeling labeling NOUN cana-4038 179 13 based base VERB cana-4038 179 14 on on ADP cana-4038 179 15 edge edge NOUN cana-4038 179 16 is be AUX cana-4038 179 17 called call VERB cana-4038 179 18 sp sp ADP cana-4038 179 19 mean mean NOUN cana-4038 179 20 e e ADJ cana-4038 179 21 - - ADJ cana-4038 179 22 cordial cordial ADJ cana-4038 179 23 labeling labeling NOUN cana-4038 179 24 and and CCONJ cana-4038 179 25 sp sp ADP cana-4038 179 26 mean mean NOUN cana-4038 179 27 e e ADJ cana-4038 179 28 - - ADJ cana-4038 179 29 cordial cordial ADJ cana-4038 179 30 graph graph NOUN cana-4038 179 31 . . PUNCT cana-4038 180 1 here here ADV cana-4038 180 2 is be AUX cana-4038 180 3 it it PRON cana-4038 180 4 verified verify VERB cana-4038 180 5 some some PRON cana-4038 180 6 of of ADP cana-4038 180 7 the the DET cana-4038 180 8 standard standard ADJ cana-4038 180 9 graphs graph NOUN cana-4038 180 10 like like ADP cana-4038 180 11 cycle cycle NOUN cana-4038 180 12 , , PUNCT cana-4038 180 13 path path NOUN cana-4038 180 14 , , PUNCT cana-4038 180 15 wheel wheel NOUN cana-4038 180 16 , , PUNCT cana-4038 180 17 crown crown NOUN cana-4038 180 18 graph graph NOUN cana-4038 180 19 , , PUNCT cana-4038 180 20 combo combo NOUN cana-4038 180 21 graph graph NOUN cana-4038 180 22 , , PUNCT cana-4038 180 23 triangular triangular NOUN cana-4038 180 24 snake snake NOUN cana-4038 180 25 graph graph NOUN cana-4038 180 26 and and CCONJ cana-4038 180 27 semi semi ADJ cana-4038 180 28 point point NOUN cana-4038 180 29 total total ADJ cana-4038 180 30 graph graph NOUN cana-4038 180 31 are be AUX cana-4038 180 32 sp sp ADP cana-4038 180 33 mean mean NOUN cana-4038 180 34 e e ADJ cana-4038 180 35 - - ADJ cana-4038 180 36 cordial cordial ADJ cana-4038 180 37 graph graph NOUN cana-4038 180 38 . . PUNCT cana-4038 181 1 in in ADP cana-4038 181 2 future future NOUN cana-4038 181 3 we we PRON cana-4038 181 4 extend extend VERB cana-4038 181 5 this this DET cana-4038 181 6 work work NOUN cana-4038 181 7 for for ADP cana-4038 181 8 some some DET cana-4038 181 9 more more ADJ cana-4038 181 10 special special ADJ cana-4038 181 11 graphs graph NOUN cana-4038 181 12 . . PUNCT cana-4038 182 1 references reference NOUN cana-4038 182 2 1 1 NUM cana-4038 182 3 . . PUNCT cana-4038 183 1 harary harary PROPN cana-4038 183 2 , , PUNCT cana-4038 183 3 f. f. PROPN cana-4038 183 4 ( ( PUNCT cana-4038 183 5 2018 2018 NUM cana-4038 183 6 ) ) PUNCT cana-4038 183 7 . . PUNCT cana-4038 184 1 graph graph NOUN cana-4038 184 2 theory theory NOUN cana-4038 184 3 ( ( PUNCT cana-4038 184 4 on on ADP cana-4038 184 5 demand demand NOUN cana-4038 184 6 printing printing NOUN cana-4038 184 7 of of ADP cana-4038 184 8 02787 02787 NUM cana-4038 184 9 ) ) PUNCT cana-4038 184 10 . . PUNCT cana-4038 185 1 crc crc PROPN cana-4038 185 2 press press PROPN cana-4038 185 3 . . PUNCT cana-4038 186 1 2 2 X cana-4038 186 2 . . X cana-4038 186 3 harary harary NOUN cana-4038 186 4 , , PUNCT cana-4038 186 5 f. f. PROPN cana-4038 186 6 ( ( PUNCT cana-4038 186 7 1972 1972 NUM cana-4038 186 8 ) ) PUNCT cana-4038 186 9 . . PUNCT cana-4038 187 1 graph graph NOUN cana-4038 187 2 theory theory NOUN cana-4038 187 3 . . PUNCT cana-4038 188 1 addison addison PROPN cana-4038 188 2 - - PUNCT cana-4038 188 3 wesley wesley PROPN cana-4038 188 4 reading read VERB cana-4038 188 5 mass mass PROPN cana-4038 188 6 . . PUNCT cana-4038 189 1 3 3 X cana-4038 189 2 . . X cana-4038 189 3 rosa rosa PROPN cana-4038 189 4 , , PUNCT cana-4038 189 5 a. a. NOUN cana-4038 189 6 ( ( PUNCT cana-4038 189 7 1967 1967 NUM cana-4038 189 8 ) ) PUNCT cana-4038 189 9 . . PUNCT cana-4038 190 1 on on ADP cana-4038 190 2 certain certain ADJ cana-4038 190 3 valuations valuation NOUN cana-4038 190 4 of of ADP cana-4038 190 5 the the DET cana-4038 190 6 vertices vertex NOUN cana-4038 190 7 of of ADP cana-4038 190 8 a a DET cana-4038 190 9 graph graph NOUN cana-4038 190 10 . . PUNCT cana-4038 191 1 in in ADP cana-4038 191 2 theory theory NOUN cana-4038 191 3 of of ADP cana-4038 191 4 graphs graph NOUN cana-4038 191 5 ( ( PUNCT cana-4038 191 6 international international ADJ cana-4038 191 7 symposium symposium NOUN cana-4038 191 8 , , PUNCT cana-4038 191 9 rome rome PROPN cana-4038 191 10 , , PUNCT cana-4038 191 11 july july PROPN cana-4038 191 12 1966 1966 NUM cana-4038 191 13 ) ) PUNCT cana-4038 191 14 ( ( PUNCT cana-4038 191 15 pp pp ADJ cana-4038 191 16 . . PUNCT cana-4038 192 1 349 349 NUM cana-4038 192 2 - - SYM cana-4038 192 3 355 355 NUM cana-4038 192 4 ) ) PUNCT cana-4038 192 5 . . PUNCT cana-4038 193 1 gordon gordon PROPN cana-4038 193 2 and and CCONJ cana-4038 193 3 breach breach PROPN cana-4038 193 4 , , PUNCT cana-4038 193 5 n.y n.y PROPN cana-4038 193 6 . . PROPN cana-4038 193 7 , , PUNCT cana-4038 193 8 and and CCONJ cana-4038 193 9 dunod dunod PROPN cana-4038 193 10 paris paris PROPN cana-4038 193 11 . . PUNCT cana-4038 194 1 4 4 X cana-4038 194 2 . . X cana-4038 194 3 gallian gallian NOUN cana-4038 194 4 , , PUNCT cana-4038 194 5 j. j. PROPN cana-4038 194 6 a. a. PROPN cana-4038 194 7 ( ( PUNCT cana-4038 194 8 2012 2012 NUM cana-4038 194 9 ) ) PUNCT cana-4038 194 10 . . PUNCT cana-4038 195 1 graph graph NOUN cana-4038 195 2 labeling labeling NOUN cana-4038 195 3 . . PUNCT cana-4038 196 1 the the DET cana-4038 196 2 electronic electronic ADJ cana-4038 196 3 journal journal NOUN cana-4038 196 4 of of ADP cana-4038 196 5 combinatorics combinatorics PROPN cana-4038 196 6 , , PUNCT cana-4038 196 7 ds6 ds6 PROPN cana-4038 196 8 - - PUNCT cana-4038 196 9 dec dec PROPN cana-4038 196 10 . . PROPN cana-4038 196 11 5 5 NUM cana-4038 196 12 . . PUNCT cana-4038 196 13 ponraj ponraj VERB cana-4038 196 14 , , PUNCT cana-4038 196 15 r. r. PROPN cana-4038 196 16 , , PUNCT cana-4038 196 17 sivakumar sivakumar PROPN cana-4038 196 18 , , PUNCT cana-4038 196 19 m. m. NOUN cana-4038 196 20 , , PUNCT cana-4038 196 21 & & CCONJ cana-4038 196 22 sundaram sundaram PROPN cana-4038 196 23 , , PUNCT cana-4038 196 24 m. m. NOUN cana-4038 196 25 ( ( PUNCT cana-4038 196 26 2012 2012 NUM cana-4038 196 27 ) ) PUNCT cana-4038 196 28 . . PUNCT cana-4038 197 1 mean mean VERB cana-4038 197 2 cordial cordial ADJ cana-4038 197 3 labeling labeling NOUN cana-4038 197 4 of of ADP cana-4038 197 5 graphs graph NOUN cana-4038 197 6 . . PUNCT cana-4038 198 1 open open ADJ cana-4038 198 2 journal journal NOUN cana-4038 198 3 of of ADP cana-4038 198 4 discrete discrete ADJ cana-4038 198 5 mathematics mathematic NOUN cana-4038 198 6 , , PUNCT cana-4038 198 7 2(4 2(4 NUM cana-4038 198 8 ) ) PUNCT cana-4038 198 9 , , PUNCT cana-4038 198 10 145 145 NUM cana-4038 198 11 . . NOUN cana-4038 198 12 6 6 NUM cana-4038 198 13 . . X cana-4038 199 1 gayathri gayathri PROPN cana-4038 199 2 , , PUNCT cana-4038 199 3 b. b. PROPN cana-4038 199 4 , , PUNCT cana-4038 199 5 & & CCONJ cana-4038 199 6 siva siva PROPN cana-4038 199 7 kumar kumar PROPN cana-4038 199 8 , , PUNCT cana-4038 199 9 c. c. PROPN cana-4038 199 10 ( ( PUNCT cana-4038 199 11 2019 2019 NUM cana-4038 199 12 ) ) PUNCT cana-4038 199 13 . . PUNCT cana-4038 200 1 some some DET cana-4038 200 2 mean mean VERB cana-4038 200 3 cordial cordial ADJ cana-4038 200 4 labeling labeling NOUN cana-4038 200 5 of of ADP cana-4038 200 6 graphs graph NOUN cana-4038 200 7 . . PUNCT cana-4038 201 1 international international ADJ cana-4038 201 2 journal journal PROPN cana-4038 201 3 of of ADP cana-4038 201 4 scientific scientific ADJ cana-4038 201 5 research research NOUN cana-4038 201 6 and and CCONJ cana-4038 201 7 review review NOUN cana-4038 201 8 , , PUNCT cana-4038 201 9 8(7 8(7 NUM cana-4038 201 10 ) ) PUNCT cana-4038 201 11 , , PUNCT cana-4038 201 12 263 263 NUM cana-4038 201 13 - - SYM cana-4038 201 14 285 285 NUM cana-4038 201 15 . . PUNCT cana-4038 202 1 7 7 X cana-4038 202 2 . . X cana-4038 202 3 boxwala boxwala PROPN cana-4038 202 4 , , PUNCT cana-4038 202 5 s. s. PROPN cana-4038 202 6 ( ( PUNCT cana-4038 202 7 n.d n.d PROPN cana-4038 202 8 . . PROPN cana-4038 202 9 ) ) PUNCT cana-4038 202 10 . . PUNCT cana-4038 203 1 some some DET cana-4038 203 2 problems problem NOUN cana-4038 203 3 in in ADP cana-4038 203 4 cordial cordial ADJ cana-4038 203 5 labeling labeling NOUN cana-4038 203 6 of of ADP cana-4038 203 7 graph graph NOUN cana-4038 203 8 . . PUNCT cana-4038 204 1 lap lap NOUN cana-4038 204 2 lambert lambert PROPN cana-4038 204 3 academic academic ADJ cana-4038 204 4 publishing publishing NOUN cana-4038 204 5 . . PUNCT cana-4038 205 1 8 8 X cana-4038 205 2 . . PUNCT cana-4038 206 1 vaidya vaidya PROPN cana-4038 206 2 , , PUNCT cana-4038 206 3 s. s. PROPN cana-4038 206 4 k. k. PROPN cana-4038 206 5 , , PUNCT cana-4038 206 6 & & CCONJ cana-4038 206 7 kothari kothari PROPN cana-4038 206 8 , , PUNCT cana-4038 206 9 n. n. PROPN cana-4038 206 10 j. j. PROPN cana-4038 206 11 ( ( PUNCT cana-4038 206 12 2011 2011 NUM cana-4038 206 13 ) ) PUNCT cana-4038 206 14 . . PUNCT cana-4038 207 1 some some DET cana-4038 207 2 new new ADJ cana-4038 207 3 families family NOUN cana-4038 207 4 of of ADP cana-4038 207 5 line line NOUN cana-4038 207 6 graceful graceful ADJ cana-4038 207 7 graphs graph NOUN cana-4038 207 8 . . PUNCT cana-4038 208 1 international international ADJ cana-4038 208 2 journal journal NOUN cana-4038 208 3 of of ADP cana-4038 208 4 mathematics mathematic NOUN cana-4038 208 5 and and CCONJ cana-4038 208 6 scientific scientific ADJ cana-4038 208 7 computing computing NOUN cana-4038 208 8 , , PUNCT cana-4038 208 9 1(2 1(2 NUM cana-4038 208 10 ) ) PUNCT cana-4038 208 11 , , PUNCT cana-4038 208 12 26 26 NUM cana-4038 208 13 - - SYM cana-4038 208 14 28 28 NUM cana-4038 208 15 . . PUNCT cana-4038 209 1 9 9 NUM cana-4038 209 2 . . X cana-4038 209 3 william william PROPN cana-4038 209 4 , , PUNCT cana-4038 209 5 a. a. PROPN cana-4038 209 6 , , PUNCT cana-4038 209 7 rajsingh rajsingh PROPN cana-4038 209 8 , , PUNCT cana-4038 209 9 i. i. PROPN cana-4038 209 10 , , PUNCT cana-4038 209 11 & & CCONJ cana-4038 209 12 roy roy PROPN cana-4038 209 13 , , PUNCT cana-4038 209 14 s. s. PROPN cana-4038 209 15 ( ( PUNCT cana-4038 209 16 2013 2013 NUM cana-4038 209 17 ) ) PUNCT cana-4038 209 18 . . PUNCT cana-4038 210 1 mean mean VERB cana-4038 210 2 cordial cordial ADJ cana-4038 210 3 labeling labeling NOUN cana-4038 210 4 of of ADP cana-4038 210 5 certain certain ADJ cana-4038 210 6 graphs graph NOUN cana-4038 210 7 . . PUNCT cana-4038 211 1 journal journal NOUN cana-4038 211 2 of of ADP cana-4038 211 3 computer computer PROPN cana-4038 211 4 & & CCONJ cana-4038 211 5 mathematical mathematical PROPN cana-4038 211 6 sciences sciences PROPN cana-4038 211 7 , , PUNCT cana-4038 211 8 4(4 4(4 NUM cana-4038 211 9 ) ) PUNCT cana-4038 211 10 , , PUNCT cana-4038 211 11 274 274 NUM cana-4038 211 12 - - SYM cana-4038 211 13 281 281 NUM cana-4038 211 14 . . PUNCT cana-4038 212 1 10 10 NUM cana-4038 212 2 . . PUNCT cana-4038 213 1 yilmaz yilmaz PROPN cana-4038 213 2 , , PUNCT cana-4038 213 3 r. r. PROPN cana-4038 213 4 , , PUNCT cana-4038 213 5 & & CCONJ cana-4038 213 6 cahit cahit PROPN cana-4038 213 7 , , PUNCT cana-4038 213 8 i. i. NOUN cana-4038 213 9 ( ( PUNCT cana-4038 213 10 1997 1997 NUM cana-4038 213 11 ) ) PUNCT cana-4038 213 12 . . PUNCT cana-4038 214 1 e e X cana-4038 214 2 - - ADJ cana-4038 214 3 cordial cordial ADJ cana-4038 214 4 graphs graph NOUN cana-4038 214 5 . . PUNCT cana-4038 215 1 ars ars PROPN cana-4038 215 2 combinatoria combinatoria PROPN cana-4038 215 3 , , PUNCT cana-4038 215 4 46 46 NUM cana-4038 215 5 , , PUNCT cana-4038 215 6 251 251 NUM cana-4038 215 7 - - SYM cana-4038 215 8 266 266 NUM cana-4038 215 9 . . PUNCT cana-4038 216 1 11 11 NUM cana-4038 216 2 . . PUNCT cana-4038 217 1 vaidya vaidya PROPN cana-4038 217 2 , , PUNCT cana-4038 217 3 s. s. PROPN cana-4038 217 4 k. k. PROPN cana-4038 217 5 , , PUNCT cana-4038 217 6 & & CCONJ cana-4038 217 7 vyas vyas PROPN cana-4038 217 8 , , PUNCT cana-4038 217 9 n. n. PROPN cana-4038 217 10 b. b. PROPN cana-4038 217 11 ( ( PUNCT cana-4038 217 12 2012 2012 NUM cana-4038 217 13 ) ) PUNCT cana-4038 217 14 . . PUNCT cana-4038 218 1 some some DET cana-4038 218 2 results result VERB cana-4038 218 3 on on ADP cana-4038 218 4 e e ADJ cana-4038 218 5 - - ADJ cana-4038 218 6 cordial cordial ADJ cana-4038 218 7 labeling labeling NOUN cana-4038 218 8 . . PUNCT cana-4038 219 1 international international ADJ cana-4038 219 2 journal journal NOUN cana-4038 219 3 of of ADP cana-4038 219 4 mathematics mathematic NOUN cana-4038 219 5 and and CCONJ cana-4038 219 6 scientific scientific ADJ cana-4038 219 7 computing computing NOUN cana-4038 219 8 , , PUNCT cana-4038 219 9 2(1 2(1 NUM cana-4038 219 10 ) ) PUNCT cana-4038 219 11 , , PUNCT cana-4038 219 12 9 9 NUM cana-4038 219 13 - - SYM cana-4038 219 14 13 13 NUM cana-4038 219 15 . . PUNCT cana-4038 219 16 12 12 NUM cana-4038 219 17 . . PUNCT cana-4038 220 1 vaidya vaidya PROPN cana-4038 220 2 , , PUNCT cana-4038 220 3 s. s. PROPN cana-4038 220 4 k. k. PROPN cana-4038 220 5 ( ( PUNCT cana-4038 220 6 2011 2011 NUM cana-4038 220 7 ) ) PUNCT cana-4038 220 8 . . PUNCT cana-4038 221 1 some some DET cana-4038 221 2 new new ADJ cana-4038 221 3 families family NOUN cana-4038 221 4 of of ADP cana-4038 221 5 e e ADJ cana-4038 221 6 - - ADJ cana-4038 221 7 cordial cordial ADJ cana-4038 221 8 graphs graph NOUN cana-4038 221 9 . . PUNCT cana-4038 222 1 13 13 NUM cana-4038 222 2 . . PUNCT cana-4038 223 1 vaidya vaidya PROPN cana-4038 223 2 , , PUNCT cana-4038 223 3 s. s. PROPN cana-4038 223 4 k. k. PROPN cana-4038 223 5 , , PUNCT cana-4038 223 6 & & CCONJ cana-4038 223 7 lekha lekha PROPN cana-4038 223 8 bijukumar bijukumar PROPN cana-4038 223 9 . . PUNCT cana-4038 224 1 ( ( PUNCT cana-4038 224 2 2011 2011 NUM cana-4038 224 3 ) ) PUNCT cana-4038 224 4 . . PUNCT cana-4038 225 1 some some DET cana-4038 225 2 new new ADJ cana-4038 225 3 results result NOUN cana-4038 225 4 on on ADP cana-4038 225 5 e e ADJ cana-4038 225 6 - - ADJ cana-4038 225 7 cordial cordial ADJ cana-4038 225 8 labeling labeling NOUN cana-4038 225 9 . . PUNCT cana-4038 226 1 international international ADJ cana-4038 226 2 journal journal PROPN cana-4038 226 3 of of ADP cana-4038 226 4 information information NOUN cana-4038 226 5 science science NOUN cana-4038 226 6 and and CCONJ cana-4038 226 7 computer computer NOUN cana-4038 226 8 mathematics mathematic NOUN cana-4038 226 9 , , PUNCT cana-4038 226 10 3 3 NUM cana-4038 226 11 , , PUNCT cana-4038 226 12 21 21 NUM cana-4038 226 13 - - SYM cana-4038 226 14 29 29 NUM cana-4038 226 15 . . PUNCT cana-4038 226 16 14 14 NUM cana-4038 226 17 . . PUNCT cana-4038 227 1 vaidya vaidya PROPN cana-4038 227 2 , , PUNCT cana-4038 227 3 s. s. PROPN cana-4038 227 4 k. k. PROPN cana-4038 227 5 , , PUNCT cana-4038 227 6 & & CCONJ cana-4038 227 7 vyas vyas PROPN cana-4038 227 8 , , PUNCT cana-4038 227 9 n. n. PROPN cana-4038 227 10 b. b. PROPN cana-4038 227 11 ( ( PUNCT cana-4038 227 12 2011 2011 NUM cana-4038 227 13 ) ) PUNCT cana-4038 227 14 . . PUNCT cana-4038 228 1 e e X cana-4038 228 2 - - ADJ cana-4038 228 3 cordial cordial ADJ cana-4038 228 4 labeling labeling NOUN cana-4038 228 5 of of ADP cana-4038 228 6 some some DET cana-4038 228 7 mirror mirror NOUN cana-4038 228 8 graphs graph NOUN cana-4038 228 9 . . PUNCT cana-4038 229 1 international international ADJ cana-4038 229 2 journal journal PROPN cana-4038 229 3 of of ADP cana-4038 229 4 contemporary contemporary ADJ cana-4038 229 5 advanced advanced ADJ cana-4038 229 6 mathematics mathematic NOUN cana-4038 229 7 , , PUNCT cana-4038 229 8 2(1 2(1 NUM cana-4038 229 9 ) ) PUNCT cana-4038 229 10 , , PUNCT cana-4038 229 11 22 22 NUM cana-4038 229 12 - - SYM cana-4038 229 13 27 27 NUM cana-4038 229 14 . . PUNCT cana-4038 229 15 15 15 NUM cana-4038 229 16 . . PUNCT cana-4038 230 1 vaidya vaidya PROPN cana-4038 230 2 , , PUNCT cana-4038 230 3 s. s. PROPN cana-4038 230 4 k. k. PROPN cana-4038 230 5 , , PUNCT cana-4038 230 6 & & CCONJ cana-4038 230 7 vyas vyas PROPN cana-4038 230 8 , , PUNCT cana-4038 230 9 n. n. PROPN cana-4038 230 10 b. b. PROPN cana-4038 230 11 ( ( PUNCT cana-4038 230 12 2011 2011 NUM cana-4038 230 13 ) ) PUNCT cana-4038 230 14 . . PUNCT cana-4038 231 1 e e X cana-4038 231 2 - - ADJ cana-4038 231 3 cordial cordial ADJ cana-4038 231 4 labeling labeling NOUN cana-4038 231 5 for for ADP cana-4038 231 6 cartesian cartesian ADJ cana-4038 231 7 product product NOUN cana-4038 231 8 of of ADP cana-4038 231 9 some some DET cana-4038 231 10 graphs graph NOUN cana-4038 231 11 . . PUNCT cana-4038 232 1 studies study NOUN cana-4038 232 2 in in ADP cana-4038 232 3 mathematical mathematical ADJ cana-4038 232 4 sciences science NOUN cana-4038 232 5 , , PUNCT cana-4038 232 6 2 2 NUM cana-4038 232 7 , , PUNCT cana-4038 232 8 11 11 NUM cana-4038 232 9 - - SYM cana-4038 232 10 15 15 NUM cana-4038 232 11 . . NOUN cana-4038 232 12 16 16 NUM cana-4038 232 13 . . PUNCT cana-4038 233 1 wavelet wavelet PROPN cana-4038 233 2 , , PUNCT cana-4038 233 3 d. d. PROPN cana-4038 233 4 ( ( PUNCT cana-4038 233 5 2010 2010 NUM cana-4038 233 6 ) ) PUNCT cana-4038 233 7 . . PUNCT cana-4038 234 1 a a DET cana-4038 234 2 study study NOUN cana-4038 234 3 on on ADP cana-4038 234 4 e e ADJ cana-4038 234 5 - - ADJ cana-4038 234 6 cordial cordial ADJ cana-4038 234 7 labeling labeling NOUN cana-4038 234 8 of of ADP cana-4038 234 9 graphs graph NOUN cana-4038 234 10 ( ( PUNCT cana-4038 234 11 ph.d ph.d PROPN cana-4038 234 12 . . PUNCT cana-4038 235 1 thesis thesis NOUN cana-4038 235 2 ) ) PUNCT cana-4038 235 3 . . PUNCT cana-4038 236 1 manonmaniam manonmaniam PROPN cana-4038 236 2 sundaranar sundaranar PROPN cana-4038 236 3 university university PROPN cana-4038 236 4 . . PUNCT cana-4038 237 1 17 17 NUM cana-4038 237 2 . . PUNCT cana-4038 238 1 cahit cahit ADJ cana-4038 238 2 , , PUNCT cana-4038 238 3 i. i. NOUN cana-4038 238 4 ( ( PUNCT cana-4038 238 5 1987 1987 NUM cana-4038 238 6 ) ) PUNCT cana-4038 238 7 . . PUNCT cana-4038 239 1 cordial cordial ADJ cana-4038 239 2 graphs graph NOUN cana-4038 239 3 β€” β€” PUNCT cana-4038 239 4 a a DET cana-4038 239 5 weaker weak ADJ cana-4038 239 6 version version NOUN cana-4038 239 7 of of ADP cana-4038 239 8 graceful graceful ADJ cana-4038 239 9 and and CCONJ cana-4038 239 10 harmonious harmonious ADJ cana-4038 239 11 graphs graph NOUN cana-4038 239 12 . . PUNCT cana-4038 240 1 ars ar NOUN cana-4038 240 2 combinatoria combinatoria PROPN cana-4038 240 3 , , PUNCT cana-4038 240 4 23 23 NUM cana-4038 240 5 , , PUNCT cana-4038 240 6 201 201 NUM cana-4038 240 7 - - SYM cana-4038 240 8 207 207 NUM cana-4038 240 9 . . PUNCT