id sid tid token lemma pos cana-3856 1 1 communications communication NOUN cana-3856 1 2 on on ADP cana-3856 1 3 applied apply VERB cana-3856 1 4 nonlinear nonlinear ADJ cana-3856 1 5 analysis analysis NOUN cana-3856 1 6 issn issn NOUN cana-3856 1 7 : : PUNCT cana-3856 1 8 1074 1074 NUM cana-3856 1 9 - - PUNCT cana-3856 1 10 133x 133x NUM cana-3856 1 11 vol vol NOUN cana-3856 1 12 32 32 NUM cana-3856 1 13 no no NOUN cana-3856 1 14 . . PUNCT cana-3856 2 1 9s 9s NUM cana-3856 2 2 ( ( PUNCT cana-3856 2 3 2025 2025 NUM cana-3856 2 4 ) ) PUNCT cana-3856 2 5 284 284 NUM cana-3856 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 2 7 on on ADP cana-3856 2 8 sombor sombor NOUN cana-3856 2 9 energy energy NOUN cana-3856 2 10 of of ADP cana-3856 2 11 graphs graph NOUN cana-3856 2 12 with with ADP cana-3856 2 13 self self NOUN cana-3856 2 14 - - PUNCT cana-3856 2 15 loops loop NOUN cana-3856 2 16 s. s. PROPN cana-3856 2 17 h. h. PROPN cana-3856 2 18 pathan pathan PROPN cana-3856 2 19 1 1 PROPN cana-3856 2 20 and and CCONJ cana-3856 2 21 s. s. PROPN cana-3856 2 22 c. c. PROPN cana-3856 2 23 patekar patekar PROPN cana-3856 2 24 2 2 NUM cana-3856 2 25 1,2 1,2 NUM cana-3856 2 26 department department NOUN cana-3856 2 27 of of ADP cana-3856 2 28 mathematics mathematic NOUN cana-3856 2 29 , , PUNCT cana-3856 2 30 savitribai savitribai VERB cana-3856 2 31 phule phule PROPN cana-3856 2 32 pune pune PROPN cana-3856 2 33 university university PROPN cana-3856 2 34 , , PUNCT cana-3856 2 35 pune pune NOUN cana-3856 2 36 ( ( PUNCT cana-3856 2 37 india india PROPN cana-3856 2 38 ) ) PUNCT cana-3856 2 39 1samrinpathan508@gmail.com 1samrinpathan508@gmail.com NUM cana-3856 2 40 , , PUNCT cana-3856 2 41 2shri82patekar@gmail.com 2shri82patekar@gmail.com NUM cana-3856 2 42 article article NOUN cana-3856 2 43 history history NOUN cana-3856 2 44 : : PUNCT cana-3856 2 45 received receive VERB cana-3856 2 46 : : PUNCT cana-3856 2 47 14 14 NUM cana-3856 2 48 - - SYM cana-3856 2 49 11 11 NUM cana-3856 2 50 - - PUNCT cana-3856 2 51 2024 2024 NUM cana-3856 2 52 revised:26 revised:26 PROPN cana-3856 2 53 - - PUNCT cana-3856 2 54 12 12 NUM cana-3856 2 55 - - PUNCT cana-3856 2 56 2024 2024 NUM cana-3856 2 57 accepted:10 accepted:10 PROPN cana-3856 2 58 - - PUNCT cana-3856 2 59 01 01 NUM cana-3856 2 60 - - PUNCT cana-3856 2 61 2025 2025 NUM cana-3856 2 62 abstract abstract NOUN cana-3856 2 63 : : PUNCT cana-3856 2 64 the the DET cana-3856 2 65 goal goal NOUN cana-3856 2 66 of of ADP cana-3856 2 67 this this DET cana-3856 2 68 paper paper NOUN cana-3856 2 69 is be AUX cana-3856 2 70 to to PART cana-3856 2 71 broaden broaden VERB cana-3856 2 72 the the DET cana-3856 2 73 concept concept NOUN cana-3856 2 74 of of ADP cana-3856 2 75 sombor sombor NOUN cana-3856 2 76 energy energy NOUN cana-3856 2 77 from from ADP cana-3856 2 78 a a DET cana-3856 2 79 simple simple ADJ cana-3856 2 80 graph graph NOUN cana-3856 2 81 to to ADP cana-3856 2 82 one one NUM cana-3856 2 83 containing contain VERB cana-3856 2 84 self self NOUN cana-3856 2 85 - - PUNCT cana-3856 2 86 loops loop NOUN cana-3856 2 87 . . PUNCT cana-3856 3 1 let let VERB cana-3856 3 2 g g PRON cana-3856 3 3 be be AUX cana-3856 3 4 a a DET cana-3856 3 5 simple simple ADJ cana-3856 3 6 nth nth NOUN cana-3856 3 7 - - PUNCT cana-3856 3 8 order order NOUN cana-3856 3 9 graph graph NOUN cana-3856 3 10 , , PUNCT cana-3856 3 11 and and CCONJ cana-3856 3 12 𝐺𝑠 𝐺𝑠 PROPN cana-3856 3 13 be be AUX cana-3856 3 14 the the DET cana-3856 3 15 graph graph NOUN cana-3856 3 16 generated generate VERB cana-3856 3 17 by by ADP cana-3856 3 18 adding add VERB cana-3856 3 19 𝜎 𝜎 PRON cana-3856 3 20 self self NOUN cana-3856 3 21 - - PUNCT cana-3856 3 22 loops loop NOUN cana-3856 3 23 to to ADP cana-3856 3 24 g. g. PROPN cana-3856 3 25 sombor sombor NOUN cana-3856 3 26 matrix matrix NOUN cana-3856 3 27 of of ADP cana-3856 3 28 𝐺𝑠 𝐺𝑠 PROPN cana-3856 3 29 is be AUX cana-3856 3 30 defined define VERB cana-3856 3 31 as as ADP cana-3856 3 32 𝐴𝑆𝑂(𝐺𝑠 𝐴𝑆𝑂(𝐺𝑠 PROPN cana-3856 3 33 ) ) PUNCT cana-3856 3 34 = = SYM cana-3856 3 35 ( ( PUNCT cana-3856 3 36 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-3856 3 37 ) ) PUNCT cana-3856 3 38 = = PRON cana-3856 3 39 { { PUNCT cana-3856 3 40 βˆšπ‘‘π‘– βˆšπ‘‘π‘– NOUN cana-3856 3 41 2 2 NUM cana-3856 3 42 + + CCONJ cana-3856 3 43 𝑑𝑗 𝑑𝑗 PROPN cana-3856 3 44 2 2 NUM cana-3856 3 45 ; ; PUNCT cana-3856 3 46 𝑖𝑓𝑣𝑖 𝑖𝑓𝑣𝑖 PROPN cana-3856 3 47 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-3856 3 48 π‘£π‘—π‘Žπ‘Ÿπ‘’ π‘£π‘—π‘Žπ‘Ÿπ‘’ PROPN cana-3856 3 49 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ PROPN cana-3856 3 50 √2𝑑𝑖 √2𝑑𝑖 PROPN cana-3856 3 51 ; ; PUNCT cana-3856 3 52 𝑖𝑓 𝑖𝑓 ADP cana-3856 3 53 𝑖 𝑖 X cana-3856 4 1 = = PUNCT cana-3856 4 2 𝑗 𝑗 PROPN cana-3856 4 3 0 0 NUM cana-3856 4 4 ; ; PUNCT cana-3856 4 5 𝑖𝑓𝑣𝑖 𝑖𝑓𝑣𝑖 X cana-3856 4 6 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-3856 4 7 𝑣𝑗 𝑣𝑗 ADP cana-3856 4 8 π‘Žπ‘Ÿπ‘’ π‘Žπ‘Ÿπ‘’ PROPN cana-3856 4 9 π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ X cana-3856 4 10 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘. π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘. NOUN cana-3856 4 11 if if SCONJ cana-3856 4 12 πœ†1(𝐺𝑠 πœ†1(𝐺𝑠 NOUN cana-3856 4 13 ) ) PUNCT cana-3856 4 14 , , PUNCT cana-3856 4 15 πœ†2(𝐺𝑠 πœ†2(𝐺𝑠 NUM cana-3856 4 16 ) ) PUNCT cana-3856 4 17 , , PUNCT cana-3856 4 18 . . PUNCT cana-3856 4 19 . . PUNCT cana-3856 4 20 . . PUNCT cana-3856 5 1 , , PUNCT cana-3856 5 2 πœ†π‘›(𝐺𝑠 πœ†π‘›(𝐺𝑠 NOUN cana-3856 5 3 ) ) PUNCT cana-3856 5 4 are be AUX cana-3856 5 5 eigenvalues eigenvalue NOUN cana-3856 5 6 of of ADP cana-3856 5 7 𝐴𝑆𝑂(𝐺𝑠 𝐴𝑆𝑂(𝐺𝑠 NOUN cana-3856 5 8 ) ) PUNCT cana-3856 5 9 , , PUNCT cana-3856 5 10 then then ADV cana-3856 5 11 sombor sombor VERB cana-3856 5 12 energy energy NOUN cana-3856 5 13 of of ADP cana-3856 5 14 𝐺𝑠 𝐺𝑠 PROPN cana-3856 5 15 is be AUX cana-3856 5 16 defined define VERB cana-3856 5 17 as as ADP cana-3856 5 18 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 NOUN cana-3856 5 19 ) ) PUNCT cana-3856 5 20 = = PUNCT cana-3856 5 21 βˆ‘ βˆ‘ PUNCT cana-3856 5 22 𝑛 𝑛 DET cana-3856 5 23 𝑖=1 𝑖=1 PROPN cana-3856 5 24 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 NOUN cana-3856 5 25 ) ) PUNCT cana-3856 5 26 βˆ’ βˆ’ PROPN cana-3856 5 27 √2βˆ‘ √2βˆ‘ NOUN cana-3856 5 28 𝜎 𝜎 PROPN cana-3856 5 29 𝑗=1𝑑𝑗 𝑗=1𝑑𝑗 X cana-3856 6 1 𝑛 𝑛 ADP cana-3856 6 2 | | INTJ cana-3856 6 3 where where SCONJ cana-3856 6 4 𝑑𝑗 𝑑𝑗 PROPN cana-3856 6 5 is be AUX cana-3856 6 6 the the DET cana-3856 6 7 degree degree NOUN cana-3856 6 8 of of ADP cana-3856 6 9 vertex vertex NOUN cana-3856 6 10 𝑣𝑗 𝑣𝑗 ADP cana-3856 6 11 with with ADP cana-3856 6 12 self self NOUN cana-3856 6 13 loop loop NOUN cana-3856 6 14 keywords keyword NOUN cana-3856 6 15 : : PUNCT cana-3856 6 16 eigenvalue eigenvalue NOUN cana-3856 6 17 , , PUNCT cana-3856 6 18 sombor sombor NOUN cana-3856 6 19 energy energy NOUN cana-3856 6 20 , , PUNCT cana-3856 6 21 self self NOUN cana-3856 6 22 - - PUNCT cana-3856 6 23 loops loop NOUN cana-3856 6 24 . . PUNCT cana-3856 7 1 1 1 X cana-3856 7 2 . . X cana-3856 7 3 introduction introduction NOUN cana-3856 7 4 let let VERB cana-3856 7 5 𝐺 𝐺 PROPN cana-3856 7 6 be be AUX cana-3856 7 7 a a DET cana-3856 7 8 simple simple ADJ cana-3856 7 9 graph graph NOUN cana-3856 7 10 with with ADP cana-3856 7 11 vertex vertex NOUN cana-3856 7 12 set set VERB cana-3856 7 13 𝑉(𝐺 𝑉(𝐺 NOUN cana-3856 7 14 ) ) PUNCT cana-3856 7 15 and and CCONJ cana-3856 7 16 edge edge VERB cana-3856 7 17 set set VERB cana-3856 7 18 𝐸(𝐺 𝐸(𝐺 PROPN cana-3856 7 19 ) ) PUNCT cana-3856 7 20 , , PUNCT cana-3856 7 21 |𝑉(𝐺)| |𝑉(𝐺)| X cana-3856 7 22 = = SYM cana-3856 7 23 𝑛. 𝑛. NOUN cana-3856 7 24 if if SCONJ cana-3856 7 25 the the DET cana-3856 7 26 vertices vertex NOUN cana-3856 7 27 𝑒 𝑒 PART cana-3856 7 28 , , PUNCT cana-3856 7 29 𝑣 𝑣 DET cana-3856 7 30 ∈ ∈ PROPN cana-3856 7 31 𝑉(𝐺 𝑉(𝐺 NOUN cana-3856 7 32 ) ) PUNCT cana-3856 7 33 are be AUX cana-3856 7 34 adjacent adjacent ADJ cana-3856 7 35 , , PUNCT cana-3856 7 36 then then ADV cana-3856 7 37 edge edge VERB cana-3856 7 38 with with ADP cana-3856 7 39 end end NOUN cana-3856 7 40 points point NOUN cana-3856 7 41 are be AUX cana-3856 7 42 𝑒 𝑒 PRON cana-3856 7 43 and and CCONJ cana-3856 7 44 𝑣 𝑣 PROPN cana-3856 7 45 is be AUX cana-3856 7 46 denoted denote VERB cana-3856 7 47 by by ADP cana-3856 7 48 𝑒𝑣. 𝑒𝑣. PROPN cana-3856 7 49 the the DET cana-3856 7 50 neighbor neighbor NOUN cana-3856 7 51 of of ADP cana-3856 7 52 𝑒 𝑒 PROPN cana-3856 7 53 is be AUX cana-3856 7 54 denoted denote VERB cana-3856 7 55 by by ADP cana-3856 7 56 𝑁(𝑒 𝑁(𝑒 NOUN cana-3856 7 57 ) ) PUNCT cana-3856 7 58 that that PRON cana-3856 7 59 is be AUX cana-3856 7 60 the the DET cana-3856 7 61 set set NOUN cana-3856 7 62 of of ADP cana-3856 7 63 vertices vertex NOUN cana-3856 7 64 adjacent adjacent ADJ cana-3856 7 65 to to ADP cana-3856 7 66 𝑒 𝑒 NOUN cana-3856 7 67 , , PUNCT cana-3856 7 68 |𝑁(𝑒)| |𝑁(𝑒)| PROPN cana-3856 7 69 is be AUX cana-3856 7 70 total total ADJ cana-3856 7 71 number number NOUN cana-3856 7 72 of of ADP cana-3856 7 73 adjacent adjacent ADJ cana-3856 7 74 vertices vertex NOUN cana-3856 7 75 to to ADP cana-3856 7 76 𝑒 𝑒 PRON cana-3856 7 77 and and CCONJ cana-3856 7 78 is be AUX cana-3856 7 79 called call VERB cana-3856 7 80 degree degree NOUN cana-3856 7 81 of of ADP cana-3856 7 82 𝑒 𝑒 PRON cana-3856 7 83 and and CCONJ cana-3856 7 84 denoted denote VERB cana-3856 7 85 by by ADP cana-3856 7 86 𝑑𝐺(𝑒 𝑑𝐺(𝑒 PROPN cana-3856 7 87 ) ) PUNCT cana-3856 7 88 . . PUNCT cana-3856 8 1 let let VERB cana-3856 8 2 𝐴(𝐺 𝐴(𝐺 PROPN cana-3856 8 3 ) ) PUNCT cana-3856 8 4 be be VERB cana-3856 8 5 the the DET cana-3856 8 6 adjacency adjacency NOUN cana-3856 8 7 matrix matrix NOUN cana-3856 8 8 of of ADP cana-3856 8 9 a a DET cana-3856 8 10 simple simple ADJ cana-3856 8 11 graph graph NOUN cana-3856 8 12 𝐺 𝐺 PROPN cana-3856 8 13 with with ADP cana-3856 8 14 vertices vertex NOUN cana-3856 8 15 𝑣1 𝑣1 PROPN cana-3856 8 16 , , PUNCT cana-3856 8 17 𝑣2 𝑣2 PROPN cana-3856 8 18 , , PUNCT cana-3856 8 19 . . PUNCT cana-3856 8 20 . . PUNCT cana-3856 9 1 . . PUNCT cana-3856 10 1 , , PUNCT cana-3856 10 2 𝑣𝑛 𝑣𝑛 NOUN cana-3856 10 3 , , PUNCT cana-3856 10 4 elements element NOUN cana-3856 10 5 of of ADP cana-3856 10 6 adjacency adjacency NOUN cana-3856 10 7 matrix matrix NOUN cana-3856 10 8 are be AUX cana-3856 10 9 defined define VERB cana-3856 10 10 by by ADP cana-3856 10 11 𝐴(𝐺 𝐴(𝐺 NOUN cana-3856 10 12 ) ) PUNCT cana-3856 11 1 = = SYM cana-3856 11 2 ( ( PUNCT cana-3856 11 3 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-3856 11 4 ) ) PUNCT cana-3856 11 5 = = NOUN cana-3856 11 6 { { PUNCT cana-3856 11 7 1 1 NUM cana-3856 11 8 ; ; PUNCT cana-3856 11 9 𝑖𝑓 𝑖𝑓 NUM cana-3856 11 10 𝑣𝑖 𝑣𝑖 ADV cana-3856 11 11 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-3856 11 12 𝑣𝑗 𝑣𝑗 ADP cana-3856 11 13 π‘Žπ‘Ÿπ‘’ π‘Žπ‘Ÿπ‘’ PROPN cana-3856 11 14 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ PROPN cana-3856 11 15 0 0 NUM cana-3856 11 16 ; ; PUNCT cana-3856 11 17 𝑖𝑓 𝑖𝑓 NUM cana-3856 11 18 𝑣𝑖 𝑣𝑖 ADV cana-3856 11 19 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-3856 11 20 𝑣𝑗 𝑣𝑗 ADP cana-3856 11 21 π‘Žπ‘Ÿπ‘’ π‘Žπ‘Ÿπ‘’ PROPN cana-3856 11 22 π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ PROPN cana-3856 11 23 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘. π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘. NOUN cana-3856 11 24 and and CCONJ cana-3856 11 25 let let VERB cana-3856 11 26 πœ†1 πœ†1 NOUN cana-3856 11 27 , , PUNCT cana-3856 11 28 πœ†2 πœ†2 PROPN cana-3856 11 29 , , PUNCT cana-3856 11 30 . . PUNCT cana-3856 11 31 . . PUNCT cana-3856 11 32 . . PUNCT cana-3856 12 1 , , PUNCT cana-3856 12 2 πœ†π‘› πœ†π‘› ADP cana-3856 12 3 be be AUX cana-3856 12 4 the the DET cana-3856 12 5 eigenvalues eigenvalue NOUN cana-3856 12 6 of of ADP cana-3856 12 7 matrix matrix NOUN cana-3856 12 8 𝐴(𝐺 𝐴(𝐺 NOUN cana-3856 12 9 ) ) PUNCT cana-3856 12 10 . . PUNCT cana-3856 13 1 the the DET cana-3856 13 2 energy energy NOUN cana-3856 13 3 of of ADP cana-3856 13 4 simple simple ADJ cana-3856 13 5 graph graph NOUN cana-3856 13 6 , , PUNCT cana-3856 13 7 introduced introduce VERB cana-3856 13 8 by by ADP cana-3856 13 9 i. i. PROPN cana-3856 13 10 gutman gutman PROPN cana-3856 13 11 [ [ X cana-3856 13 12 3 3 NUM cana-3856 13 13 ] ] PUNCT cana-3856 13 14 , , PUNCT cana-3856 13 15 is be AUX cana-3856 13 16 defined define VERB cana-3856 13 17 as as ADP cana-3856 13 18 πœ€(𝐺 πœ€(𝐺 VERB cana-3856 13 19 ) ) PUNCT cana-3856 14 1 = = SYM cana-3856 14 2 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 14 3 |πœ†π‘–| |πœ†π‘–| PROPN cana-3856 14 4 . . PUNCT cana-3856 15 1 let let VERB cana-3856 15 2 s s PRON cana-3856 15 3 be be AUX cana-3856 15 4 a a DET cana-3856 15 5 subset subset NOUN cana-3856 15 6 of of ADP cana-3856 15 7 v(g v(g PROPN cana-3856 15 8 ) ) PUNCT cana-3856 15 9 . . PUNCT cana-3856 16 1 the the DET cana-3856 16 2 number number NOUN cana-3856 16 3 of of ADP cana-3856 16 4 element element NOUN cana-3856 16 5 of of ADP cana-3856 16 6 s s PRON cana-3856 16 7 will will AUX cana-3856 16 8 be be AUX cana-3856 16 9 denoted denote VERB cana-3856 16 10 by by ADP cana-3856 16 11 𝜎. 𝜎. NOUN cana-3856 16 12 let let VERB cana-3856 16 13 𝐺𝑆 𝐺𝑆 PROPN cana-3856 16 14 is be AUX cana-3856 16 15 graph graph NOUN cana-3856 16 16 with with ADP cana-3856 16 17 𝜎 𝜎 PRON cana-3856 16 18 self self NOUN cana-3856 16 19 - - PUNCT cana-3856 16 20 loops loop NOUN cana-3856 16 21 . . PUNCT cana-3856 17 1 because because SCONJ cana-3856 17 2 graphs graph NOUN cana-3856 17 3 containing contain VERB cana-3856 17 4 self self NOUN cana-3856 17 5 - - PUNCT cana-3856 17 6 loops loop NOUN cana-3856 17 7 are be AUX cana-3856 17 8 useful useful ADJ cana-3856 17 9 in in ADP cana-3856 17 10 chemistry chemistry NOUN cana-3856 17 11 , , PUNCT cana-3856 17 12 [ [ X cana-3856 17 13 heteroatoms heteroatom NOUN cana-3856 17 14 , , PUNCT cana-3856 17 15 heteroconjugated heteroconjugate VERB cana-3856 17 16 , , PUNCT cana-3856 17 17 chemistry chemistry NOUN cana-3856 17 18 , , PUNCT cana-3856 17 19 molecules molecule NOUN cana-3856 17 20 ] ] PUNCT cana-3856 17 21 . . PUNCT cana-3856 18 1 in in ADP cana-3856 18 2 2021 2021 NUM cana-3856 18 3 , , PUNCT cana-3856 18 4 gutman gutman NOUN cana-3856 18 5 defined define VERB cana-3856 18 6 the the DET cana-3856 18 7 adjacency adjacency NOUN cana-3856 18 8 matrix matrix NOUN cana-3856 18 9 𝐴(𝐺𝑆 𝐴(𝐺𝑆 PROPN cana-3856 18 10 ) ) PUNCT cana-3856 19 1 [ [ X cana-3856 19 2 9 9 NUM cana-3856 19 3 ] ] PUNCT cana-3856 19 4 of of ADP cana-3856 19 5 the the DET cana-3856 19 6 graph graph NOUN cana-3856 19 7 𝐺𝑆. 𝐺𝑆. ADP cana-3856 19 8 mailto:samrinpathan508@gmail.com mailto:samrinpathan508@gmail.com X cana-3856 19 9 mailto:shri82patekar@gmail.com mailto:shri82patekar@gmail.com NOUN cana-3856 19 10 communications communication NOUN cana-3856 19 11 on on ADP cana-3856 19 12 applied apply VERB cana-3856 19 13 nonlinear nonlinear ADJ cana-3856 19 14 analysis analysis NOUN cana-3856 19 15 issn issn NOUN cana-3856 19 16 : : PUNCT cana-3856 19 17 1074 1074 NUM cana-3856 19 18 - - PUNCT cana-3856 19 19 133x 133x NUM cana-3856 19 20 vol vol NOUN cana-3856 19 21 32 32 NUM cana-3856 19 22 no no NOUN cana-3856 19 23 . . PUNCT cana-3856 20 1 9s 9s NUM cana-3856 20 2 ( ( PUNCT cana-3856 20 3 2025 2025 NUM cana-3856 20 4 ) ) PUNCT cana-3856 20 5 285 285 NUM cana-3856 20 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 20 7 𝐴(𝐺𝑆 𝐴(𝐺𝑆 PROPN cana-3856 20 8 ) ) PUNCT cana-3856 20 9 = = SYM cana-3856 20 10 ( ( PUNCT cana-3856 20 11 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-3856 20 12 ) ) PUNCT cana-3856 20 13 = = NOUN cana-3856 20 14 { { PUNCT cana-3856 20 15 1 1 NUM cana-3856 20 16 ; ; PUNCT cana-3856 20 17 𝑖𝑓 𝑖𝑓 NUM cana-3856 20 18 𝑣𝑖 𝑣𝑖 ADV cana-3856 20 19 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-3856 20 20 𝑣𝑗 𝑣𝑗 ADP cana-3856 20 21 π‘Žπ‘Ÿπ‘’ π‘Žπ‘Ÿπ‘’ PROPN cana-3856 20 22 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ PROPN cana-3856 20 23 1 1 NUM cana-3856 20 24 ; ; PUNCT cana-3856 20 25 𝑖𝑓𝑣𝑖 𝑖𝑓𝑣𝑖 PROPN cana-3856 20 26 ∈ ∈ PROPN cana-3856 20 27 𝑆 𝑆 PROPN cana-3856 20 28 0 0 NUM cana-3856 20 29 ; ; PUNCT cana-3856 20 30 𝑖𝑓 𝑖𝑓 NUM cana-3856 20 31 𝑣𝑖 𝑣𝑖 ADV cana-3856 20 32 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-3856 20 33 𝑣𝑗 𝑣𝑗 ADP cana-3856 20 34 π‘Žπ‘Ÿπ‘’ π‘Žπ‘Ÿπ‘’ PROPN cana-3856 20 35 π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ PROPN cana-3856 20 36 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘. π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘. PROPN cana-3856 20 37 definition definition NOUN cana-3856 20 38 1 1 X cana-3856 20 39 . . PUNCT cana-3856 21 1 [ [ X cana-3856 21 2 9 9 NUM cana-3856 21 3 ] ] PUNCT cana-3856 21 4 let let VERB cana-3856 21 5 πœ†1 πœ†1 NOUN cana-3856 21 6 , , PUNCT cana-3856 21 7 πœ†2 πœ†2 PROPN cana-3856 21 8 , , PUNCT cana-3856 21 9 . . PUNCT cana-3856 21 10 . . PUNCT cana-3856 22 1 . . PUNCT cana-3856 23 1 , , PUNCT cana-3856 23 2 πœ†π‘› πœ†π‘› ADP cana-3856 23 3 be be AUX cana-3856 23 4 the the DET cana-3856 23 5 eigenvalues eigenvalue NOUN cana-3856 23 6 of of ADP cana-3856 23 7 matrix matrix NOUN cana-3856 23 8 𝐴(𝐺𝑆 𝐴(𝐺𝑆 NOUN cana-3856 23 9 ) ) PUNCT cana-3856 23 10 such such ADJ cana-3856 23 11 that that SCONJ cana-3856 23 12 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 23 13 πœ†π‘– πœ†π‘– PROPN cana-3856 23 14 = = SYM cana-3856 23 15 𝜎 𝜎 PROPN cana-3856 23 16 then then ADV cana-3856 23 17 energy energy NOUN cana-3856 23 18 of of ADP cana-3856 23 19 graph graph NOUN cana-3856 23 20 with with ADP cana-3856 23 21 𝜎 𝜎 PROPN cana-3856 23 22 self self NOUN cana-3856 23 23 loop loop NOUN cana-3856 23 24 is be AUX cana-3856 23 25 given give VERB cana-3856 23 26 by by ADP cana-3856 23 27 𝐸(𝐺𝑆 𝐸(𝐺𝑆 NOUN cana-3856 23 28 ) ) PUNCT cana-3856 24 1 = = SYM cana-3856 24 2 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 24 3 |πœ†π‘– |πœ†π‘– ADJ cana-3856 24 4 βˆ’ βˆ’ PROPN cana-3856 24 5 𝜎 𝜎 VERB cana-3856 25 1 𝑛 𝑛 VERB cana-3856 25 2 | | NOUN cana-3856 25 3 . . PUNCT cana-3856 26 1 definition definition NOUN cana-3856 26 2 2 2 NUM cana-3856 26 3 . . PUNCT cana-3856 27 1 [ [ X cana-3856 27 2 7 7 X cana-3856 27 3 ] ] X cana-3856 27 4 let let VERB cana-3856 27 5 𝐺 𝐺 PROPN cana-3856 27 6 be be AUX cana-3856 27 7 a a DET cana-3856 27 8 graph graph NOUN cana-3856 27 9 . . PUNCT cana-3856 28 1 if if SCONJ cana-3856 28 2 𝑒 𝑒 X cana-3856 28 3 , , PUNCT cana-3856 28 4 𝑣 𝑣 PRON cana-3856 28 5 ∈ ∈ PROPN cana-3856 28 6 𝑉(𝐺 𝑉(𝐺 NOUN cana-3856 28 7 ) ) PUNCT cana-3856 28 8 and and CCONJ cana-3856 28 9 𝑒𝑣 𝑒𝑣 PROPN cana-3856 28 10 ∈ ∈ PROPN cana-3856 28 11 𝐸(𝐺 𝐸(𝐺 NOUN cana-3856 28 12 ) ) PUNCT cana-3856 28 13 , , PUNCT cana-3856 28 14 then then ADV cana-3856 28 15 sombor sombor NOUN cana-3856 28 16 index index NOUN cana-3856 28 17 of of ADP cana-3856 28 18 graph graph NOUN cana-3856 28 19 𝐺 𝐺 PROPN cana-3856 28 20 is be AUX cana-3856 28 21 defined define VERB cana-3856 28 22 by by ADP cana-3856 28 23 𝑆𝑂(𝐺 𝑆𝑂(𝐺 NOUN cana-3856 28 24 ) ) PUNCT cana-3856 28 25 = = SYM cana-3856 29 1 βˆ‘π‘’π‘£βˆˆπΈ(𝐺 βˆ‘π‘’π‘£βˆˆπΈ(𝐺 X cana-3856 29 2 ) ) PUNCT cana-3856 29 3 βˆšπ‘‘πΊ(𝑒)2 βˆšπ‘‘πΊ(𝑒)2 PROPN cana-3856 30 1 + + SYM cana-3856 30 2 𝑑𝐺(𝑣)2 𝑑𝐺(𝑣)2 PROPN cana-3856 30 3 definition definition NOUN cana-3856 30 4 3 3 NUM cana-3856 30 5 . . PUNCT cana-3856 31 1 [ [ X cana-3856 31 2 8 8 NUM cana-3856 31 3 ] ] PUNCT cana-3856 31 4 the the DET cana-3856 31 5 sombor sombor NOUN cana-3856 31 6 matrix matrix NOUN cana-3856 31 7 of of ADP cana-3856 31 8 g g PROPN cana-3856 31 9 is be AUX cana-3856 31 10 defined define VERB cana-3856 31 11 by by ADP cana-3856 31 12 𝑆(𝐺 𝑆(𝐺 NOUN cana-3856 31 13 ) ) PUNCT cana-3856 32 1 = = PRON cana-3856 32 2 ( ( PUNCT cana-3856 32 3 𝑠𝑖𝑗)𝑛×𝑛 𝑠𝑖𝑗)𝑛×𝑛 PROPN cana-3856 32 4 = = SYM cana-3856 32 5 { { PUNCT cana-3856 32 6 βˆšπ‘‘π‘’2 βˆšπ‘‘π‘’2 PROPN cana-3856 32 7 + + CCONJ cana-3856 32 8 𝑑𝑣2 𝑑𝑣2 ADJ cana-3856 32 9 ; ; PUNCT cana-3856 32 10 if if SCONJ cana-3856 32 11 u u NOUN cana-3856 32 12 and and CCONJ cana-3856 32 13 vare vare VERB cana-3856 32 14 adjacent adjacent ADJ cana-3856 32 15 0 0 NUM cana-3856 32 16 ; ; PUNCT cana-3856 32 17 otherwise otherwise ADV cana-3856 32 18 we we PRON cana-3856 32 19 denote denote VERB cana-3856 32 20 the the DET cana-3856 32 21 eigenvalues eigenvalue NOUN cana-3856 32 22 of of ADP cana-3856 32 23 s(g s(g PROPN cana-3856 32 24 ) ) PUNCT cana-3856 32 25 by by ADP cana-3856 32 26 πœ‡π‘–β€²π‘  πœ‡π‘–β€²π‘  PROPN cana-3856 32 27 such such ADJ cana-3856 32 28 that that DET cana-3856 32 29 πœ‡1 πœ‡1 PROPN cana-3856 32 30 β‰₯ β‰₯ NOUN cana-3856 32 31 πœ‡2 πœ‡2 PROPN cana-3856 32 32 β‰₯. β‰₯. NOUN cana-3856 32 33 . . PUNCT cana-3856 32 34 . . PUNCT cana-3856 33 1 β‰₯ β‰₯ PROPN cana-3856 33 2 πœ‡π‘›. πœ‡π‘›. VERB cana-3856 33 3 the the DET cana-3856 33 4 set set NOUN cana-3856 33 5 of of ADP cana-3856 33 6 all all DET cana-3856 33 7 eigenvalues eigenvalue NOUN cana-3856 33 8 of of ADP cana-3856 33 9 s(g s(g PROPN cana-3856 33 10 ) ) PUNCT cana-3856 33 11 is be AUX cana-3856 33 12 called call VERB cana-3856 33 13 sombor sombor NOUN cana-3856 33 14 spectrum spectrum NOUN cana-3856 33 15 and and CCONJ cana-3856 33 16 πœ‡1 πœ‡1 NOUN cana-3856 33 17 is be AUX cana-3856 33 18 the the DET cana-3856 33 19 sombor sombor NOUN cana-3856 33 20 spectral spectral ADJ cana-3856 33 21 radius radius NOUN cana-3856 33 22 of of ADP cana-3856 33 23 g. g. PROPN cana-3856 33 24 the the DET cana-3856 33 25 sombor sombor NOUN cana-3856 33 26 energy energy NOUN cana-3856 34 1 [ [ X cana-3856 34 2 8 8 NUM cana-3856 34 3 ] ] PUNCT cana-3856 34 4 is be AUX cana-3856 34 5 defined define VERB cana-3856 34 6 by by ADP cana-3856 34 7 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 34 8 ) ) PUNCT cana-3856 34 9 = = PUNCT cana-3856 34 10 βˆ‘ βˆ‘ PUNCT cana-3856 34 11 𝑛 𝑛 DET cana-3856 34 12 𝑖=1 𝑖=1 PROPN cana-3856 34 13 |πœ‡π‘–| |πœ‡π‘–| PROPN cana-3856 34 14 . . PUNCT cana-3856 35 1 the the DET cana-3856 35 2 sum sum NOUN cana-3856 35 3 of of ADP cana-3856 35 4 squares square NOUN cana-3856 35 5 of of ADP cana-3856 35 6 eigenvalues eigenvalue NOUN cana-3856 35 7 of of ADP cana-3856 35 8 s(g s(g PROPN cana-3856 35 9 ) ) PUNCT cana-3856 35 10 satisfies satisfie NOUN cana-3856 35 11 following follow VERB cana-3856 35 12 equation equation NOUN cana-3856 35 13 is be AUX cana-3856 35 14 given give VERB cana-3856 35 15 by by ADP cana-3856 35 16 [ [ X cana-3856 35 17 2 2 NUM cana-3856 35 18 ] ] X cana-3856 35 19 2𝐹 2𝐹 NOUN cana-3856 35 20 = = SYM cana-3856 35 21 πœ‡1 πœ‡1 PROPN cana-3856 35 22 2 2 NUM cana-3856 35 23 + + CCONJ cana-3856 35 24 πœ‡2 πœ‡2 PROPN cana-3856 35 25 2 2 NUM cana-3856 35 26 + + CCONJ cana-3856 35 27 πœ‡3 πœ‡3 NOUN cana-3856 35 28 2 2 NUM cana-3856 35 29 + + PROPN cana-3856 35 30 . . PUNCT cana-3856 35 31 . . PUNCT cana-3856 35 32 . . PUNCT cana-3856 36 1 + + ADV cana-3856 36 2 πœ‡π‘› πœ‡π‘› PROPN cana-3856 36 3 2 2 NUM cana-3856 36 4 ( ( PUNCT cana-3856 36 5 1 1 NUM cana-3856 36 6 ) ) PUNCT cana-3856 36 7 where where SCONJ cana-3856 36 8 𝐹 𝐹 PROPN cana-3856 36 9 = = SYM cana-3856 36 10 𝐹(𝐺 𝐹(𝐺 NUM cana-3856 36 11 ) ) PUNCT cana-3856 36 12 = = SYM cana-3856 36 13 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 36 14 𝑑𝑣𝑖 𝑑𝑣𝑖 NOUN cana-3856 36 15 3 3 NUM cana-3856 36 16 = = SYM cana-3856 36 17 βˆ‘π‘£π‘–βˆΌπ‘£π‘— βˆ‘π‘£π‘–βˆΌπ‘£π‘— PROPN cana-3856 36 18 ( ( PUNCT cana-3856 36 19 𝑑𝑖 𝑑𝑖 PROPN cana-3856 36 20 2 2 NUM cana-3856 36 21 + + CCONJ cana-3856 36 22 𝑑𝑗 𝑑𝑗 PROPN cana-3856 36 23 2 2 NUM cana-3856 36 24 ) ) PUNCT cana-3856 36 25 is be AUX cana-3856 36 26 forgotten forget VERB cana-3856 36 27 topological topological ADJ cana-3856 36 28 index index NOUN cana-3856 36 29 of of ADP cana-3856 36 30 g g PROPN cana-3856 37 1 [ [ X cana-3856 37 2 1 1 NUM cana-3856 37 3 ] ] PUNCT cana-3856 37 4 . . PUNCT cana-3856 38 1 definition definition NOUN cana-3856 38 2 4 4 NUM cana-3856 38 3 . . PUNCT cana-3856 39 1 let let VERB cana-3856 39 2 s s PRON cana-3856 39 3 be be AUX cana-3856 39 4 a a DET cana-3856 39 5 subset subset NOUN cana-3856 39 6 of of ADP cana-3856 39 7 v(g v(g PROPN cana-3856 39 8 ) ) PUNCT cana-3856 39 9 . . PUNCT cana-3856 40 1 the the DET cana-3856 40 2 number number NOUN cana-3856 40 3 of of ADP cana-3856 40 4 element element NOUN cana-3856 40 5 of of ADP cana-3856 40 6 s s PRON cana-3856 40 7 will will AUX cana-3856 40 8 be be AUX cana-3856 40 9 denoted denote VERB cana-3856 40 10 by by ADP cana-3856 40 11 𝜎. 𝜎. NOUN cana-3856 40 12 the the DET cana-3856 40 13 sombor sombor NOUN cana-3856 40 14 matrix matrix NOUN cana-3856 40 15 of of ADP cana-3856 40 16 graph graph NOUN cana-3856 40 17 g g PROPN cana-3856 40 18 with with ADP cana-3856 40 19 𝜎 𝜎 PROPN cana-3856 40 20 self self NOUN cana-3856 40 21 loop loop NOUN cana-3856 40 22 is be AUX cana-3856 40 23 defined define VERB cana-3856 40 24 by by ADP cana-3856 40 25 𝐴𝑆𝑂(𝐺𝑠 𝐴𝑆𝑂(𝐺𝑠 PROPN cana-3856 40 26 ) ) PUNCT cana-3856 40 27 = = SYM cana-3856 40 28 ( ( PUNCT cana-3856 40 29 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-3856 40 30 ) ) PUNCT cana-3856 40 31 = = PRON cana-3856 40 32 { { PUNCT cana-3856 40 33 βˆšπ‘‘π‘– βˆšπ‘‘π‘– NOUN cana-3856 40 34 2 2 NUM cana-3856 40 35 + + CCONJ cana-3856 40 36 𝑑𝑗 𝑑𝑗 PROPN cana-3856 40 37 2 2 NUM cana-3856 40 38 𝑖𝑓𝑣𝑖 𝑖𝑓𝑣𝑖 NOUN cana-3856 40 39 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-3856 40 40 π‘£π‘—π‘Žπ‘Ÿπ‘’ π‘£π‘—π‘Žπ‘Ÿπ‘’ PROPN cana-3856 40 41 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ PROPN cana-3856 40 42 √2𝑑𝑖 √2𝑑𝑖 PROPN cana-3856 40 43 𝑖𝑓 𝑖𝑓 ADP cana-3856 40 44 𝑖 𝑖 NOUN cana-3856 41 1 = = PUNCT cana-3856 41 2 𝑗 𝑗 PRON cana-3856 41 3 0 0 NUM cana-3856 41 4 𝑖𝑓𝑣𝑖 𝑖𝑓𝑣𝑖 NOUN cana-3856 41 5 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-3856 41 6 π‘£π‘—π‘Žπ‘Ÿπ‘’ π‘£π‘—π‘Žπ‘Ÿπ‘’ PROPN cana-3856 41 7 π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ PROPN cana-3856 41 8 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ PROPN cana-3856 41 9 communications communication NOUN cana-3856 41 10 on on ADP cana-3856 41 11 applied apply VERB cana-3856 41 12 nonlinear nonlinear ADJ cana-3856 41 13 analysis analysis NOUN cana-3856 41 14 issn issn NOUN cana-3856 41 15 : : PUNCT cana-3856 41 16 1074 1074 NUM cana-3856 41 17 - - PUNCT cana-3856 41 18 133x 133x NUM cana-3856 41 19 vol vol NOUN cana-3856 41 20 32 32 NUM cana-3856 41 21 no no NOUN cana-3856 41 22 . . PUNCT cana-3856 42 1 9s 9s NUM cana-3856 42 2 ( ( PUNCT cana-3856 42 3 2025 2025 NUM cana-3856 42 4 ) ) PUNCT cana-3856 42 5 286 286 NUM cana-3856 42 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 42 7 if if SCONJ cana-3856 42 8 πœ†1(𝐺𝑠 πœ†1(𝐺𝑠 NOUN cana-3856 42 9 ) ) PUNCT cana-3856 42 10 , , PUNCT cana-3856 42 11 πœ†2(𝐺𝑠 πœ†2(𝐺𝑠 NUM cana-3856 42 12 ) ) PUNCT cana-3856 42 13 , , PUNCT cana-3856 42 14 . . PUNCT cana-3856 42 15 . . PUNCT cana-3856 42 16 . . PUNCT cana-3856 43 1 πœ†π‘›(𝐺𝑠 πœ†π‘›(𝐺𝑠 NOUN cana-3856 43 2 ) ) PUNCT cana-3856 43 3 are be AUX cana-3856 43 4 eigenvalues eigenvalue NOUN cana-3856 43 5 of of ADP cana-3856 43 6 𝐴𝑆𝑂(𝐺𝑠 𝐴𝑆𝑂(𝐺𝑠 NOUN cana-3856 43 7 ) ) PUNCT cana-3856 43 8 , , PUNCT cana-3856 43 9 then then ADV cana-3856 43 10 the the DET cana-3856 43 11 sombor sombor NOUN cana-3856 43 12 energy energy NOUN cana-3856 43 13 of of ADP cana-3856 43 14 𝐺𝑠 𝐺𝑠 PROPN cana-3856 43 15 ( ( PUNCT cana-3856 43 16 which which PRON cana-3856 43 17 is be AUX cana-3856 43 18 analogous analogous ADJ cana-3856 43 19 to to ADP cana-3856 43 20 the the DET cana-3856 43 21 energy energy NOUN cana-3856 43 22 of of ADP cana-3856 43 23 any any DET cana-3856 43 24 matrix matrix NOUN cana-3856 43 25 with with ADP cana-3856 43 26 a a DET cana-3856 43 27 non non ADJ cana-3856 43 28 - - ADJ cana-3856 43 29 zero zero ADJ cana-3856 43 30 diagonal diagonal ADJ cana-3856 43 31 [ [ X cana-3856 43 32 laplace laplace NOUN cana-3856 43 33 energy energy NOUN cana-3856 43 34 , , PUNCT cana-3856 43 35 laplace laplace NOUN cana-3856 43 36 energy energy NOUN cana-3856 43 37 and and CCONJ cana-3856 43 38 radiac radiac ADJ cana-3856 43 39 energy energy NOUN cana-3856 43 40 , , PUNCT cana-3856 43 41 new new ADJ cana-3856 43 42 spectral spectral NOUN cana-3856 43 43 ] ] X cana-3856 43 44 ) ) PUNCT cana-3856 43 45 must must AUX cana-3856 43 46 be be AUX cana-3856 43 47 defined define VERB cana-3856 43 48 as as ADP cana-3856 43 49 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 NOUN cana-3856 43 50 ) ) PUNCT cana-3856 43 51 = = PUNCT cana-3856 44 1 βˆ‘ βˆ‘ PUNCT cana-3856 44 2 𝑛 𝑛 PRON cana-3856 44 3 𝑖=1 𝑖=1 PROPN cana-3856 44 4 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 NOUN cana-3856 44 5 ) ) PUNCT cana-3856 44 6 βˆ’ βˆ’ PROPN cana-3856 45 1 √2βˆ‘πœŽπ‘—=1𝑑𝑗 √2βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 45 2 𝑛 𝑛 VERB cana-3856 45 3 | | ADV cana-3856 45 4 where where SCONJ cana-3856 45 5 𝑑𝑗 𝑑𝑗 PROPN cana-3856 45 6 is be AUX cana-3856 45 7 the the DET cana-3856 45 8 degree degree NOUN cana-3856 45 9 of of ADP cana-3856 45 10 vertex vertex NOUN cana-3856 45 11 𝑣𝑖 𝑣𝑖 NOUN cana-3856 45 12 with with ADP cana-3856 45 13 self self NOUN cana-3856 45 14 loop loop NOUN cana-3856 45 15 . . PUNCT cana-3856 46 1 2 2 X cana-3856 46 2 . . X cana-3856 46 3 main main ADJ cana-3856 46 4 results result NOUN cana-3856 46 5 proposition proposition NOUN cana-3856 46 6 1 1 NUM cana-3856 46 7 . . PUNCT cana-3856 47 1 let let VERB cana-3856 47 2 g g NOUN cana-3856 47 3 be be AUX cana-3856 47 4 graph graph NOUN cana-3856 47 5 with with ADP cana-3856 47 6 n n ADP cana-3856 47 7 vertices vertex NOUN cana-3856 47 8 and and CCONJ cana-3856 47 9 s s AUX cana-3856 47 10 be be AUX cana-3856 47 11 any any DET cana-3856 47 12 subset subset NOUN cana-3856 47 13 of of ADP cana-3856 47 14 v(g v(g PROPN cana-3856 47 15 ) ) PUNCT cana-3856 47 16 with with ADP cana-3856 47 17 𝜎 𝜎 PROPN cana-3856 47 18 elements element NOUN cana-3856 47 19 . . PUNCT cana-3856 48 1 if if SCONJ cana-3856 48 2 𝜎 𝜎 PRON cana-3856 48 3 = = SYM cana-3856 48 4 0 0 NUM cana-3856 48 5 , , PUNCT cana-3856 48 6 then then ADV cana-3856 48 7 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NUM cana-3856 48 8 ) ) PUNCT cana-3856 48 9 = = SYM cana-3856 48 10 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 48 11 ) ) PUNCT cana-3856 48 12 . . PUNCT cana-3856 49 1 proof proof NOUN cana-3856 49 2 . . PUNCT cana-3856 50 1 it it PRON cana-3856 50 2 is be AUX cana-3856 50 3 trivially trivially ADV cana-3856 50 4 obvious obvious ADJ cana-3856 50 5 , , PUNCT cana-3856 50 6 since since SCONJ cana-3856 50 7 for for ADP cana-3856 50 8 𝜎 𝜎 NOUN cana-3856 50 9 = = SYM cana-3856 50 10 0 0 PROPN cana-3856 50 11 , , PUNCT cana-3856 50 12 the the DET cana-3856 50 13 graphs graph NOUN cana-3856 50 14 𝐺𝑆 𝐺𝑆 PROPN cana-3856 50 15 and and CCONJ cana-3856 50 16 g g NOUN cana-3856 50 17 coincide coincide NOUN cana-3856 50 18 then then ADV cana-3856 50 19 sombor sombor NOUN cana-3856 50 20 matrix matrix NOUN cana-3856 50 21 of of ADP cana-3856 50 22 𝐺𝑆 𝐺𝑆 PROPN cana-3856 50 23 and and CCONJ cana-3856 50 24 𝐺 𝐺 PROPN cana-3856 50 25 are be AUX cana-3856 50 26 same same ADJ cana-3856 50 27 and and CCONJ cana-3856 50 28 hence hence ADV cana-3856 50 29 sombor sombor NOUN cana-3856 50 30 energy energy NOUN cana-3856 50 31 are be AUX cana-3856 50 32 also also ADV cana-3856 50 33 same same ADJ cana-3856 50 34 . . PUNCT cana-3856 51 1 let let VERB cana-3856 51 2 𝑆(𝐺 𝑆(𝐺 NOUN cana-3856 51 3 ) ) PUNCT cana-3856 51 4 be be AUX cana-3856 51 5 sombor sombor NOUN cana-3856 51 6 matrix matrix NOUN cana-3856 51 7 of of ADP cana-3856 51 8 g g PROPN cana-3856 51 9 and and CCONJ cana-3856 51 10 𝐷(𝐺𝑆 𝐷(𝐺𝑆 NOUN cana-3856 51 11 ) ) PUNCT cana-3856 51 12 is be AUX cana-3856 51 13 the the DET cana-3856 51 14 diagonal diagonal ADJ cana-3856 51 15 matrix matrix NOUN cana-3856 51 16 with with ADP cana-3856 51 17 diagonal diagonal ADJ cana-3856 51 18 entries entry NOUN cana-3856 51 19 degree degree NOUN cana-3856 51 20 of of ADP cana-3856 51 21 vertex vertex NOUN cana-3856 51 22 having have VERB cana-3856 51 23 self self NOUN cana-3856 51 24 loop loop VERB cana-3856 51 25 then then ADV cana-3856 51 26 sombor sombor NOUN cana-3856 51 27 matrix matrix NOUN cana-3856 51 28 of of ADP cana-3856 51 29 𝐺𝑆 𝐺𝑆 PROPN cana-3856 51 30 is be AUX cana-3856 51 31 equal equal ADJ cana-3856 51 32 to to ADP cana-3856 51 33 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 PROPN cana-3856 51 34 ) ) PUNCT cana-3856 51 35 = = SYM cana-3856 52 1 𝑆(𝐺 𝑆(𝐺 NOUN cana-3856 52 2 ) ) PUNCT cana-3856 53 1 + + CCONJ cana-3856 53 2 √2𝐷(𝐺𝑆 √2𝐷(𝐺𝑆 PROPN cana-3856 53 3 ) ) PUNCT cana-3856 53 4 ( ( PUNCT cana-3856 53 5 2 2 X cana-3856 53 6 ) ) PUNCT cana-3856 53 7 proposition proposition NOUN cana-3856 53 8 2 2 NUM cana-3856 53 9 . . PUNCT cana-3856 54 1 let let VERB cana-3856 54 2 g g PRON cana-3856 54 3 be be AUX cana-3856 54 4 a a DET cana-3856 54 5 graph graph NOUN cana-3856 54 6 with with ADP cana-3856 54 7 n n ADP cana-3856 54 8 vertices vertex NOUN cana-3856 54 9 and and CCONJ cana-3856 54 10 m m PRON cana-3856 54 11 edges edge NOUN cana-3856 54 12 . . PUNCT cana-3856 55 1 if if SCONJ cana-3856 55 2 sβŠ‚v sβŠ‚v ADJ cana-3856 55 3 with with ADP cana-3856 55 4 𝜎 𝜎 PROPN cana-3856 55 5 elements element NOUN cana-3856 55 6 , , PUNCT cana-3856 55 7 then then ADV cana-3856 55 8 eigenvalues eigenvalue VERB cana-3856 55 9 πœ†1(𝐺𝑠 πœ†1(𝐺𝑠 X cana-3856 55 10 ) ) PUNCT cana-3856 55 11 , , PUNCT cana-3856 55 12 πœ†2(𝐺𝑠 πœ†2(𝐺𝑠 NUM cana-3856 55 13 ) ) PUNCT cana-3856 55 14 , , PUNCT cana-3856 55 15 . . PUNCT cana-3856 55 16 . . PUNCT cana-3856 56 1 . . PUNCT cana-3856 56 2 , , PUNCT cana-3856 57 1 πœ†π‘›(𝐺𝑠 πœ†π‘›(𝐺𝑠 NOUN cana-3856 57 2 ) ) PUNCT cana-3856 57 3 of of ADP cana-3856 57 4 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 PROPN cana-3856 57 5 ) ) PUNCT cana-3856 57 6 satisfy satisfy NOUN cana-3856 57 7 , , PUNCT cana-3856 57 8 1 1 X cana-3856 57 9 . . X cana-3856 57 10 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 57 11 πœ†π‘– πœ†π‘– PROPN cana-3856 57 12 2(𝐺𝑠 2(𝐺𝑠 NUM cana-3856 57 13 ) ) PUNCT cana-3856 58 1 = = VERB cana-3856 59 1 2𝐹 2𝐹 NOUN cana-3856 59 2 + + CCONJ cana-3856 59 3 4βˆ‘π‘£π‘–βˆˆπ‘† 4βˆ‘π‘£π‘–βˆˆπ‘† NUM cana-3856 59 4 𝑑𝑖 𝑑𝑖 PART cana-3856 59 5 2 2 NUM cana-3856 59 6 2 2 NUM cana-3856 59 7 . . PUNCT cana-3856 60 1 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 60 2 [ [ X cana-3856 60 3 πœ†π‘–(𝐺𝑠 πœ†π‘–(𝐺𝑠 NOUN cana-3856 60 4 ) ) PUNCT cana-3856 60 5 βˆ’ βˆ’ PUNCT cana-3856 61 1 √2βˆ‘πœŽπ‘—=1𝑑𝑗 √2βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 61 2 𝑛 𝑛 X cana-3856 61 3 ] ] SYM cana-3856 61 4 2 2 NUM cana-3856 61 5 = = SYM cana-3856 61 6 2𝐹 2𝐹 NOUN cana-3856 61 7 + + CCONJ cana-3856 61 8 4βˆ‘π‘£π‘–βˆˆπ‘† 4βˆ‘π‘£π‘–βˆˆπ‘† NUM cana-3856 61 9 𝑑𝑖 𝑑𝑖 PART cana-3856 61 10 2 2 NUM cana-3856 61 11 βˆ’ βˆ’ PROPN cana-3856 61 12 2(βˆ‘π‘£π‘–βˆˆπ‘†π‘‘π‘– 2(βˆ‘π‘£π‘–βˆˆπ‘†π‘‘π‘– NUM cana-3856 61 13 ) ) PUNCT cana-3856 61 14 2 2 NUM cana-3856 61 15 𝑛 𝑛 NOUN cana-3856 61 16 . . PUNCT cana-3856 62 1 proof proof NOUN cana-3856 62 2 . . PUNCT cana-3856 63 1 1 1 X cana-3856 63 2 . . X cana-3856 63 3 from from ADP cana-3856 63 4 equation equation NOUN cana-3856 63 5 ( ( PUNCT cana-3856 63 6 2 2 NUM cana-3856 63 7 ) ) PUNCT cana-3856 63 8 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 PROPN cana-3856 63 9 ) ) PUNCT cana-3856 63 10 = = SYM cana-3856 63 11 𝑆(𝐺 𝑆(𝐺 NOUN cana-3856 63 12 ) ) PUNCT cana-3856 64 1 + + CCONJ cana-3856 64 2 √2𝐷(𝐺𝑆 √2𝐷(𝐺𝑆 PROPN cana-3856 64 3 ) ) PUNCT cana-3856 64 4 therefore therefore ADV cana-3856 64 5 , , PUNCT cana-3856 64 6 βˆ‘ βˆ‘ ADV cana-3856 64 7 𝑛 𝑛 PRON cana-3856 64 8 𝑖=1 𝑖=1 AUX cana-3856 64 9 πœ†π‘– πœ†π‘– PROPN cana-3856 64 10 2(𝐺𝑠 2(𝐺𝑠 NUM cana-3856 64 11 ) ) PUNCT cana-3856 65 1 = = NOUN cana-3856 65 2 βˆ‘ βˆ‘ PUNCT cana-3856 65 3 𝑛 𝑛 PRON cana-3856 65 4 𝑖=1 𝑖=1 PROPN cana-3856 66 1 [ [ X cana-3856 66 2 ( ( PUNCT cana-3856 66 3 𝑆(𝐺 𝑆(𝐺 NOUN cana-3856 66 4 ) ) PUNCT cana-3856 66 5 + + CCONJ cana-3856 66 6 √2𝐷(𝐺𝑆 √2𝐷(𝐺𝑆 PROPN cana-3856 66 7 ) ) PUNCT cana-3856 66 8 ) ) PUNCT cana-3856 67 1 2]𝑖𝑖 2]𝑖𝑖 NUM cana-3856 68 1 = = SYM cana-3856 68 2 βˆ‘ βˆ‘ PUNCT cana-3856 68 3 𝑛 𝑛 PRON cana-3856 68 4 𝑖=1 𝑖=1 PROPN cana-3856 69 1 [ [ X cana-3856 69 2 ( ( PUNCT cana-3856 69 3 𝑆(𝐺)2)𝑖𝑖 𝑆(𝐺)2)𝑖𝑖 NOUN cana-3856 69 4 + + CCONJ cana-3856 69 5 2√2[𝑆(𝐺)𝐷(𝐺𝑆)]𝑖𝑖 2√2[𝑆(𝐺)𝐷(𝐺𝑆)]𝑖𝑖 NUM cana-3856 69 6 + + CCONJ cana-3856 69 7 2[(𝐷(𝐺𝑆 2[(𝐷(𝐺𝑆 NUM cana-3856 69 8 ) ) PUNCT cana-3856 69 9 ) ) PUNCT cana-3856 70 1 2]𝑖𝑖 2]𝑖𝑖 NUM cana-3856 70 2 ] ] PUNCT cana-3856 70 3 from from ADP cana-3856 70 4 equation equation NOUN cana-3856 70 5 ( ( PUNCT cana-3856 70 6 1 1 NUM cana-3856 70 7 ) ) PUNCT cana-3856 70 8 , , PUNCT cana-3856 70 9 2𝐹 2𝐹 NOUN cana-3856 70 10 = = SYM cana-3856 70 11 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 70 12 ( ( PUNCT cana-3856 70 13 𝑆(𝐺 𝑆(𝐺 PROPN cana-3856 70 14 ) ) PUNCT cana-3856 70 15 2)𝑖𝑖 2)𝑖𝑖 PROPN cana-3856 70 16 where where SCONJ cana-3856 70 17 𝐹 𝐹 PROPN cana-3856 70 18 = = SYM cana-3856 70 19 βˆ‘π‘£π‘–βˆΌπ‘£π‘— βˆ‘π‘£π‘–βˆΌπ‘£π‘— PROPN cana-3856 70 20 ( ( PUNCT cana-3856 70 21 𝑑𝑖 𝑑𝑖 PROPN cana-3856 70 22 2 2 NUM cana-3856 70 23 + + CCONJ cana-3856 70 24 𝑑𝑗 𝑑𝑗 PROPN cana-3856 70 25 2 2 NUM cana-3856 70 26 ) ) PUNCT cana-3856 70 27 as as SCONJ cana-3856 70 28 the the DET cana-3856 70 29 diagonal diagonal ADJ cana-3856 70 30 entries entry NOUN cana-3856 70 31 of of ADP cana-3856 70 32 diagonal diagonal ADJ cana-3856 70 33 matrix matrix NOUN cana-3856 70 34 are be AUX cana-3856 70 35 nonzero nonzero NOUN cana-3856 70 36 for for SCONJ cana-3856 70 37 𝜎 𝜎 PROPN cana-3856 70 38 self self NOUN cana-3856 70 39 loop loop NOUN cana-3856 70 40 and and CCONJ cana-3856 70 41 diagonal diagonal ADJ cana-3856 70 42 entries entry NOUN cana-3856 70 43 of of ADP cana-3856 70 44 sombor sombor NOUN cana-3856 70 45 matrix matrix NOUN cana-3856 70 46 is be AUX cana-3856 70 47 zero zero NUM cana-3856 70 48 . . PUNCT cana-3856 71 1 hence hence ADV cana-3856 71 2 , , PUNCT cana-3856 71 3 diagonal diagonal ADJ cana-3856 71 4 entries entry NOUN cana-3856 71 5 of of ADP cana-3856 71 6 𝑆(𝐺)𝐷(𝐺𝑆 𝑆(𝐺)𝐷(𝐺𝑆 NOUN cana-3856 71 7 ) ) PUNCT cana-3856 71 8 are be AUX cana-3856 71 9 zero zero NUM cana-3856 71 10 . . PUNCT cana-3856 72 1 therefore therefore ADV cana-3856 72 2 , , PUNCT cana-3856 72 3 communications communication NOUN cana-3856 72 4 on on ADP cana-3856 72 5 applied apply VERB cana-3856 72 6 nonlinear nonlinear ADJ cana-3856 72 7 analysis analysis NOUN cana-3856 72 8 issn issn NOUN cana-3856 72 9 : : PUNCT cana-3856 72 10 1074 1074 NUM cana-3856 72 11 - - PUNCT cana-3856 72 12 133x 133x NUM cana-3856 72 13 vol vol NOUN cana-3856 72 14 32 32 NUM cana-3856 72 15 no no NOUN cana-3856 72 16 . . PUNCT cana-3856 73 1 9s 9s NUM cana-3856 73 2 ( ( PUNCT cana-3856 73 3 2025 2025 NUM cana-3856 73 4 ) ) PUNCT cana-3856 73 5 287 287 NUM cana-3856 73 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 73 7 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 74 1 [ [ X cana-3856 74 2 𝑆(𝐺)𝐷(𝐺𝑆)]𝑖𝑖 𝑆(𝐺)𝐷(𝐺𝑆)]𝑖𝑖 X cana-3856 74 3 = = SYM cana-3856 74 4 0 0 NUM cana-3856 74 5 also also ADV cana-3856 74 6 , , PUNCT cana-3856 74 7 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 75 1 [ [ X cana-3856 75 2 𝐷(𝐺𝑆 𝐷(𝐺𝑆 X cana-3856 75 3 ) ) PUNCT cana-3856 75 4 2 2 NUM cana-3856 75 5 ] ] SYM cana-3856 75 6 = = SYM cana-3856 75 7 2βˆ‘π‘£π‘–βˆˆπ‘† 2βˆ‘π‘£π‘–βˆˆπ‘† NUM cana-3856 75 8 𝑑𝑖 𝑑𝑖 PRON cana-3856 75 9 2 2 NUM cana-3856 75 10 βˆ‘ βˆ‘ NOUN cana-3856 75 11 𝑛 𝑛 PRON cana-3856 75 12 𝑖=1 𝑖=1 PROPN cana-3856 75 13 πœ†π‘– πœ†π‘– PROPN cana-3856 75 14 2(𝐺𝑠 2(𝐺𝑠 NUM cana-3856 75 15 ) ) PUNCT cana-3856 76 1 = = NOUN cana-3856 76 2 βˆ‘ βˆ‘ PART cana-3856 76 3 𝑛 𝑛 PRON cana-3856 76 4 𝑖=1 𝑖=1 PROPN cana-3856 76 5 ( ( PUNCT cana-3856 76 6 𝑆(𝐺)2)𝑖𝑖 𝑆(𝐺)2)𝑖𝑖 NOUN cana-3856 76 7 + + CCONJ cana-3856 76 8 2√2βˆ‘ 2√2βˆ‘ NOUN cana-3856 76 9 𝑛 𝑛 PROPN cana-3856 76 10 𝑖=1 𝑖=1 PROPN cana-3856 77 1 [ [ X cana-3856 77 2 𝑆(𝐺)𝐷(𝐺𝑆)]𝑖𝑖 𝑆(𝐺)𝐷(𝐺𝑆)]𝑖𝑖 X cana-3856 77 3 + + NUM cana-3856 77 4 2βˆ‘ 2βˆ‘ NUM cana-3856 77 5 𝑛 𝑛 PRON cana-3856 77 6 𝑖=1 𝑖=1 PUNCT cana-3856 77 7 ( ( PUNCT cana-3856 77 8 𝐷(𝐺𝑆 𝐷(𝐺𝑆 PROPN cana-3856 77 9 ) ) PUNCT cana-3856 77 10 2)𝑖𝑖 2)𝑖𝑖 PUNCT cana-3856 78 1 = = PUNCT cana-3856 79 1 2𝐹 2𝐹 NOUN cana-3856 80 1 + + CCONJ cana-3856 80 2 2(2βˆ‘ 2(2βˆ‘ NUM cana-3856 80 3 π‘£π‘–βˆˆπ‘† π‘£π‘–βˆˆπ‘† PROPN cana-3856 80 4 𝑑𝑖 𝑑𝑖 PROPN cana-3856 80 5 2 2 NUM cana-3856 80 6 ) ) PUNCT cana-3856 80 7 = = PUNCT cana-3856 81 1 2𝐹 2𝐹 NOUN cana-3856 82 1 + + CCONJ cana-3856 82 2 4βˆ‘π‘£π‘–βˆˆπ‘† 4βˆ‘π‘£π‘–βˆˆπ‘† NUM cana-3856 82 3 𝑑𝑖 𝑑𝑖 PART cana-3856 82 4 2 2 NUM cana-3856 82 5 2 2 NUM cana-3856 82 6 . . PUNCT cana-3856 83 1 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 83 2 [ [ X cana-3856 83 3 πœ†π‘–(𝐺𝑠 πœ†π‘–(𝐺𝑠 NOUN cana-3856 83 4 ) ) PUNCT cana-3856 83 5 βˆ’ βˆ’ PUNCT cana-3856 84 1 √2βˆ‘πœŽπ‘—=1𝑑𝑗 √2βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 84 2 𝑛 𝑛 X cana-3856 84 3 ] ] SYM cana-3856 84 4 2 2 X cana-3856 84 5 = = SYM cana-3856 84 6 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 85 1 [ [ X cana-3856 85 2 πœ†π‘–(𝐺𝑆 πœ†π‘–(𝐺𝑆 NOUN cana-3856 85 3 ) ) PUNCT cana-3856 85 4 2 2 NUM cana-3856 85 5 βˆ’ βˆ’ NOUN cana-3856 85 6 2√2πœ†π‘–(𝐺𝑠 2√2πœ†π‘–(𝐺𝑠 NOUN cana-3856 85 7 ) ) PUNCT cana-3856 85 8 √2βˆ‘πœŽπ‘—=1𝑑𝑗 √2βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 85 9 𝑛 𝑛 X cana-3856 85 10 + + ADJ cana-3856 85 11 2 2 NUM cana-3856 85 12 ( ( PUNCT cana-3856 85 13 βˆ‘πœŽπ‘—=1𝑑𝑗 βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 85 14 ) ) PUNCT cana-3856 85 15 2 2 NUM cana-3856 85 16 𝑛2 𝑛2 NOUN cana-3856 85 17 ] ] PUNCT cana-3856 86 1 = = NOUN cana-3856 86 2 βˆ‘ βˆ‘ PUNCT cana-3856 86 3 𝑛 𝑛 PRON cana-3856 86 4 𝑖=1 𝑖=1 PROPN cana-3856 87 1 [ [ X cana-3856 87 2 πœ†π‘– πœ†π‘– X cana-3856 87 3 2(𝐺𝑆 2(𝐺𝑆 NUM cana-3856 87 4 ) ) PUNCT cana-3856 87 5 βˆ’ βˆ’ PROPN cana-3856 88 1 2√2 2√2 NUM cana-3856 88 2 𝑛 𝑛 PRON cana-3856 88 3 βˆ‘ βˆ‘ PUNCT cana-3856 88 4 π‘£π‘–βˆˆπ‘† π‘£π‘–βˆˆπ‘† PROPN cana-3856 88 5 π‘‘π‘–βˆ‘ π‘‘π‘–βˆ‘ PROPN cana-3856 88 6 𝑛 𝑛 DET cana-3856 88 7 𝑖=1 𝑖=1 PROPN cana-3856 88 8 πœ†π‘–(𝐺𝑆 πœ†π‘–(𝐺𝑆 NOUN cana-3856 88 9 ) ) PUNCT cana-3856 89 1 + + CCONJ cana-3856 89 2 2𝑛 2𝑛 NUM cana-3856 89 3 ( ( PUNCT cana-3856 89 4 βˆ‘πœŽπ‘—=1 βˆ‘πœŽπ‘—=1 NOUN cana-3856 89 5 𝑑𝑗 𝑑𝑗 NOUN cana-3856 89 6 ) ) PUNCT cana-3856 89 7 2 2 NUM cana-3856 89 8 𝑛2 𝑛2 NOUN cana-3856 89 9 ] ] PUNCT cana-3856 89 10 as as ADP cana-3856 89 11 βˆ‘π‘›π‘–=1 βˆ‘π‘›π‘–=1 PROPN cana-3856 89 12 πœ†π‘–(𝐺𝑆 πœ†π‘–(𝐺𝑆 PROPN cana-3856 89 13 ) ) PUNCT cana-3856 90 1 = = SYM cana-3856 90 2 √2βˆ‘π‘£π‘–βˆˆπ‘† √2βˆ‘π‘£π‘–βˆˆπ‘† NOUN cana-3856 90 3 𝑑𝑖 𝑑𝑖 VERB cana-3856 90 4 , , PUNCT cana-3856 90 5 = = SYM cana-3856 90 6 2𝐹 2𝐹 NOUN cana-3856 90 7 + + CCONJ cana-3856 90 8 4βˆ‘ 4βˆ‘ PROPN cana-3856 90 9 π‘£π‘–βˆˆπ‘† π‘£π‘–βˆˆπ‘† PROPN cana-3856 90 10 𝑑𝑖 𝑑𝑖 NUM cana-3856 90 11 2 2 NUM cana-3856 90 12 βˆ’ βˆ’ PROPN cana-3856 90 13 2√2 2√2 NUM cana-3856 90 14 𝑛 𝑛 DET cana-3856 90 15 βˆ‘ βˆ‘ PUNCT cana-3856 90 16 π‘£π‘–βˆˆπ‘† π‘£π‘–βˆˆπ‘† PROPN cana-3856 90 17 π‘‘π‘–βˆš2βˆ‘ π‘‘π‘–βˆš2βˆ‘ PROPN cana-3856 90 18 π‘£π‘–βˆˆπ‘† π‘£π‘–βˆˆπ‘† PROPN cana-3856 90 19 𝑑𝑖 𝑑𝑖 PART cana-3856 90 20 + + CCONJ cana-3856 90 21 2 2 NUM cana-3856 90 22 ( ( PUNCT cana-3856 90 23 βˆ‘πœŽπ‘—=1 βˆ‘πœŽπ‘—=1 NOUN cana-3856 90 24 𝑑𝑗 𝑑𝑗 NOUN cana-3856 90 25 ) ) PUNCT cana-3856 90 26 2 2 NUM cana-3856 90 27 𝑛 𝑛 NOUN cana-3856 90 28 = = SYM cana-3856 90 29 2𝐹 2𝐹 NOUN cana-3856 90 30 + + CCONJ cana-3856 90 31 4βˆ‘ 4βˆ‘ PROPN cana-3856 90 32 π‘£π‘–βˆˆπ‘† π‘£π‘–βˆˆπ‘† PROPN cana-3856 90 33 𝑑𝑖 𝑑𝑖 NUM cana-3856 90 34 2 2 NUM cana-3856 90 35 βˆ’ βˆ’ PROPN cana-3856 90 36 4 4 NUM cana-3856 90 37 𝑛 𝑛 NOUN cana-3856 90 38 ( ( PUNCT cana-3856 90 39 βˆ‘ βˆ‘ PROPN cana-3856 90 40 π‘£π‘–βˆˆπ‘† π‘£π‘–βˆˆπ‘† PROPN cana-3856 90 41 𝑑𝑖 𝑑𝑖 PROPN cana-3856 90 42 ) ) PUNCT cana-3856 90 43 2 2 NUM cana-3856 91 1 + + NUM cana-3856 91 2 2 2 NUM cana-3856 91 3 ( ( PUNCT cana-3856 91 4 βˆ‘πœŽπ‘—=1 βˆ‘πœŽπ‘—=1 NOUN cana-3856 91 5 𝑑𝑗 𝑑𝑗 NOUN cana-3856 91 6 ) ) PUNCT cana-3856 91 7 2 2 NUM cana-3856 92 1 𝑛 𝑛 NOUN cana-3856 92 2 = = SYM cana-3856 92 3 2𝐹 2𝐹 NOUN cana-3856 92 4 + + CCONJ cana-3856 92 5 4βˆ‘π‘£π‘–βˆˆπ‘† 4βˆ‘π‘£π‘–βˆˆπ‘† NUM cana-3856 92 6 𝑑𝑖 𝑑𝑖 PART cana-3856 92 7 2 2 NUM cana-3856 92 8 βˆ’ βˆ’ PROPN cana-3856 92 9 2 2 NUM cana-3856 92 10 𝑛 𝑛 PROPN cana-3856 92 11 ( ( PUNCT cana-3856 92 12 βˆ‘π‘£π‘–βˆˆπ‘† βˆ‘π‘£π‘–βˆˆπ‘† NOUN cana-3856 92 13 𝑑𝑖 𝑑𝑖 PART cana-3856 92 14 ) ) PUNCT cana-3856 92 15 2 2 NUM cana-3856 92 16 ( ( PUNCT cana-3856 92 17 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 PROPN cana-3856 92 18 βˆ‘π‘£π‘–βˆˆπ‘† βˆ‘π‘£π‘–βˆˆπ‘† PROPN cana-3856 92 19 𝑑𝑖 𝑑𝑖 PART cana-3856 92 20 = = SYM cana-3856 92 21 βˆ‘πœŽπ‘—=1 βˆ‘πœŽπ‘—=1 PROPN cana-3856 92 22 𝑑𝑗 𝑑𝑗 NOUN cana-3856 92 23 ) ) PUNCT cana-3856 92 24 . . PUNCT cana-3856 93 1 lemma lemma PROPN cana-3856 93 2 1 1 X cana-3856 93 3 . . PUNCT cana-3856 94 1 let let VERB cana-3856 95 1 𝐺 𝐺 NOUN cana-3856 95 2 = = SYM cana-3856 95 3 𝐾𝑛 𝐾𝑛 PROPN cana-3856 95 4 be be AUX cana-3856 95 5 a a DET cana-3856 95 6 complete complete ADJ cana-3856 95 7 graph graph NOUN cana-3856 95 8 with with ADP cana-3856 95 9 n n ADP cana-3856 95 10 vertices vertex NOUN cana-3856 95 11 , , PUNCT cana-3856 95 12 if if SCONJ cana-3856 95 13 𝐺𝑆 𝐺𝑆 PROPN cana-3856 95 14 is be AUX cana-3856 95 15 a a DET cana-3856 95 16 graph graph NOUN cana-3856 95 17 obtained obtain VERB cana-3856 95 18 from from ADP cana-3856 95 19 g g NOUN cana-3856 95 20 by by ADP cana-3856 95 21 adding add VERB cana-3856 95 22 n-1 n-1 NOUN cana-3856 95 23 self self NOUN cana-3856 95 24 loops loop NOUN cana-3856 95 25 , , PUNCT cana-3856 95 26 then then ADV cana-3856 95 27 sombor sombor NOUN cana-3856 95 28 eigenvalues eigenvalue NOUN cana-3856 95 29 of of ADP cana-3856 95 30 𝐺𝑆 𝐺𝑆 PROPN cana-3856 95 31 are be AUX cana-3856 95 32 πœ†1 πœ†1 ADJ cana-3856 95 33 = = PUNCT cana-3856 95 34 ( ( PUNCT cana-3856 95 35 𝑛2βˆ’1)√2+√2𝑛4 𝑛2βˆ’1)√2+√2𝑛4 ADJ cana-3856 95 36 + + ADJ cana-3856 95 37 8𝑛3βˆ’12𝑛2 8𝑛3βˆ’12𝑛2 NUM cana-3856 95 38 + + ADJ cana-3856 95 39 8π‘›βˆ’6 8π‘›βˆ’6 NOUN cana-3856 95 40 2 2 NUM cana-3856 95 41 , , PUNCT cana-3856 95 42 πœ†2 πœ†2 NOUN cana-3856 95 43 = = PUNCT cana-3856 95 44 ( ( PUNCT cana-3856 95 45 𝑛2βˆ’1)√2βˆ’βˆš2𝑛4 𝑛2βˆ’1)√2βˆ’βˆš2𝑛4 VERB cana-3856 95 46 + + ADJ cana-3856 95 47 8𝑛3βˆ’12𝑛2 8𝑛3βˆ’12𝑛2 NUM cana-3856 95 48 + + ADJ cana-3856 95 49 8π‘›βˆ’6 8π‘›βˆ’6 NOUN cana-3856 95 50 2 2 NUM cana-3856 95 51 , , PUNCT cana-3856 95 52 and and CCONJ cana-3856 95 53 πœ†π‘– πœ†π‘– X cana-3856 95 54 = = NOUN cana-3856 95 55 0 0 NUM cana-3856 96 1 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN cana-3856 96 2 𝑖 𝑖 PRON cana-3856 96 3 β‰₯ β‰₯ NOUN cana-3856 96 4 3 3 NUM cana-3856 96 5 . . PUNCT cana-3856 97 1 proof proof NOUN cana-3856 97 2 . . PUNCT cana-3856 98 1 let let VERB cana-3856 98 2 j j PROPN cana-3856 98 3 be be AUX cana-3856 98 4 the the DET cana-3856 98 5 𝑛 𝑛 ADJ cana-3856 98 6 Γ— Γ— NOUN cana-3856 98 7 𝑛 𝑛 DET cana-3856 98 8 matrix matrix NOUN cana-3856 98 9 with with ADP cana-3856 98 10 all all DET cana-3856 98 11 entries entry NOUN cana-3856 98 12 one one NUM cana-3856 98 13 and and CCONJ cana-3856 98 14 0 0 NUM cana-3856 98 15 is be AUX cana-3856 98 16 the the DET cana-3856 98 17 matrix matrix NOUN cana-3856 98 18 with with ADP cana-3856 98 19 all all DET cana-3856 98 20 zero zero NUM cana-3856 98 21 entries entry NOUN cana-3856 98 22 . . PUNCT cana-3856 99 1 sombor sombor NOUN cana-3856 99 2 matrix matrix NOUN cana-3856 99 3 of of ADP cana-3856 99 4 𝐺𝑆 𝐺𝑆 PROPN cana-3856 99 5 is be AUX cana-3856 99 6 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 99 7 ) ) PUNCT cana-3856 100 1 = = PRON cana-3856 100 2 [ [ PUNCT cana-3856 100 3 ( ( PUNCT cana-3856 100 4 𝑛 𝑛 PROPN cana-3856 100 5 + + NUM cana-3856 100 6 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 NUM cana-3856 100 7 √2(𝑛2 √2(𝑛2 NOUN cana-3856 100 8 + + PUNCT cana-3856 100 9 1)π½π‘›βˆ’1Γ—1 1)π½π‘›βˆ’1Γ—1 NUM cana-3856 100 10 √2(𝑛2 √2(𝑛2 NOUN cana-3856 100 11 + + CCONJ cana-3856 100 12 1)𝐽1Γ—π‘›βˆ’1 1)𝐽1Γ—π‘›βˆ’1 NOUN cana-3856 100 13 0 0 NUM cana-3856 100 14 ] ] PUNCT cana-3856 101 1 𝑛×𝑛 𝑛×𝑛 PROPN cana-3856 101 2 then then ADV cana-3856 101 3 , , PUNCT cana-3856 101 4 communications communication NOUN cana-3856 101 5 on on ADP cana-3856 101 6 applied apply VERB cana-3856 101 7 nonlinear nonlinear ADJ cana-3856 101 8 analysis analysis NOUN cana-3856 101 9 issn issn NOUN cana-3856 101 10 : : PUNCT cana-3856 101 11 1074 1074 NUM cana-3856 101 12 - - PUNCT cana-3856 101 13 133x 133x NUM cana-3856 101 14 vol vol NOUN cana-3856 101 15 32 32 NUM cana-3856 101 16 no no NOUN cana-3856 101 17 . . PUNCT cana-3856 102 1 9s 9s NUM cana-3856 102 2 ( ( PUNCT cana-3856 102 3 2025 2025 NUM cana-3856 102 4 ) ) PUNCT cana-3856 102 5 288 288 NUM cana-3856 102 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 102 7 𝑑𝑒𝑑(πœ†πΌ 𝑑𝑒𝑑(πœ†πΌ NOUN cana-3856 102 8 βˆ’ βˆ’ NOUN cana-3856 102 9 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 102 10 ) ) PUNCT cana-3856 102 11 ) ) PUNCT cana-3856 103 1 = = SYM cana-3856 103 2 𝑑𝑒𝑑 𝑑𝑒𝑑 NOUN cana-3856 103 3 [ [ PUNCT cana-3856 103 4 πœ†πΌπ‘›βˆ’1Γ—π‘›βˆ’1 πœ†πΌπ‘›βˆ’1Γ—π‘›βˆ’1 NOUN cana-3856 103 5 βˆ’ βˆ’ PROPN cana-3856 103 6 ( ( PUNCT cana-3856 103 7 𝑛 𝑛 PROPN cana-3856 103 8 + + NUM cana-3856 103 9 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 NUM cana-3856 103 10 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 PROPN cana-3856 103 11 + + PUNCT cana-3856 103 12 1)π½π‘›βˆ’1Γ—1 1)π½π‘›βˆ’1Γ—1 NUM cana-3856 103 13 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 104 1 + + CCONJ cana-3856 104 2 1)𝐽1Γ—π‘›βˆ’1 1)𝐽1Γ—π‘›βˆ’1 PROPN cana-3856 104 3 πœ†πΌ1Γ—1 πœ†πΌ1Γ—1 NOUN cana-3856 104 4 ] ] PUNCT cana-3856 104 5 𝑛×𝑛 𝑛×𝑛 PROPN cana-3856 104 6 if if SCONJ cana-3856 104 7 m m NOUN cana-3856 104 8 is be AUX cana-3856 104 9 non non ADJ cana-3856 104 10 - - ADJ cana-3856 104 11 singular singular ADJ cana-3856 104 12 square square ADJ cana-3856 104 13 matrix matrix NOUN cana-3856 104 14 𝑑𝑒𝑑 𝑑𝑒𝑑 NOUN cana-3856 104 15 [ [ PUNCT cana-3856 104 16 𝑀 𝑀 NOUN cana-3856 104 17 𝑁 𝑁 PROPN cana-3856 104 18 𝑃 𝑃 NOUN cana-3856 104 19 𝑄 𝑄 PRON cana-3856 104 20 ] ] PUNCT cana-3856 104 21 = = SYM cana-3856 104 22 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 NOUN cana-3856 104 23 βˆ’ βˆ’ PROPN cana-3856 104 24 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 104 25 ) ) PUNCT cana-3856 104 26 here here ADV cana-3856 104 27 𝑀 𝑀 PROPN cana-3856 104 28 = = PUNCT cana-3856 105 1 πœ†πΌπ‘›βˆ’1Γ—π‘›βˆ’1 πœ†πΌπ‘›βˆ’1Γ—π‘›βˆ’1 NOUN cana-3856 105 2 βˆ’ βˆ’ PROPN cana-3856 105 3 ( ( PUNCT cana-3856 105 4 𝑛 𝑛 PROPN cana-3856 105 5 + + NUM cana-3856 105 6 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 NOUN cana-3856 105 7 , , PUNCT cana-3856 105 8 𝑁 𝑁 PROPN cana-3856 105 9 = = SYM cana-3856 105 10 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 PROPN cana-3856 105 11 + + NUM cana-3856 105 12 1)π½π‘›βˆ’1Γ—1 1)π½π‘›βˆ’1Γ—1 NUM cana-3856 105 13 , , PUNCT cana-3856 105 14 𝑃 𝑃 NOUN cana-3856 105 15 = = SYM cana-3856 105 16 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 PROPN cana-3856 105 17 + + CCONJ cana-3856 105 18 1)𝐽1Γ—π‘›βˆ’1 1)𝐽1Γ—π‘›βˆ’1 NUM cana-3856 105 19 and and CCONJ cana-3856 105 20 𝑄 𝑄 PROPN cana-3856 105 21 = = SYM cana-3856 105 22 πœ†πΌ1Γ—1 πœ†πΌ1Γ—1 PROPN cana-3856 105 23 , , PUNCT cana-3856 105 24 then then ADV cana-3856 105 25 𝑑𝑒𝑑(𝑀 𝑑𝑒𝑑(𝑀 NUM cana-3856 105 26 ) ) PUNCT cana-3856 105 27 = = SYM cana-3856 105 28 πœ†π‘›βˆ’2(πœ† πœ†π‘›βˆ’2(πœ† PROPN cana-3856 106 1 βˆ’ βˆ’ PROPN cana-3856 106 2 ( ( PUNCT cana-3856 106 3 𝑛2 𝑛2 NOUN cana-3856 106 4 βˆ’ βˆ’ PROPN cana-3856 106 5 1)√2 1)√2 NUM cana-3856 106 6 ) ) PUNCT cana-3856 106 7 π‘€βˆ’1 π‘€βˆ’1 NOUN cana-3856 106 8 = = SYM cana-3856 106 9 1 1 NUM cana-3856 106 10 πœ†π‘›βˆ’2(πœ† πœ†π‘›βˆ’2(πœ† NOUN cana-3856 106 11 βˆ’ βˆ’ PROPN cana-3856 106 12 ( ( PUNCT cana-3856 106 13 𝑛2 𝑛2 NOUN cana-3856 106 14 βˆ’ βˆ’ PROPN cana-3856 106 15 1)√2 1)√2 NUM cana-3856 106 16 ) ) PUNCT cana-3856 106 17 ( ( PUNCT cana-3856 106 18 πœ†π‘›βˆ’3(πœ† πœ†π‘›βˆ’3(πœ† PROPN cana-3856 107 1 βˆ’ βˆ’ PROPN cana-3856 107 2 ( ( PUNCT cana-3856 107 3 𝑛2 𝑛2 NOUN cana-3856 107 4 βˆ’ βˆ’ PROPN cana-3856 107 5 1)√2πΌπ‘›βˆ’1Γ—π‘›βˆ’1 1)√2πΌπ‘›βˆ’1Γ—π‘›βˆ’1 NUM cana-3856 107 6 ) ) PUNCT cana-3856 107 7 + + CCONJ cana-3856 107 8 πœ† πœ† X cana-3856 107 9 π‘›βˆ’3(𝑛 π‘›βˆ’3(𝑛 ADJ cana-3856 107 10 + + CCONJ cana-3856 107 11 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 NOUN cana-3856 107 12 ) ) PUNCT cana-3856 107 13 π‘ƒπ‘€βˆ’1 π‘ƒπ‘€βˆ’1 NOUN cana-3856 107 14 = = PUNCT cana-3856 108 1 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 109 1 + + CCONJ cana-3856 109 2 1)𝐽1Γ—π‘›βˆ’1 1)𝐽1Γ—π‘›βˆ’1 NUM cana-3856 109 3 Γ— Γ— NOUN cana-3856 109 4 1 1 NUM cana-3856 109 5 πœ†π‘›βˆ’2(πœ† πœ†π‘›βˆ’2(πœ† NOUN cana-3856 109 6 βˆ’ βˆ’ PROPN cana-3856 109 7 ( ( PUNCT cana-3856 109 8 𝑛2 𝑛2 NOUN cana-3856 109 9 βˆ’ βˆ’ PROPN cana-3856 109 10 1)√2 1)√2 NUM cana-3856 109 11 ) ) PUNCT cana-3856 109 12 ( ( PUNCT cana-3856 109 13 πœ†π‘›βˆ’3(πœ† πœ†π‘›βˆ’3(πœ† PROPN cana-3856 110 1 βˆ’ βˆ’ PROPN cana-3856 110 2 ( ( PUNCT cana-3856 110 3 𝑛2 𝑛2 NOUN cana-3856 110 4 βˆ’ βˆ’ PROPN cana-3856 110 5 1)√2πΌπ‘›βˆ’1Γ—π‘›βˆ’1 1)√2πΌπ‘›βˆ’1Γ—π‘›βˆ’1 NUM cana-3856 110 6 ) ) PUNCT cana-3856 111 1 + + NUM cana-3856 111 2 πœ†π‘›βˆ’3(𝑛 πœ†π‘›βˆ’3(𝑛 NOUN cana-3856 112 1 + + CCONJ cana-3856 112 2 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 1)√2π½π‘›βˆ’1Γ—π‘›βˆ’1 NUM cana-3856 112 3 ) ) PUNCT cana-3856 112 4 = = PUNCT cana-3856 113 1 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 113 2 + + CCONJ cana-3856 113 3 1)𝐽1Γ—π‘›βˆ’1 1)𝐽1Γ—π‘›βˆ’1 NUM cana-3856 113 4 Γ— Γ— NOUN cana-3856 113 5 [ [ PUNCT cana-3856 113 6 1 1 NUM cana-3856 113 7 πœ† πœ† DET cana-3856 113 8 πΌπ‘›βˆ’1Γ—π‘›βˆ’1 πΌπ‘›βˆ’1Γ—π‘›βˆ’1 NOUN cana-3856 113 9 + + CCONJ cana-3856 113 10 ( ( PUNCT cana-3856 113 11 𝑛 𝑛 PRON cana-3856 113 12 + + SYM cana-3856 113 13 1)√2 1)√2 NUM cana-3856 113 14 πœ†(πœ† πœ†(πœ† PROPN cana-3856 113 15 βˆ’ βˆ’ PROPN cana-3856 113 16 ( ( PUNCT cana-3856 113 17 𝑛2 𝑛2 NOUN cana-3856 113 18 βˆ’ βˆ’ PROPN cana-3856 113 19 1)√2 1)√2 NUM cana-3856 113 20 ) ) PUNCT cana-3856 113 21 π½π‘›βˆ’1Γ—π‘›βˆ’1 π½π‘›βˆ’1Γ—π‘›βˆ’1 NOUN cana-3856 113 22 ] ] X cana-3856 113 23 = = PUNCT cana-3856 114 1 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 114 2 + + CCONJ cana-3856 114 3 1 1 X cana-3856 114 4 ) ) PUNCT cana-3856 114 5 πœ† πœ† DET cana-3856 114 6 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 NOUN cana-3856 114 7 βˆ’ βˆ’ NOUN cana-3856 114 8 √2(𝑛2 √2(𝑛2 NOUN cana-3856 114 9 + + PUNCT cana-3856 115 1 1)(𝑛 1)(𝑛 NUM cana-3856 115 2 + + SYM cana-3856 115 3 1)√2 1)√2 NUM cana-3856 115 4 πœ†(πœ† πœ†(πœ† PROPN cana-3856 115 5 βˆ’ βˆ’ PROPN cana-3856 115 6 ( ( PUNCT cana-3856 115 7 𝑛2 𝑛2 NOUN cana-3856 115 8 βˆ’ βˆ’ PROPN cana-3856 115 9 1)√2 1)√2 NUM cana-3856 115 10 ) ) PUNCT cana-3856 115 11 Γ— Γ— NOUN cana-3856 115 12 ( ( PUNCT cana-3856 115 13 𝑛 𝑛 PROPN cana-3856 115 14 βˆ’ βˆ’ PROPN cana-3856 115 15 1)𝐽1Γ—π‘›βˆ’1 1)𝐽1Γ—π‘›βˆ’1 NOUN cana-3856 115 16 = = SYM cana-3856 116 1 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 116 2 + + CCONJ cana-3856 116 3 1 1 X cana-3856 116 4 ) ) PUNCT cana-3856 116 5 πœ† πœ† PRON cana-3856 117 1 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 NUM cana-3856 118 1 βˆ’ βˆ’ PROPN cana-3856 118 2 2(𝑛2 2(𝑛2 NUM cana-3856 118 3 βˆ’ βˆ’ NOUN cana-3856 118 4 1)√(𝑛2 1)√(𝑛2 NUM cana-3856 119 1 + + CCONJ cana-3856 119 2 1 1 X cana-3856 119 3 ) ) PUNCT cana-3856 119 4 πœ†(πœ† πœ†(πœ† PROPN cana-3856 119 5 βˆ’ βˆ’ PROPN cana-3856 119 6 ( ( PUNCT cana-3856 119 7 𝑛2 𝑛2 NOUN cana-3856 119 8 βˆ’ βˆ’ PROPN cana-3856 119 9 1)√2 1)√2 NUM cana-3856 119 10 ) ) PUNCT cana-3856 119 11 Γ— Γ— NOUN cana-3856 119 12 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 NOUN cana-3856 120 1 = = PUNCT cana-3856 120 2 [ [ PUNCT cana-3856 120 3 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 120 4 + + CCONJ cana-3856 120 5 1 1 X cana-3856 120 6 ) ) PUNCT cana-3856 120 7 πœ† πœ† DET cana-3856 120 8 βˆ’ βˆ’ PROPN cana-3856 120 9 2(𝑛2 2(𝑛2 NUM cana-3856 120 10 βˆ’ βˆ’ NOUN cana-3856 120 11 1)√(𝑛2 1)√(𝑛2 NUM cana-3856 121 1 + + CCONJ cana-3856 121 2 1 1 X cana-3856 121 3 ) ) PUNCT cana-3856 121 4 πœ†(πœ† πœ†(πœ† PROPN cana-3856 121 5 βˆ’ βˆ’ PROPN cana-3856 121 6 ( ( PUNCT cana-3856 121 7 𝑛2 𝑛2 NOUN cana-3856 121 8 βˆ’ βˆ’ PROPN cana-3856 121 9 1)√2 1)√2 NUM cana-3856 121 10 ) ) PUNCT cana-3856 121 11 ] ] PUNCT cana-3856 121 12 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 X cana-3856 122 1 = = X cana-3856 122 2 [ [ PUNCT cana-3856 122 3 βˆ’(πœ† βˆ’(πœ† NOUN cana-3856 122 4 βˆ’ βˆ’ PROPN cana-3856 122 5 ( ( PUNCT cana-3856 122 6 𝑛2 𝑛2 NOUN cana-3856 122 7 βˆ’ βˆ’ PROPN cana-3856 122 8 1)√2)√2(𝑛2 1)√2)√2(𝑛2 NUM cana-3856 122 9 + + CCONJ cana-3856 122 10 1 1 NUM cana-3856 122 11 ) ) PUNCT cana-3856 122 12 βˆ’ βˆ’ PROPN cana-3856 122 13 2(𝑛2 2(𝑛2 NUM cana-3856 122 14 βˆ’ βˆ’ NOUN cana-3856 122 15 1)√(𝑛2 1)√(𝑛2 NUM cana-3856 123 1 + + CCONJ cana-3856 123 2 1 1 X cana-3856 123 3 ) ) PUNCT cana-3856 123 4 πœ†(πœ† πœ†(πœ† PROPN cana-3856 123 5 βˆ’ βˆ’ PROPN cana-3856 123 6 ( ( PUNCT cana-3856 123 7 𝑛2 𝑛2 NOUN cana-3856 123 8 βˆ’ βˆ’ PROPN cana-3856 123 9 1)√2 1)√2 NUM cana-3856 123 10 ) ) PUNCT cana-3856 123 11 ] ] PUNCT cana-3856 123 12 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 X cana-3856 124 1 = = X cana-3856 124 2 [ [ PUNCT cana-3856 124 3 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 124 4 + + PUNCT cana-3856 124 5 1)πœ† 1)πœ† NUM cana-3856 124 6 πœ†(πœ† πœ†(πœ† PROPN cana-3856 124 7 βˆ’ βˆ’ PROPN cana-3856 124 8 ( ( PUNCT cana-3856 124 9 𝑛2 𝑛2 NOUN cana-3856 124 10 βˆ’ βˆ’ PROPN cana-3856 124 11 1)√2 1)√2 NUM cana-3856 124 12 ) ) PUNCT cana-3856 124 13 ] ] PUNCT cana-3856 124 14 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 X cana-3856 125 1 = = X cana-3856 125 2 [ [ PUNCT cana-3856 125 3 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 125 4 + + CCONJ cana-3856 125 5 1 1 X cana-3856 125 6 ) ) PUNCT cana-3856 125 7 ( ( PUNCT cana-3856 125 8 πœ† πœ† X cana-3856 125 9 βˆ’ βˆ’ PROPN cana-3856 125 10 ( ( PUNCT cana-3856 125 11 𝑛2 𝑛2 NOUN cana-3856 125 12 βˆ’ βˆ’ PROPN cana-3856 125 13 1)√2 1)√2 NUM cana-3856 125 14 ) ) PUNCT cana-3856 125 15 ] ] PUNCT cana-3856 125 16 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 NOUN cana-3856 125 17 communications communication NOUN cana-3856 125 18 on on ADP cana-3856 125 19 applied apply VERB cana-3856 125 20 nonlinear nonlinear ADJ cana-3856 125 21 analysis analysis NOUN cana-3856 125 22 issn issn NOUN cana-3856 125 23 : : PUNCT cana-3856 125 24 1074 1074 NUM cana-3856 125 25 - - PUNCT cana-3856 125 26 133x 133x NUM cana-3856 125 27 vol vol NOUN cana-3856 125 28 32 32 NUM cana-3856 126 1 no no NOUN cana-3856 126 2 . . PUNCT cana-3856 127 1 9s 9s NUM cana-3856 127 2 ( ( PUNCT cana-3856 127 3 2025 2025 NUM cana-3856 127 4 ) ) PUNCT cana-3856 127 5 289 289 NUM cana-3856 127 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 127 7 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 127 8 = = PUNCT cana-3856 127 9 [ [ PUNCT cana-3856 127 10 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 127 11 + + CCONJ cana-3856 127 12 1 1 X cana-3856 127 13 ) ) PUNCT cana-3856 127 14 ( ( PUNCT cana-3856 127 15 πœ† πœ† X cana-3856 127 16 βˆ’ βˆ’ PROPN cana-3856 127 17 ( ( PUNCT cana-3856 127 18 𝑛2 𝑛2 NOUN cana-3856 127 19 βˆ’ βˆ’ PROPN cana-3856 127 20 1)√2 1)√2 NUM cana-3856 127 21 ) ) PUNCT cana-3856 127 22 ] ] X cana-3856 127 23 𝐽1Γ—π‘›βˆ’1 𝐽1Γ—π‘›βˆ’1 X cana-3856 128 1 Γ— Γ— NOUN cana-3856 128 2 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 PROPN cana-3856 128 3 + + PUNCT cana-3856 128 4 1)π½π‘›βˆ’1Γ—1 1)π½π‘›βˆ’1Γ—1 NUM cana-3856 128 5 = = SYM cana-3856 129 1 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 129 2 + + CCONJ cana-3856 129 3 1 1 X cana-3856 129 4 ) ) PUNCT cana-3856 129 5 Γ— Γ— NOUN cana-3856 129 6 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 129 7 + + PUNCT cana-3856 130 1 1)(𝑛 1)(𝑛 NUM cana-3856 130 2 βˆ’ βˆ’ NOUN cana-3856 130 3 1 1 NUM cana-3856 130 4 ) ) PUNCT cana-3856 130 5 ( ( PUNCT cana-3856 130 6 πœ† πœ† X cana-3856 130 7 βˆ’ βˆ’ PROPN cana-3856 130 8 ( ( PUNCT cana-3856 130 9 𝑛2 𝑛2 NOUN cana-3856 130 10 βˆ’ βˆ’ PROPN cana-3856 130 11 1)√2 1)√2 NUM cana-3856 130 12 ) ) PUNCT cana-3856 130 13 𝐽1Γ—1 𝐽1Γ—1 NOUN cana-3856 130 14 = = SYM cana-3856 130 15 2(𝑛2 2(𝑛2 NUM cana-3856 130 16 + + CCONJ cana-3856 130 17 1)(𝑛 1)(𝑛 NUM cana-3856 130 18 βˆ’ βˆ’ NOUN cana-3856 130 19 1 1 NUM cana-3856 130 20 ) ) PUNCT cana-3856 130 21 ( ( PUNCT cana-3856 130 22 πœ† πœ† X cana-3856 130 23 βˆ’ βˆ’ PROPN cana-3856 130 24 ( ( PUNCT cana-3856 130 25 𝑛2 𝑛2 NOUN cana-3856 130 26 βˆ’ βˆ’ PROPN cana-3856 130 27 1)√2 1)√2 NUM cana-3856 130 28 ) ) PUNCT cana-3856 130 29 𝐽1Γ—1 𝐽1Γ—1 VERB cana-3856 130 30 𝑄 𝑄 PROPN cana-3856 130 31 βˆ’ βˆ’ NOUN cana-3856 130 32 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 130 33 = = PUNCT cana-3856 130 34 πœ†πΌ1Γ—1 πœ†πΌ1Γ—1 NOUN cana-3856 130 35 βˆ’ βˆ’ PROPN cana-3856 130 36 2(𝑛2 2(𝑛2 NUM cana-3856 130 37 + + CCONJ cana-3856 130 38 1)(𝑛 1)(𝑛 NUM cana-3856 130 39 βˆ’ βˆ’ NOUN cana-3856 130 40 1 1 NUM cana-3856 130 41 ) ) PUNCT cana-3856 130 42 ( ( PUNCT cana-3856 130 43 πœ† πœ† X cana-3856 130 44 βˆ’ βˆ’ PROPN cana-3856 130 45 ( ( PUNCT cana-3856 130 46 𝑛2 𝑛2 NOUN cana-3856 130 47 βˆ’ βˆ’ PROPN cana-3856 130 48 1)√2 1)√2 NUM cana-3856 130 49 ) ) PUNCT cana-3856 130 50 𝐽1Γ—1 𝐽1Γ—1 NOUN cana-3856 130 51 therefore therefore ADV cana-3856 130 52 , , PUNCT cana-3856 130 53 𝑑𝑒𝑑(𝑄 𝑑𝑒𝑑(𝑄 NUM cana-3856 130 54 βˆ’ βˆ’ PROPN cana-3856 130 55 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 130 56 ) ) PUNCT cana-3856 130 57 = = PUNCT cana-3856 131 1 πœ† πœ† PRON cana-3856 131 2 βˆ’ βˆ’ PROPN cana-3856 131 3 2(𝑛2 2(𝑛2 NUM cana-3856 132 1 + + CCONJ cana-3856 133 1 1)(𝑛 1)(𝑛 NUM cana-3856 133 2 βˆ’ βˆ’ NOUN cana-3856 133 3 1 1 NUM cana-3856 133 4 ) ) PUNCT cana-3856 133 5 ( ( PUNCT cana-3856 133 6 πœ† πœ† X cana-3856 133 7 βˆ’ βˆ’ PROPN cana-3856 133 8 ( ( PUNCT cana-3856 133 9 𝑛2 𝑛2 NOUN cana-3856 133 10 βˆ’ βˆ’ PROPN cana-3856 133 11 1)√2 1)√2 NUM cana-3856 133 12 ) ) PUNCT cana-3856 133 13 𝑑𝑒𝑑(πœ†πΌ 𝑑𝑒𝑑(πœ†πΌ NOUN cana-3856 133 14 βˆ’ βˆ’ NOUN cana-3856 133 15 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 133 16 ) ) PUNCT cana-3856 133 17 ) ) PUNCT cana-3856 134 1 = = PUNCT cana-3856 134 2 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 NOUN cana-3856 134 3 βˆ’ βˆ’ PROPN cana-3856 134 4 𝑃𝑀 𝑃𝑀 PROPN cana-3856 134 5 βˆ’1𝑁 βˆ’1𝑁 PROPN cana-3856 134 6 ) ) PUNCT cana-3856 134 7 = = PUNCT cana-3856 135 1 πœ†π‘›βˆ’2(πœ† πœ†π‘›βˆ’2(πœ† PROPN cana-3856 135 2 βˆ’ βˆ’ PROPN cana-3856 135 3 ( ( PUNCT cana-3856 135 4 𝑛2 𝑛2 NOUN cana-3856 135 5 βˆ’ βˆ’ PROPN cana-3856 135 6 1)√2 1)√2 NUM cana-3856 135 7 ) ) PUNCT cana-3856 135 8 Γ— Γ— NOUN cana-3856 136 1 [ [ X cana-3856 136 2 πœ† πœ† X cana-3856 136 3 βˆ’ βˆ’ PROPN cana-3856 136 4 2(𝑛2 2(𝑛2 NUM cana-3856 137 1 + + CCONJ cana-3856 137 2 1)(𝑛 1)(𝑛 NUM cana-3856 137 3 βˆ’ βˆ’ NOUN cana-3856 137 4 1 1 NUM cana-3856 137 5 ) ) PUNCT cana-3856 137 6 ( ( PUNCT cana-3856 137 7 πœ† πœ† X cana-3856 137 8 βˆ’ βˆ’ PROPN cana-3856 137 9 ( ( PUNCT cana-3856 137 10 𝑛2 𝑛2 NOUN cana-3856 137 11 βˆ’ βˆ’ PROPN cana-3856 137 12 1)√2 1)√2 NUM cana-3856 137 13 ) ) PUNCT cana-3856 137 14 ] ] PUNCT cana-3856 138 1 = = PUNCT cana-3856 138 2 πœ†π‘›βˆ’2(πœ† πœ†π‘›βˆ’2(πœ† PROPN cana-3856 138 3 βˆ’ βˆ’ PROPN cana-3856 139 1 ( ( PUNCT cana-3856 139 2 𝑛2 𝑛2 NOUN cana-3856 139 3 βˆ’ βˆ’ PROPN cana-3856 139 4 1)√2)(πœ† 1)√2)(πœ† NUM cana-3856 139 5 βˆ’ βˆ’ PROPN cana-3856 139 6 2(𝑛2 2(𝑛2 NUM cana-3856 140 1 + + CCONJ cana-3856 140 2 1)(𝑛 1)(𝑛 NUM cana-3856 140 3 βˆ’ βˆ’ NOUN cana-3856 140 4 1 1 NUM cana-3856 140 5 ) ) PUNCT cana-3856 140 6 ( ( PUNCT cana-3856 140 7 πœ† πœ† X cana-3856 140 8 βˆ’ βˆ’ PROPN cana-3856 140 9 ( ( PUNCT cana-3856 140 10 𝑛2 𝑛2 NOUN cana-3856 140 11 βˆ’ βˆ’ PROPN cana-3856 140 12 1)√2 1)√2 NUM cana-3856 140 13 ) ) PUNCT cana-3856 140 14 ) ) PUNCT cana-3856 141 1 = = PUNCT cana-3856 142 1 πœ†π‘›βˆ’2(πœ†2 πœ†π‘›βˆ’2(πœ†2 PROPN cana-3856 142 2 βˆ’ βˆ’ PROPN cana-3856 142 3 ( ( PUNCT cana-3856 142 4 𝑛2 𝑛2 NOUN cana-3856 142 5 βˆ’ βˆ’ PROPN cana-3856 142 6 1)√2πœ† 1)√2πœ† NUM cana-3856 142 7 βˆ’ βˆ’ PROPN cana-3856 142 8 2(𝑛2 2(𝑛2 NUM cana-3856 142 9 + + CCONJ cana-3856 143 1 1)(𝑛 1)(𝑛 NUM cana-3856 143 2 βˆ’ βˆ’ NOUN cana-3856 143 3 1 1 NUM cana-3856 143 4 ) ) PUNCT cana-3856 143 5 ) ) PUNCT cana-3856 144 1 = = PUNCT cana-3856 145 1 πœ†π‘›βˆ’2(πœ† πœ†π‘›βˆ’2(πœ† PROPN cana-3856 145 2 βˆ’ βˆ’ PROPN cana-3856 145 3 ( ( PUNCT cana-3856 145 4 𝑛2 𝑛2 NOUN cana-3856 145 5 βˆ’ βˆ’ PROPN cana-3856 145 6 1)√2 1)√2 NOUN cana-3856 145 7 βˆ’ βˆ’ PROPN cana-3856 145 8 √2𝑛4 √2𝑛4 NOUN cana-3856 145 9 + + CCONJ cana-3856 145 10 8𝑛3 8𝑛3 NUM cana-3856 145 11 βˆ’ βˆ’ PROPN cana-3856 145 12 12𝑛2 12𝑛2 NUM cana-3856 145 13 + + NUM cana-3856 145 14 8𝑛 8𝑛 NOUN cana-3856 145 15 βˆ’ βˆ’ NUM cana-3856 145 16 6 6 NUM cana-3856 145 17 2 2 NUM cana-3856 145 18 ) ) PUNCT cana-3856 145 19 ( ( PUNCT cana-3856 145 20 πœ† πœ† PROPN cana-3856 145 21 βˆ’ βˆ’ PROPN cana-3856 145 22 ( ( PUNCT cana-3856 145 23 𝑛2 𝑛2 NOUN cana-3856 145 24 βˆ’ βˆ’ PROPN cana-3856 145 25 1)√2 1)√2 PROPN cana-3856 145 26 + + CCONJ cana-3856 145 27 √2𝑛4 √2𝑛4 ADJ cana-3856 145 28 + + CCONJ cana-3856 145 29 8𝑛3 8𝑛3 NUM cana-3856 145 30 βˆ’ βˆ’ PROPN cana-3856 145 31 12𝑛2 12𝑛2 NUM cana-3856 145 32 + + NUM cana-3856 145 33 8𝑛 8𝑛 NOUN cana-3856 145 34 βˆ’ βˆ’ NUM cana-3856 145 35 6 6 NUM cana-3856 145 36 2 2 NUM cana-3856 145 37 ) ) PUNCT cana-3856 145 38 hence hence ADV cana-3856 145 39 , , PUNCT cana-3856 145 40 eigenvalues eigenvalue NOUN cana-3856 145 41 of of ADP cana-3856 145 42 𝐺𝑆 𝐺𝑆 PROPN cana-3856 145 43 are be AUX cana-3856 145 44 πœ†1 πœ†1 ADJ cana-3856 145 45 = = PUNCT cana-3856 145 46 ( ( PUNCT cana-3856 145 47 𝑛2βˆ’1)√2+√2𝑛4 𝑛2βˆ’1)√2+√2𝑛4 ADJ cana-3856 145 48 + + ADJ cana-3856 145 49 8𝑛3βˆ’12𝑛2 8𝑛3βˆ’12𝑛2 NUM cana-3856 145 50 + + ADJ cana-3856 145 51 8π‘›βˆ’6 8π‘›βˆ’6 NOUN cana-3856 145 52 2 2 NUM cana-3856 145 53 , , PUNCT cana-3856 145 54 πœ†2 πœ†2 NOUN cana-3856 145 55 = = PUNCT cana-3856 145 56 ( ( PUNCT cana-3856 145 57 𝑛2βˆ’1)√2βˆ’βˆš2𝑛4 𝑛2βˆ’1)√2βˆ’βˆš2𝑛4 VERB cana-3856 145 58 + + ADJ cana-3856 145 59 8𝑛3βˆ’12𝑛2 8𝑛3βˆ’12𝑛2 NUM cana-3856 145 60 + + ADJ cana-3856 145 61 8π‘›βˆ’6 8π‘›βˆ’6 NOUN cana-3856 145 62 2 2 NUM cana-3856 145 63 , , PUNCT cana-3856 145 64 and and CCONJ cana-3856 145 65 πœ†π‘– πœ†π‘– X cana-3856 145 66 = = NOUN cana-3856 145 67 0 0 NUM cana-3856 145 68 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN cana-3856 145 69 𝑖 𝑖 ADP cana-3856 145 70 β‰₯ β‰₯ NOUN cana-3856 145 71 3 3 NUM cana-3856 145 72 . . PUNCT cana-3856 145 73 theorem theorem NOUN cana-3856 145 74 3 3 X cana-3856 145 75 . . PUNCT cana-3856 146 1 let let VERB cana-3856 147 1 𝐺 𝐺 NOUN cana-3856 147 2 = = SYM cana-3856 147 3 𝐾𝑛 𝐾𝑛 PROPN cana-3856 147 4 be be AUX cana-3856 147 5 a a DET cana-3856 147 6 complete complete ADJ cana-3856 147 7 graph graph NOUN cana-3856 147 8 with with ADP cana-3856 147 9 n n ADP cana-3856 147 10 vertices vertex NOUN cana-3856 147 11 . . PUNCT cana-3856 148 1 if if SCONJ cana-3856 148 2 𝐺𝑆 𝐺𝑆 PROPN cana-3856 148 3 is be AUX cana-3856 148 4 a a DET cana-3856 148 5 graph graph NOUN cana-3856 148 6 obtained obtain VERB cana-3856 148 7 from from ADP cana-3856 148 8 𝐺 𝐺 PROPN cana-3856 148 9 by by ADP cana-3856 148 10 adding add VERB cana-3856 148 11 𝑛 𝑛 DET cana-3856 148 12 βˆ’ βˆ’ NUM cana-3856 148 13 1 1 NUM cana-3856 148 14 self self NOUN cana-3856 148 15 loops loop NOUN cana-3856 148 16 , , PUNCT cana-3856 148 17 then then ADV cana-3856 148 18 sombor sombor NOUN cana-3856 148 19 energy energy NOUN cana-3856 148 20 of of ADP cana-3856 148 21 𝐺𝑆 𝐺𝑆 PROPN cana-3856 148 22 is be AUX cana-3856 148 23 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NOUN cana-3856 148 24 ) ) PUNCT cana-3856 148 25 = = NOUN cana-3856 148 26 ( ( PUNCT cana-3856 148 27 𝑛 𝑛 PRON cana-3856 148 28 βˆ’ βˆ’ NUM cana-3856 148 29 2)√2 2)√2 NUM cana-3856 148 30 𝑛2βˆ’1 𝑛2βˆ’1 NOUN cana-3856 148 31 𝑛 𝑛 PROPN cana-3856 148 32 + + X cana-3856 148 33 √2𝑛4 √2𝑛4 ADV cana-3856 148 34 + + CCONJ cana-3856 148 35 8𝑛3 8𝑛3 NUM cana-3856 148 36 βˆ’ βˆ’ PROPN cana-3856 148 37 12𝑛2 12𝑛2 NUM cana-3856 148 38 + + NUM cana-3856 148 39 8𝑛 8𝑛 NOUN cana-3856 148 40 βˆ’ βˆ’ NUM cana-3856 148 41 6 6 NUM cana-3856 148 42 . . PUNCT cana-3856 149 1 proof proof NOUN cana-3856 149 2 . . PUNCT cana-3856 150 1 by by ADP cana-3856 150 2 definition definition NOUN cana-3856 150 3 sombor sombor NOUN cana-3856 150 4 energy energy NOUN cana-3856 150 5 of of ADP cana-3856 150 6 matrix matrix NOUN cana-3856 150 7 with with ADP cana-3856 150 8 self self NOUN cana-3856 150 9 loop loop NOUN cana-3856 150 10 is be AUX cana-3856 150 11 , , PUNCT cana-3856 150 12 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 NOUN cana-3856 150 13 ) ) PUNCT cana-3856 150 14 = = NOUN cana-3856 150 15 βˆ‘ βˆ‘ PROPN cana-3856 150 16 𝑛 𝑛 PRON cana-3856 150 17 𝑖=1 𝑖=1 PROPN cana-3856 150 18 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 NUM cana-3856 150 19 ) ) PUNCT cana-3856 150 20 βˆ’ βˆ’ PROPN cana-3856 151 1 √2βˆ‘πœŽπ‘—=1 √2βˆ‘πœŽπ‘—=1 ADV cana-3856 151 2 𝑑𝑗 𝑑𝑗 ADP cana-3856 151 3 𝑛 𝑛 PRON cana-3856 151 4 | | ADV cana-3856 151 5 by by ADP cana-3856 151 6 lemma lemma PROPN cana-3856 151 7 1 1 NUM cana-3856 151 8 , , PUNCT cana-3856 151 9 for for ADP cana-3856 151 10 complete complete ADJ cana-3856 151 11 graph graph NOUN cana-3856 151 12 with with ADP cana-3856 151 13 n-1 n-1 ADJ cana-3856 151 14 selfloop selfloop NOUN cana-3856 151 15 communications communication NOUN cana-3856 151 16 on on ADP cana-3856 151 17 applied apply VERB cana-3856 151 18 nonlinear nonlinear ADJ cana-3856 151 19 analysis analysis NOUN cana-3856 151 20 issn issn NOUN cana-3856 151 21 : : PUNCT cana-3856 151 22 1074 1074 NUM cana-3856 151 23 - - PUNCT cana-3856 151 24 133x 133x NUM cana-3856 151 25 vol vol NOUN cana-3856 151 26 32 32 NUM cana-3856 151 27 no no NOUN cana-3856 151 28 . . PUNCT cana-3856 152 1 9s 9s NUM cana-3856 152 2 ( ( PUNCT cana-3856 152 3 2025 2025 NUM cana-3856 152 4 ) ) PUNCT cana-3856 152 5 290 290 NUM cana-3856 153 1 https://internationalpubls.com https://internationalpubls.com X cana-3856 153 2 βˆ‘πœŽπ‘—=1 βˆ‘πœŽπ‘—=1 ADV cana-3856 153 3 𝑑𝑗 𝑑𝑗 NOUN cana-3856 153 4 = = SYM cana-3856 153 5 𝑛2 𝑛2 NOUN cana-3856 153 6 βˆ’ βˆ’ PROPN cana-3856 153 7 1 1 NUM cana-3856 153 8 and and CCONJ cana-3856 153 9 its its PRON cana-3856 153 10 eigenvalues eigenvalue NOUN cana-3856 153 11 are be AUX cana-3856 153 12 πœ†1 πœ†1 ADJ cana-3856 153 13 = = SYM cana-3856 153 14 ( ( PUNCT cana-3856 153 15 𝑛2βˆ’1)√2+√2𝑛4 𝑛2βˆ’1)√2+√2𝑛4 ADJ cana-3856 153 16 + + ADJ cana-3856 153 17 8𝑛3βˆ’12𝑛2 8𝑛3βˆ’12𝑛2 NUM cana-3856 153 18 + + ADJ cana-3856 153 19 8π‘›βˆ’6 8π‘›βˆ’6 NOUN cana-3856 153 20 2 2 NUM cana-3856 153 21 , , PUNCT cana-3856 153 22 πœ†2 πœ†2 NOUN cana-3856 153 23 = = PUNCT cana-3856 153 24 ( ( PUNCT cana-3856 153 25 𝑛2βˆ’1)√2βˆ’βˆš2𝑛4 𝑛2βˆ’1)√2βˆ’βˆš2𝑛4 VERB cana-3856 153 26 + + ADJ cana-3856 153 27 8𝑛3βˆ’12𝑛2 8𝑛3βˆ’12𝑛2 NUM cana-3856 153 28 + + ADJ cana-3856 153 29 8π‘›βˆ’6 8π‘›βˆ’6 NOUN cana-3856 153 30 2 2 NUM cana-3856 153 31 , , PUNCT cana-3856 153 32 and and CCONJ cana-3856 153 33 πœ†π‘– πœ†π‘– X cana-3856 154 1 = = NOUN cana-3856 154 2 0 0 NUM cana-3856 155 1 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN cana-3856 155 2 𝑖 𝑖 DET cana-3856 155 3 β‰₯ β‰₯ NOUN cana-3856 155 4 3 3 NUM cana-3856 155 5 . . PUNCT cana-3856 155 6 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 NOUN cana-3856 155 7 ) ) PUNCT cana-3856 156 1 = = SYM cana-3856 156 2 |πœ†1 |πœ†1 NOUN cana-3856 156 3 βˆ’ βˆ’ PROPN cana-3856 157 1 ( ( PUNCT cana-3856 157 2 𝑛2 𝑛2 NOUN cana-3856 157 3 βˆ’ βˆ’ PROPN cana-3856 157 4 1)√2 1)√2 NUM cana-3856 157 5 𝑛 𝑛 PRON cana-3856 158 1 | | ADV cana-3856 159 1 + + CCONJ cana-3856 159 2 |πœ†2 |πœ†2 NOUN cana-3856 159 3 βˆ’ βˆ’ NOUN cana-3856 159 4 ( ( PUNCT cana-3856 159 5 𝑛2 𝑛2 NOUN cana-3856 159 6 βˆ’ βˆ’ PROPN cana-3856 159 7 1)√2 1)√2 NOUN cana-3856 159 8 𝑛 𝑛 ADP cana-3856 160 1 | | ADV cana-3856 160 2 + + NOUN cana-3856 160 3 βˆ‘ βˆ‘ PROPN cana-3856 160 4 𝑛 𝑛 PRON cana-3856 160 5 𝑖=3 𝑖=3 PROPN cana-3856 160 6 | | ADV cana-3856 160 7 βˆ’ βˆ’ PROPN cana-3856 160 8 ( ( PUNCT cana-3856 160 9 𝑛2 𝑛2 NOUN cana-3856 160 10 βˆ’ βˆ’ PROPN cana-3856 160 11 1)√2 1)√2 NUM cana-3856 160 12 𝑛 𝑛 DET cana-3856 160 13 | | NOUN cana-3856 160 14 = = SYM cana-3856 161 1 | | ADV cana-3856 161 2 ( ( PUNCT cana-3856 161 3 𝑛2 𝑛2 NOUN cana-3856 161 4 βˆ’ βˆ’ PROPN cana-3856 161 5 1)√2 1)√2 PROPN cana-3856 161 6 + + CCONJ cana-3856 161 7 √2𝑛4 √2𝑛4 ADJ cana-3856 161 8 + + CCONJ cana-3856 161 9 8𝑛3 8𝑛3 NUM cana-3856 162 1 βˆ’ βˆ’ PROPN cana-3856 162 2 12𝑛2 12𝑛2 NUM cana-3856 162 3 + + NUM cana-3856 162 4 8𝑛 8𝑛 NOUN cana-3856 162 5 βˆ’ βˆ’ NUM cana-3856 162 6 6 6 NUM cana-3856 162 7 2 2 NUM cana-3856 162 8 βˆ’ βˆ’ NOUN cana-3856 162 9 ( ( PUNCT cana-3856 162 10 𝑛2 𝑛2 NOUN cana-3856 162 11 βˆ’ βˆ’ PROPN cana-3856 162 12 1)√2 1)√2 NUM cana-3856 162 13 𝑛 𝑛 ADP cana-3856 163 1 | | NOUN cana-3856 164 1 + + CCONJ cana-3856 164 2 | | ADV cana-3856 164 3 ( ( PUNCT cana-3856 164 4 𝑛2 𝑛2 NOUN cana-3856 164 5 βˆ’ βˆ’ PROPN cana-3856 164 6 1)√2 1)√2 NOUN cana-3856 164 7 βˆ’ βˆ’ PROPN cana-3856 164 8 √2𝑛4 √2𝑛4 NOUN cana-3856 164 9 + + CCONJ cana-3856 165 1 8𝑛3 8𝑛3 NUM cana-3856 165 2 βˆ’ βˆ’ PROPN cana-3856 165 3 12𝑛2 12𝑛2 NUM cana-3856 165 4 + + NUM cana-3856 165 5 8𝑛 8𝑛 NOUN cana-3856 165 6 βˆ’ βˆ’ NUM cana-3856 165 7 6 6 NUM cana-3856 165 8 2 2 NUM cana-3856 165 9 βˆ’ βˆ’ NOUN cana-3856 165 10 ( ( PUNCT cana-3856 165 11 𝑛2 𝑛2 NOUN cana-3856 165 12 βˆ’ βˆ’ PROPN cana-3856 165 13 1)√2 1)√2 NUM cana-3856 165 14 𝑛 𝑛 ADP cana-3856 165 15 | | NOUN cana-3856 165 16 + + CCONJ cana-3856 165 17 ( ( PUNCT cana-3856 165 18 𝑛 𝑛 PRON cana-3856 165 19 βˆ’ βˆ’ PROPN cana-3856 165 20 2 2 NUM cana-3856 165 21 ) ) PUNCT cana-3856 165 22 ( ( PUNCT cana-3856 165 23 𝑛2 𝑛2 NOUN cana-3856 165 24 βˆ’ βˆ’ PROPN cana-3856 165 25 1)√2 1)√2 NOUN cana-3856 165 26 𝑛 𝑛 PROPN cana-3856 165 27 = = PUNCT cana-3856 165 28 ( ( PUNCT cana-3856 165 29 𝑛2 𝑛2 NOUN cana-3856 165 30 βˆ’ βˆ’ PROPN cana-3856 165 31 1)√2(𝑛 1)√2(𝑛 NUM cana-3856 165 32 βˆ’ βˆ’ PROPN cana-3856 165 33 2 2 NUM cana-3856 165 34 ) ) PUNCT cana-3856 165 35 2𝑛 2𝑛 NOUN cana-3856 166 1 + + CCONJ cana-3856 166 2 √2𝑛4 √2𝑛4 ADV cana-3856 166 3 + + CCONJ cana-3856 166 4 8𝑛3 8𝑛3 NUM cana-3856 166 5 βˆ’ βˆ’ PROPN cana-3856 166 6 12𝑛2 12𝑛2 NUM cana-3856 166 7 + + NUM cana-3856 166 8 8𝑛 8𝑛 NOUN cana-3856 166 9 βˆ’ βˆ’ NUM cana-3856 166 10 6 6 NUM cana-3856 166 11 2 2 NUM cana-3856 166 12 + + CCONJ cana-3856 166 13 [ [ X cana-3856 166 14 βˆ’ βˆ’ X cana-3856 166 15 ( ( PUNCT cana-3856 166 16 𝑛2 𝑛2 NOUN cana-3856 166 17 βˆ’ βˆ’ PROPN cana-3856 166 18 1)√2(𝑛 1)√2(𝑛 NUM cana-3856 166 19 βˆ’ βˆ’ PROPN cana-3856 166 20 2 2 NUM cana-3856 166 21 ) ) PUNCT cana-3856 166 22 2𝑛 2𝑛 NOUN cana-3856 167 1 + + CCONJ cana-3856 167 2 √2𝑛4 √2𝑛4 ADV cana-3856 167 3 + + CCONJ cana-3856 167 4 8𝑛3 8𝑛3 NUM cana-3856 167 5 βˆ’ βˆ’ PROPN cana-3856 167 6 12𝑛2 12𝑛2 NUM cana-3856 167 7 + + NUM cana-3856 167 8 8𝑛 8𝑛 NOUN cana-3856 167 9 βˆ’ βˆ’ NUM cana-3856 167 10 6 6 NUM cana-3856 167 11 2 2 NUM cana-3856 167 12 ] ] PUNCT cana-3856 167 13 + + CCONJ cana-3856 167 14 ( ( PUNCT cana-3856 167 15 𝑛 𝑛 PRON cana-3856 167 16 βˆ’ βˆ’ PROPN cana-3856 167 17 2 2 NUM cana-3856 167 18 ) ) PUNCT cana-3856 167 19 ( ( PUNCT cana-3856 167 20 𝑛2 𝑛2 NOUN cana-3856 167 21 βˆ’ βˆ’ PROPN cana-3856 167 22 1)√2 1)√2 NOUN cana-3856 167 23 𝑛 𝑛 ADP cana-3856 167 24 = = PUNCT cana-3856 167 25 √2𝑛4 √2𝑛4 NOUN cana-3856 167 26 + + CCONJ cana-3856 167 27 8𝑛3 8𝑛3 NUM cana-3856 168 1 βˆ’ βˆ’ PROPN cana-3856 168 2 12𝑛2 12𝑛2 NUM cana-3856 168 3 + + NUM cana-3856 168 4 8𝑛 8𝑛 NOUN cana-3856 168 5 βˆ’ βˆ’ NUM cana-3856 168 6 6 6 NUM cana-3856 168 7 + + CCONJ cana-3856 168 8 ( ( PUNCT cana-3856 168 9 𝑛 𝑛 PRON cana-3856 168 10 βˆ’ βˆ’ PROPN cana-3856 168 11 2 2 NUM cana-3856 168 12 ) ) PUNCT cana-3856 168 13 ( ( PUNCT cana-3856 168 14 𝑛2βˆ’1)√2 𝑛2βˆ’1)√2 VERB cana-3856 168 15 𝑛 𝑛 PROPN cana-3856 168 16 . . PUNCT cana-3856 168 17 example example NOUN cana-3856 168 18 1 1 NUM cana-3856 168 19 . . X cana-3856 169 1 consider consider VERB cana-3856 169 2 𝐺 𝐺 NOUN cana-3856 169 3 = = PRON cana-3856 169 4 𝐾3 𝐾3 VERB cana-3856 169 5 is be AUX cana-3856 169 6 complete complete ADJ cana-3856 169 7 graph graph NOUN cana-3856 169 8 with with ADP cana-3856 169 9 3 3 NUM cana-3856 169 10 vertices vertex NOUN cana-3856 169 11 and and CCONJ cana-3856 169 12 𝐺𝑆 𝐺𝑆 NOUN cana-3856 169 13 is be AUX cana-3856 169 14 the the DET cana-3856 169 15 graph graph NOUN cana-3856 169 16 obtained obtain VERB cana-3856 169 17 by by ADP cana-3856 169 18 adding add VERB cana-3856 169 19 2 2 NUM cana-3856 169 20 loops loop NOUN cana-3856 169 21 to to PART cana-3856 169 22 graph graph VERB cana-3856 169 23 𝐺 𝐺 PROPN cana-3856 169 24 = = PUNCT cana-3856 169 25 𝐾3 𝐾3 PROPN cana-3856 169 26 . . PUNCT cana-3856 170 1 the the DET cana-3856 170 2 sombor sombor NOUN cana-3856 170 3 energy energy NOUN cana-3856 170 4 of of ADP cana-3856 170 5 graph graph NOUN cana-3856 170 6 𝐺𝑆 𝐺𝑆 PROPN cana-3856 170 7 is be AUX cana-3856 170 8 given give VERB cana-3856 170 9 by by ADP cana-3856 170 10 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NOUN cana-3856 170 11 ) ) PUNCT cana-3856 170 12 = = PUNCT cana-3856 171 1 βˆ‘ βˆ‘ PUNCT cana-3856 171 2 3 3 NUM cana-3856 171 3 𝑖=1 𝑖=1 SYM cana-3856 171 4 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 NUM cana-3856 171 5 ) ) PUNCT cana-3856 171 6 βˆ’ βˆ’ X cana-3856 172 1 √2βˆ‘πœŽπ‘—=1𝑑𝑗 √2βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 172 2 3 3 NUM cana-3856 172 3 | | NOUN cana-3856 172 4 sombor sombor NOUN cana-3856 172 5 matrix matrix NOUN cana-3856 172 6 of of ADP cana-3856 172 7 𝐺𝑆 𝐺𝑆 PROPN cana-3856 172 8 is be AUX cana-3856 172 9 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 172 10 ) ) PUNCT cana-3856 173 1 = = NOUN cana-3856 173 2 [ [ PUNCT cana-3856 173 3 4√2 4√2 PROPN cana-3856 173 4 4√2 4√2 PROPN cana-3856 173 5 2√5 2√5 NUM cana-3856 174 1 4√2 4√2 PROPN cana-3856 174 2 4√2 4√2 PROPN cana-3856 174 3 2√5 2√5 NUM cana-3856 174 4 2√5 2√5 NUM cana-3856 174 5 2√5 2√5 PROPN cana-3856 174 6 0 0 NUM cana-3856 174 7 ] ] PUNCT cana-3856 174 8 communications communication NOUN cana-3856 174 9 on on ADP cana-3856 174 10 applied apply VERB cana-3856 174 11 nonlinear nonlinear ADJ cana-3856 174 12 analysis analysis NOUN cana-3856 174 13 issn issn NOUN cana-3856 174 14 : : PUNCT cana-3856 174 15 1074 1074 NUM cana-3856 174 16 - - PUNCT cana-3856 174 17 133x 133x NUM cana-3856 174 18 vol vol NOUN cana-3856 174 19 32 32 NUM cana-3856 174 20 no no NOUN cana-3856 174 21 . . PUNCT cana-3856 175 1 9s 9s NUM cana-3856 175 2 ( ( PUNCT cana-3856 175 3 2025 2025 NUM cana-3856 175 4 ) ) PUNCT cana-3856 175 5 291 291 NUM cana-3856 175 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 175 7 here here ADV cana-3856 175 8 πœ†1 πœ†1 X cana-3856 175 9 = = PUNCT cana-3856 176 1 10√2 10√2 NUM cana-3856 176 2 , , PUNCT cana-3856 176 3 πœ†2 πœ†2 PROPN cana-3856 176 4 = = SYM cana-3856 176 5 βˆ’2√2 βˆ’2√2 PROPN cana-3856 176 6 , , PUNCT cana-3856 176 7 πœ†3 πœ†3 NOUN cana-3856 176 8 = = NOUN cana-3856 176 9 0 0 NUM cana-3856 176 10 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NUM cana-3856 176 11 ) ) PUNCT cana-3856 176 12 = = SYM cana-3856 177 1 |10√2 |10√2 PROPN cana-3856 177 2 βˆ’ βˆ’ NOUN cana-3856 178 1 8√2 8√2 NUM cana-3856 178 2 3 3 NUM cana-3856 179 1 | | ADV cana-3856 180 1 + + CCONJ cana-3856 180 2 | | ADV cana-3856 180 3 βˆ’ βˆ’ PROPN cana-3856 180 4 2√2 2√2 NUM cana-3856 180 5 βˆ’ βˆ’ NOUN cana-3856 181 1 8√2 8√2 NUM cana-3856 181 2 3 3 NUM cana-3856 182 1 | | ADV cana-3856 182 2 + + NUM cana-3856 182 3 |0 |0 NUM cana-3856 182 4 βˆ’ βˆ’ NOUN cana-3856 182 5 8√2 8√2 NUM cana-3856 182 6 3 3 NUM cana-3856 182 7 | | ADV cana-3856 182 8 = = SYM cana-3856 182 9 20.741798915 20.741798915 NUM cana-3856 182 10 in in ADP cana-3856 182 11 above above ADP cana-3856 182 12 theorem theorem NOUN cana-3856 182 13 putting put VERB cana-3856 182 14 n=3 n=3 PUNCT cana-3856 182 15 we we PRON cana-3856 182 16 get get VERB cana-3856 182 17 , , PUNCT cana-3856 182 18 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NUM cana-3856 182 19 ) ) PUNCT cana-3856 182 20 = = NOUN cana-3856 183 1 ( ( PUNCT cana-3856 183 2 3 3 NUM cana-3856 183 3 βˆ’ βˆ’ NUM cana-3856 183 4 2)√2 2)√2 NOUN cana-3856 183 5 32 32 NUM cana-3856 183 6 βˆ’ βˆ’ NOUN cana-3856 183 7 1 1 NUM cana-3856 183 8 3 3 NUM cana-3856 183 9 + + NUM cana-3856 183 10 √2(3)4 √2(3)4 NOUN cana-3856 183 11 + + NOUN cana-3856 183 12 8(3)3 8(3)3 VERB cana-3856 183 13 βˆ’ βˆ’ PROPN cana-3856 183 14 12(3)2 12(3)2 NOUN cana-3856 183 15 + + SYM cana-3856 183 16 8(3 8(3 NUM cana-3856 183 17 ) ) PUNCT cana-3856 183 18 βˆ’ βˆ’ PROPN cana-3856 184 1 6 6 NUM cana-3856 184 2 = = SYM cana-3856 184 3 √2 √2 NOUN cana-3856 184 4 8 8 NUM cana-3856 184 5 3 3 NUM cana-3856 184 6 + + NUM cana-3856 184 7 √288 √288 X cana-3856 184 8 = = SYM cana-3856 184 9 √2 √2 NOUN cana-3856 184 10 8 8 NUM cana-3856 184 11 3 3 NUM cana-3856 184 12 + + NUM cana-3856 184 13 12√2 12√2 NUM cana-3856 184 14 = = SYM cana-3856 184 15 20.741798915 20.741798915 NUM cana-3856 184 16 . . PUNCT cana-3856 185 1 lemma lemma PROPN cana-3856 185 2 2 2 X cana-3856 185 3 . . PUNCT cana-3856 186 1 let let VERB cana-3856 187 1 𝐺 𝐺 NOUN cana-3856 187 2 = = SYM cana-3856 187 3 𝐾𝑛 𝐾𝑛 PROPN cana-3856 187 4 be be AUX cana-3856 187 5 a a DET cana-3856 187 6 complete complete ADJ cana-3856 187 7 graph graph NOUN cana-3856 187 8 with with ADP cana-3856 187 9 n n ADP cana-3856 187 10 vertices vertex NOUN cana-3856 187 11 . . PUNCT cana-3856 188 1 if if SCONJ cana-3856 188 2 𝐺𝑆 𝐺𝑆 PROPN cana-3856 188 3 is be AUX cana-3856 188 4 a a DET cana-3856 188 5 graph graph NOUN cana-3856 188 6 obtained obtain VERB cana-3856 188 7 from from ADP cana-3856 188 8 𝐺 𝐺 PROPN cana-3856 188 9 by by ADP cana-3856 188 10 adding add VERB cana-3856 188 11 𝜎 𝜎 PRON cana-3856 188 12 self self NOUN cana-3856 188 13 loops loop NOUN cana-3856 188 14 , , PUNCT cana-3856 188 15 then then ADV cana-3856 188 16 sombor sombor NOUN cana-3856 188 17 eigenvalues eigenvalue NOUN cana-3856 188 18 of of ADP cana-3856 188 19 sombor sombor NOUN cana-3856 188 20 matrix matrix NOUN cana-3856 188 21 𝐺𝑆 𝐺𝑆 PROPN cana-3856 188 22 are be AUX cana-3856 188 23 0 0 NUM cana-3856 188 24 with with ADP cana-3856 188 25 multiplicity multiplicity NOUN cana-3856 188 26 𝜎 𝜎 NOUN cana-3856 188 27 βˆ’ βˆ’ PROPN cana-3856 188 28 1 1 NUM cana-3856 188 29 , , PUNCT cana-3856 188 30 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-3856 189 1 βˆ’ βˆ’ PROPN cana-3856 189 2 1)√2 1)√2 NOUN cana-3856 189 3 with with ADP cana-3856 189 4 multiplicity multiplicity NOUN cana-3856 189 5 𝑛 𝑛 PRON cana-3856 189 6 βˆ’ βˆ’ PROPN cana-3856 189 7 𝜎 𝜎 NOUN cana-3856 189 8 βˆ’ βˆ’ PROPN cana-3856 189 9 1 1 NUM cana-3856 189 10 , , PUNCT cana-3856 189 11 √2((π‘›βˆ’1)2 √2((π‘›βˆ’1)2 VERB cana-3856 189 12 + + NOUN cana-3856 189 13 2𝜎)+√2((π‘›βˆ’1)2 2𝜎)+√2((π‘›βˆ’1)2 NUM cana-3856 189 14 + + NOUN cana-3856 189 15 2𝜎)2 2𝜎)2 NUM cana-3856 189 16 + + NOUN cana-3856 189 17 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ NUM cana-3856 189 18 ) ) PUNCT cana-3856 189 19 ) ) PUNCT cana-3856 190 1 2 2 NUM cana-3856 190 2 with with ADP cana-3856 190 3 multiplicity multiplicity NOUN cana-3856 190 4 1 1 NUM cana-3856 190 5 , , PUNCT cana-3856 190 6 and and CCONJ cana-3856 190 7 √2((π‘›βˆ’1)2 √2((π‘›βˆ’1)2 VERB cana-3856 190 8 + + NOUN cana-3856 190 9 2𝜎)βˆ’βˆš2((π‘›βˆ’1)2 2𝜎)βˆ’βˆš2((π‘›βˆ’1)2 ADJ cana-3856 190 10 + + ADJ cana-3856 190 11 2𝜎)2 2𝜎)2 NUM cana-3856 190 12 + + NOUN cana-3856 190 13 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ NUM cana-3856 190 14 ) ) PUNCT cana-3856 190 15 ) ) PUNCT cana-3856 190 16 2 2 NUM cana-3856 190 17 with with ADP cana-3856 190 18 multiplicity multiplicity NOUN cana-3856 190 19 1 1 NUM cana-3856 190 20 . . PUNCT cana-3856 191 1 proof proof NOUN cana-3856 191 2 . . PUNCT cana-3856 192 1 let let VERB cana-3856 192 2 j j PROPN cana-3856 192 3 be be AUX cana-3856 192 4 the the DET cana-3856 192 5 𝑛 𝑛 ADJ cana-3856 192 6 Γ— Γ— NOUN cana-3856 192 7 𝑛 𝑛 DET cana-3856 192 8 matrix matrix NOUN cana-3856 192 9 with with ADP cana-3856 192 10 all all DET cana-3856 192 11 entries entry NOUN cana-3856 192 12 one one NUM cana-3856 192 13 , , PUNCT cana-3856 192 14 i i PRON cana-3856 192 15 be be VERB cana-3856 192 16 𝑛 𝑛 DET cana-3856 192 17 Γ— Γ— NOUN cana-3856 192 18 𝑛 𝑛 PRON cana-3856 192 19 identity identity NOUN cana-3856 192 20 matrix matrix NOUN cana-3856 192 21 . . PUNCT cana-3856 193 1 sombor sombor NOUN cana-3856 193 2 matrix matrix NOUN cana-3856 193 3 of of ADP cana-3856 193 4 complete complete ADJ cana-3856 193 5 graph graph NOUN cana-3856 193 6 𝐺𝑆 𝐺𝑆 PROPN cana-3856 193 7 with with ADP cana-3856 193 8 𝜎 𝜎 PROPN cana-3856 193 9 self self NOUN cana-3856 193 10 loop loop NOUN cana-3856 193 11 is be AUX cana-3856 193 12 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 193 13 ) ) PUNCT cana-3856 194 1 = = PRON cana-3856 194 2 [ [ PUNCT cana-3856 194 3 ( ( PUNCT cana-3856 194 4 𝑛 𝑛 PROPN cana-3856 194 5 + + NUM cana-3856 194 6 1)√2π½πœŽΓ—πœŽ 1)√2π½πœŽΓ—πœŽ NUM cana-3856 194 7 √2(𝑛2 √2(𝑛2 NOUN cana-3856 194 8 + + PUNCT cana-3856 194 9 1)π½πœŽΓ—π‘›βˆ’πœŽ 1)π½πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 194 10 √2(𝑛2 √2(𝑛2 NOUN cana-3856 194 11 + + PUNCT cana-3856 194 12 1)π½π‘›βˆ’πœŽΓ—πœŽ 1)π½π‘›βˆ’πœŽΓ—πœŽ NUM cana-3856 194 13 ( ( PUNCT cana-3856 194 14 𝑛 𝑛 PROPN cana-3856 194 15 βˆ’ βˆ’ PROPN cana-3856 194 16 1)√2(𝐽 1)√2(𝐽 PRON cana-3856 194 17 βˆ’ βˆ’ PROPN cana-3856 195 1 𝐼)π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 𝐼)π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ X cana-3856 195 2 ] ] PUNCT cana-3856 195 3 𝑛×𝑛 𝑛×𝑛 PROPN cana-3856 195 4 then then ADV cana-3856 195 5 , , PUNCT cana-3856 195 6 det(πœ†πΌ det(πœ†πΌ NOUN cana-3856 195 7 βˆ’ βˆ’ NOUN cana-3856 195 8 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 195 9 ) ) PUNCT cana-3856 195 10 ) ) PUNCT cana-3856 196 1 = = PRON cana-3856 196 2 𝑑𝑒𝑑 𝑑𝑒𝑑 NOUN cana-3856 196 3 | | ADV cana-3856 196 4 πœ†πΌπœŽΓ—πœŽ πœ†πΌπœŽΓ—πœŽ X cana-3856 197 1 βˆ’ βˆ’ PROPN cana-3856 197 2 ( ( PUNCT cana-3856 197 3 𝑛 𝑛 PROPN cana-3856 197 4 + + NUM cana-3856 197 5 1)√2π½πœŽΓ—πœŽ 1)√2π½πœŽΓ—πœŽ NUM cana-3856 197 6 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 197 7 + + PUNCT cana-3856 197 8 1)π½πœŽΓ—π‘›βˆ’πœŽ 1)π½πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 197 9 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 198 1 + + CCONJ cana-3856 198 2 1)π½π‘›βˆ’πœŽΓ—πœŽ 1)π½π‘›βˆ’πœŽΓ—πœŽ NUM cana-3856 198 3 πœ†πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ πœ†πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NOUN cana-3856 198 4 βˆ’ βˆ’ PROPN cana-3856 198 5 ( ( PUNCT cana-3856 198 6 𝑛 𝑛 PROPN cana-3856 198 7 βˆ’ βˆ’ PROPN cana-3856 199 1 1)√2(𝐽 1)√2(𝐽 PRON cana-3856 200 1 βˆ’ βˆ’ NOUN cana-3856 201 1 𝐼)π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 𝐼)π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ PROPN cana-3856 201 2 | | ADJ cana-3856 201 3 𝑛×𝑛 𝑛×𝑛 PROPN cana-3856 201 4 det(πœ†πΌ det(πœ†πΌ NOUN cana-3856 201 5 βˆ’ βˆ’ NOUN cana-3856 201 6 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 201 7 ) ) PUNCT cana-3856 201 8 ) ) PUNCT cana-3856 202 1 = = PUNCT cana-3856 202 2 | | ADV cana-3856 202 3 πœ†πΌπœŽΓ—πœŽ πœ†πΌπœŽΓ—πœŽ X cana-3856 202 4 βˆ’ βˆ’ PROPN cana-3856 203 1 ( ( PUNCT cana-3856 203 2 𝑛 𝑛 PROPN cana-3856 203 3 + + NUM cana-3856 203 4 1)√2π½πœŽΓ—πœŽ 1)√2π½πœŽΓ—πœŽ NUM cana-3856 203 5 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 203 6 + + PUNCT cana-3856 203 7 1)π½πœŽΓ—π‘›βˆ’πœŽ 1)π½πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 203 8 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 204 1 + + CCONJ cana-3856 205 1 1)π½π‘›βˆ’πœŽΓ—πœŽ 1)π½π‘›βˆ’πœŽΓ—πœŽ NUM cana-3856 205 2 ( ( PUNCT cana-3856 205 3 πœ† πœ† PROPN cana-3856 205 4 + + X cana-3856 205 5 ( ( PUNCT cana-3856 205 6 𝑛 𝑛 PROPN cana-3856 205 7 βˆ’ βˆ’ PROPN cana-3856 205 8 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 205 9 βˆ’ βˆ’ PROPN cana-3856 205 10 ( ( PUNCT cana-3856 205 11 𝑛 𝑛 PROPN cana-3856 205 12 βˆ’ βˆ’ PROPN cana-3856 205 13 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 205 14 | | NOUN cana-3856 205 15 𝑛×𝑛 𝑛×𝑛 PROPN cana-3856 205 16 if if SCONJ cana-3856 205 17 m m NOUN cana-3856 205 18 is be AUX cana-3856 205 19 non non ADJ cana-3856 205 20 - - ADJ cana-3856 205 21 singular singular ADJ cana-3856 205 22 square square ADJ cana-3856 205 23 matrix matrix NOUN cana-3856 205 24 , , PUNCT cana-3856 205 25 𝑑𝑒𝑑 𝑑𝑒𝑑 NOUN cana-3856 205 26 [ [ PUNCT cana-3856 205 27 𝑀 𝑀 NOUN cana-3856 205 28 𝑁 𝑁 PROPN cana-3856 205 29 𝑃 𝑃 NOUN cana-3856 205 30 𝑄 𝑄 PRON cana-3856 205 31 ] ] PUNCT cana-3856 205 32 = = SYM cana-3856 205 33 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 NOUN cana-3856 205 34 βˆ’ βˆ’ PROPN cana-3856 205 35 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 205 36 ) ) PUNCT cana-3856 205 37 here here ADV cana-3856 205 38 , , PUNCT cana-3856 205 39 𝑀 𝑀 PROPN cana-3856 205 40 = = PUNCT cana-3856 205 41 πœ†πΌπœŽΓ—πœŽ πœ†πΌπœŽΓ—πœŽ X cana-3856 206 1 βˆ’ βˆ’ PROPN cana-3856 206 2 ( ( PUNCT cana-3856 206 3 𝑛 𝑛 PROPN cana-3856 206 4 + + NUM cana-3856 206 5 1)√2π½πœŽΓ—πœŽ 1)√2π½πœŽΓ—πœŽ INTJ cana-3856 206 6 , , PUNCT cana-3856 206 7 𝑁 𝑁 PROPN cana-3856 206 8 = = SYM cana-3856 206 9 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 PROPN cana-3856 206 10 + + CCONJ cana-3856 206 11 1)π½πœŽΓ—π‘›βˆ’πœŽ 1)π½πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 206 12 , , PUNCT cana-3856 206 13 𝑃 𝑃 NOUN cana-3856 206 14 = = SYM cana-3856 206 15 βˆ’βˆš2(𝑛2 βˆ’βˆš2(𝑛2 NOUN cana-3856 206 16 + + CCONJ cana-3856 206 17 1)π½π‘›βˆ’πœŽΓ—πœŽ 1)π½π‘›βˆ’πœŽΓ—πœŽ NUM cana-3856 206 18 and and CCONJ cana-3856 206 19 𝑄 𝑄 NOUN cana-3856 206 20 = = SYM cana-3856 206 21 ( ( PUNCT cana-3856 206 22 πœ† πœ† PROPN cana-3856 206 23 + + X cana-3856 206 24 ( ( PUNCT cana-3856 206 25 𝑛 𝑛 PROPN cana-3856 206 26 βˆ’ βˆ’ PROPN cana-3856 206 27 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 206 28 βˆ’ βˆ’ PROPN cana-3856 206 29 ( ( PUNCT cana-3856 206 30 𝑛 𝑛 PROPN cana-3856 206 31 βˆ’ βˆ’ PROPN cana-3856 206 32 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 206 33 , , PUNCT cana-3856 206 34 then then ADV cana-3856 206 35 , , PUNCT cana-3856 206 36 𝑑𝑒𝑑(𝑀 𝑑𝑒𝑑(𝑀 PROPN cana-3856 206 37 ) ) PUNCT cana-3856 206 38 = = SYM cana-3856 207 1 πœ†πœŽβˆ’1(πœ† πœ†πœŽβˆ’1(πœ† ADJ cana-3856 207 2 βˆ’ βˆ’ PROPN cana-3856 207 3 ( ( PUNCT cana-3856 207 4 𝑛 𝑛 PROPN cana-3856 207 5 + + NOUN cana-3856 207 6 1)√2𝜎 1)√2𝜎 NUM cana-3856 207 7 ) ) PUNCT cana-3856 207 8 . . PUNCT cana-3856 208 1 therefore therefore ADV cana-3856 208 2 communications communication NOUN cana-3856 208 3 on on ADP cana-3856 208 4 applied apply VERB cana-3856 208 5 nonlinear nonlinear ADJ cana-3856 208 6 analysis analysis NOUN cana-3856 208 7 issn issn NOUN cana-3856 208 8 : : PUNCT cana-3856 208 9 1074 1074 NUM cana-3856 208 10 - - PUNCT cana-3856 208 11 133x 133x NUM cana-3856 208 12 vol vol NOUN cana-3856 208 13 32 32 NUM cana-3856 208 14 no no NOUN cana-3856 208 15 . . PUNCT cana-3856 209 1 9s 9s NUM cana-3856 209 2 ( ( PUNCT cana-3856 209 3 2025 2025 NUM cana-3856 209 4 ) ) PUNCT cana-3856 209 5 292 292 NUM cana-3856 210 1 https://internationalpubls.com https://internationalpubls.com X cana-3856 210 2 π‘€βˆ’1 π‘€βˆ’1 NOUN cana-3856 210 3 = = PUNCT cana-3856 210 4 1 1 NUM cana-3856 210 5 πœ†πœŽβˆ’1(πœ† πœ†πœŽβˆ’1(πœ† NOUN cana-3856 210 6 βˆ’ βˆ’ PROPN cana-3856 210 7 ( ( PUNCT cana-3856 210 8 𝑛 𝑛 PROPN cana-3856 210 9 + + NOUN cana-3856 210 10 1)√2𝜎 1)√2𝜎 NUM cana-3856 210 11 ) ) PUNCT cana-3856 210 12 ( ( PUNCT cana-3856 210 13 πœ†πœŽβˆ’2(πœ† πœ†πœŽβˆ’2(πœ† NOUN cana-3856 210 14 βˆ’ βˆ’ PROPN cana-3856 210 15 ( ( PUNCT cana-3856 210 16 𝑛 𝑛 PROPN cana-3856 210 17 + + NOUN cana-3856 210 18 1)√2πœŽπΌπœŽΓ—πœŽ 1)√2πœŽπΌπœŽΓ—πœŽ NUM cana-3856 210 19 ) ) PUNCT cana-3856 211 1 + + CCONJ cana-3856 211 2 πœ† πœ† DET cana-3856 211 3 πœŽβˆ’2(𝑛 πœŽβˆ’2(𝑛 PROPN cana-3856 211 4 + + CCONJ cana-3856 211 5 1)√2π½πœŽΓ—πœŽ 1)√2π½πœŽΓ—πœŽ X cana-3856 211 6 ) ) PUNCT cana-3856 211 7 π‘ƒπ‘€βˆ’1 π‘ƒπ‘€βˆ’1 NOUN cana-3856 211 8 = = PUNCT cana-3856 211 9 βˆ’ βˆ’ PROPN cana-3856 211 10 √2(𝑛2 √2(𝑛2 NOUN cana-3856 211 11 + + PROPN cana-3856 211 12 1 1 NUM cana-3856 211 13 ) ) PUNCT cana-3856 211 14 ( ( PUNCT cana-3856 211 15 πœ†βˆ’(𝑛+1)√2𝜎 πœ†βˆ’(𝑛+1)√2𝜎 NOUN cana-3856 211 16 ) ) PUNCT cana-3856 211 17 π½π‘›βˆ’πœŽΓ—πœŽ π½π‘›βˆ’πœŽΓ—πœŽ NOUN cana-3856 211 18 and and CCONJ cana-3856 211 19 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 PROPN cana-3856 211 20 = = SYM cana-3856 211 21 2(𝑛2 2(𝑛2 PROPN cana-3856 211 22 + + SYM cana-3856 211 23 1)𝜎 1)𝜎 NUM cana-3856 211 24 ( ( PUNCT cana-3856 211 25 πœ†βˆ’(𝑛+1)√2𝜎 πœ†βˆ’(𝑛+1)√2𝜎 NOUN cana-3856 211 26 ) ) PUNCT cana-3856 211 27 π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ. π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ. NOUN cana-3856 212 1 so so SCONJ cana-3856 212 2 𝑄 𝑄 PROPN cana-3856 212 3 βˆ’ βˆ’ PROPN cana-3856 212 4 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 212 5 = = PUNCT cana-3856 212 6 ( ( PUNCT cana-3856 212 7 πœ† πœ† PROPN cana-3856 212 8 + + X cana-3856 212 9 ( ( PUNCT cana-3856 212 10 𝑛 𝑛 PROPN cana-3856 212 11 βˆ’ βˆ’ PROPN cana-3856 212 12 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 212 13 βˆ’ βˆ’ PROPN cana-3856 212 14 ( ( PUNCT cana-3856 212 15 𝑛 𝑛 PROPN cana-3856 212 16 βˆ’ βˆ’ PROPN cana-3856 212 17 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 212 18 βˆ’ βˆ’ NOUN cana-3856 212 19 2(𝑛2 2(𝑛2 NUM cana-3856 213 1 + + CCONJ cana-3856 213 2 1)𝜎 1)𝜎 NUM cana-3856 213 3 ( ( PUNCT cana-3856 213 4 πœ† πœ† PROPN cana-3856 213 5 βˆ’ βˆ’ PROPN cana-3856 213 6 ( ( PUNCT cana-3856 213 7 𝑛 𝑛 PROPN cana-3856 213 8 + + NOUN cana-3856 213 9 1)√2𝜎 1)√2𝜎 NUM cana-3856 213 10 ) ) PUNCT cana-3856 213 11 π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ PRON cana-3856 213 12 the the DET cana-3856 213 13 eigenvalues eigenvalues PROPN cana-3856 213 14 of of ADP cana-3856 213 15 ( ( PUNCT cana-3856 213 16 πœ† πœ† PROPN cana-3856 213 17 + + X cana-3856 213 18 ( ( PUNCT cana-3856 213 19 𝑛 𝑛 DET cana-3856 213 20 βˆ’ βˆ’ PROPN cana-3856 213 21 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2)πΌπ‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 213 22 are be AUX cana-3856 213 23 ( ( PUNCT cana-3856 213 24 πœ† πœ† X cana-3856 213 25 + + X cana-3856 213 26 ( ( PUNCT cana-3856 213 27 𝑛 𝑛 DET cana-3856 213 28 βˆ’ βˆ’ PROPN cana-3856 213 29 1)√2 1)√2 NUM cana-3856 213 30 ) ) PUNCT cana-3856 213 31 with with ADP cana-3856 213 32 n-𝜎 n-𝜎 PROPN cana-3856 213 33 multiplicity multiplicity NOUN cana-3856 213 34 and and CCONJ cana-3856 213 35 eigenvalues eigenvalue NOUN cana-3856 213 36 of of ADP cana-3856 213 37 ( ( PUNCT cana-3856 213 38 𝑛 𝑛 PROPN cana-3856 213 39 βˆ’ βˆ’ PROPN cana-3856 213 40 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ 1)√2π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ NUM cana-3856 213 41 + + NUM cana-3856 213 42 2(𝑛2 2(𝑛2 NUM cana-3856 213 43 + + SYM cana-3856 213 44 1)𝜎 1)𝜎 NUM cana-3856 213 45 ( ( PUNCT cana-3856 213 46 πœ†βˆ’(𝑛+1)√2𝜎 πœ†βˆ’(𝑛+1)√2𝜎 PROPN cana-3856 213 47 ) ) PUNCT cana-3856 213 48 π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ π½π‘›βˆ’πœŽΓ—π‘›βˆ’πœŽ PRON cana-3856 213 49 are be AUX cana-3856 213 50 0 0 NUM cana-3856 213 51 with with ADP cana-3856 213 52 multiplicity multiplicity NOUN cana-3856 213 53 𝑛 𝑛 PRON cana-3856 213 54 βˆ’ βˆ’ PROPN cana-3856 213 55 𝜎 𝜎 NOUN cana-3856 213 56 βˆ’ βˆ’ NOUN cana-3856 213 57 1 1 NUM cana-3856 213 58 and and CCONJ cana-3856 213 59 ( ( PUNCT cana-3856 213 60 πœ†(π‘›βˆ’1)√2 πœ†(π‘›βˆ’1)√2 X cana-3856 213 61 + + NOUN cana-3856 213 62 4𝜎)(π‘›βˆ’πœŽ 4𝜎)(π‘›βˆ’πœŽ NUM cana-3856 213 63 ) ) PUNCT cana-3856 213 64 πœ†βˆ’(𝑛+1)√2𝜎 πœ†βˆ’(𝑛+1)√2𝜎 PROPN cana-3856 213 65 ) ) PUNCT cana-3856 213 66 with with ADP cana-3856 213 67 multiplicity multiplicity NOUN cana-3856 213 68 1 1 NUM cana-3856 213 69 . . PUNCT cana-3856 214 1 hence hence ADV cana-3856 214 2 , , PUNCT cana-3856 214 3 the the DET cana-3856 214 4 eigenvalues eigenvalue NOUN cana-3856 214 5 of of ADP cana-3856 214 6 𝑄 𝑄 PROPN cana-3856 214 7 βˆ’ βˆ’ PROPN cana-3856 214 8 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 214 9 are be AUX cana-3856 214 10 ( ( PUNCT cana-3856 214 11 πœ† πœ† X cana-3856 214 12 + + X cana-3856 214 13 ( ( PUNCT cana-3856 214 14 𝑛 𝑛 DET cana-3856 214 15 βˆ’ βˆ’ PROPN cana-3856 214 16 1)√2 1)√2 NUM cana-3856 214 17 ) ) PUNCT cana-3856 214 18 βˆ’ βˆ’ PROPN cana-3856 215 1 ( ( PUNCT cana-3856 215 2 πœ†(π‘›βˆ’1)√2 πœ†(π‘›βˆ’1)√2 X cana-3856 215 3 + + NOUN cana-3856 215 4 4𝜎)(π‘›βˆ’πœŽ 4𝜎)(π‘›βˆ’πœŽ NUM cana-3856 215 5 ) ) PUNCT cana-3856 215 6 πœ†βˆ’(𝑛+1)√2𝜎 πœ†βˆ’(𝑛+1)√2𝜎 PROPN cana-3856 215 7 ) ) PUNCT cana-3856 215 8 with with ADP cana-3856 215 9 multiplicity multiplicity NOUN cana-3856 215 10 1 1 NUM cana-3856 215 11 and and CCONJ cana-3856 215 12 ( ( PUNCT cana-3856 215 13 πœ† πœ† PROPN cana-3856 215 14 + + X cana-3856 215 15 ( ( PUNCT cana-3856 215 16 𝑛 𝑛 DET cana-3856 215 17 βˆ’ βˆ’ PROPN cana-3856 215 18 1)√2 1)√2 NUM cana-3856 215 19 ) ) PUNCT cana-3856 215 20 with with ADP cana-3856 215 21 multiplicity multiplicity NOUN cana-3856 215 22 𝑛 𝑛 PRON cana-3856 215 23 βˆ’ βˆ’ PROPN cana-3856 215 24 𝜎 𝜎 NOUN cana-3856 216 1 βˆ’ βˆ’ NOUN cana-3856 216 2 1 1 NUM cana-3856 216 3 . . PUNCT cana-3856 217 1 then then ADV cana-3856 217 2 𝑑𝑒𝑑(𝑄 𝑑𝑒𝑑(𝑄 NUM cana-3856 217 3 βˆ’ βˆ’ PROPN cana-3856 217 4 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 217 5 ) ) PUNCT cana-3856 217 6 = = PUNCT cana-3856 218 1 ( ( PUNCT cana-3856 218 2 πœ† πœ† PROPN cana-3856 218 3 + + X cana-3856 218 4 ( ( PUNCT cana-3856 218 5 𝑛 𝑛 DET cana-3856 218 6 βˆ’ βˆ’ PROPN cana-3856 218 7 1)√2)π‘›βˆ’πœŽβˆ’1 1)√2)π‘›βˆ’πœŽβˆ’1 NUM cana-3856 218 8 ( ( PUNCT cana-3856 218 9 πœ†2 πœ†2 NOUN cana-3856 218 10 βˆ’ βˆ’ PROPN cana-3856 218 11 √2πœ†((𝑛 √2πœ†((𝑛 VERB cana-3856 218 12 βˆ’ βˆ’ PROPN cana-3856 218 13 1)2 1)2 NUM cana-3856 219 1 + + NUM cana-3856 219 2 2𝜎 2𝜎 NUM cana-3856 219 3 ) ) PUNCT cana-3856 220 1 βˆ’ βˆ’ PROPN cana-3856 220 2 𝜎(2(𝑛2 𝜎(2(𝑛2 CCONJ cana-3856 221 1 βˆ’ βˆ’ NUM cana-3856 221 2 1 1 NUM cana-3856 221 3 ) ) PUNCT cana-3856 221 4 + + CCONJ cana-3856 221 5 4(𝑛 4(𝑛 NUM cana-3856 221 6 βˆ’ βˆ’ PROPN cana-3856 221 7 𝜎 𝜎 NOUN cana-3856 221 8 ) ) PUNCT cana-3856 221 9 ) ) PUNCT cana-3856 222 1 πœ† πœ† X cana-3856 222 2 βˆ’ βˆ’ PROPN cana-3856 222 3 ( ( PUNCT cana-3856 222 4 𝑛 𝑛 PROPN cana-3856 222 5 + + NOUN cana-3856 222 6 1)√2𝜎 1)√2𝜎 NUM cana-3856 222 7 ) ) PUNCT cana-3856 222 8 ) ) PUNCT cana-3856 223 1 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 𝑑𝑒𝑑(𝑀)𝑑𝑒𝑑(𝑄 VERB cana-3856 223 2 βˆ’ βˆ’ PROPN cana-3856 223 3 π‘ƒπ‘€βˆ’1𝑁 π‘ƒπ‘€βˆ’1𝑁 NOUN cana-3856 223 4 ) ) PUNCT cana-3856 223 5 = = SYM cana-3856 224 1 πœ†πœŽβˆ’1(πœ† πœ†πœŽβˆ’1(πœ† PROPN cana-3856 225 1 + + CCONJ cana-3856 225 2 ( ( PUNCT cana-3856 225 3 𝑛 𝑛 PRON cana-3856 225 4 βˆ’ βˆ’ PROPN cana-3856 225 5 1)√2)π‘›βˆ’πœŽβˆ’1(πœ†2 1)√2)π‘›βˆ’πœŽβˆ’1(πœ†2 NUM cana-3856 225 6 βˆ’ βˆ’ NOUN cana-3856 225 7 √2πœ†((𝑛 √2πœ†((𝑛 NOUN cana-3856 225 8 βˆ’ βˆ’ PROPN cana-3856 226 1 1)2 1)2 NUM cana-3856 226 2 + + NUM cana-3856 226 3 2𝜎 2𝜎 NUM cana-3856 226 4 ) ) PUNCT cana-3856 227 1 βˆ’ βˆ’ PROPN cana-3856 227 2 𝜎(2(𝑛2 𝜎(2(𝑛2 CCONJ cana-3856 228 1 βˆ’ βˆ’ NUM cana-3856 228 2 1 1 NUM cana-3856 228 3 ) ) PUNCT cana-3856 228 4 + + CCONJ cana-3856 228 5 4(𝑛 4(𝑛 NUM cana-3856 228 6 βˆ’ βˆ’ PROPN cana-3856 228 7 𝜎 𝜎 NOUN cana-3856 228 8 ) ) PUNCT cana-3856 228 9 ) ) PUNCT cana-3856 228 10 ) ) PUNCT cana-3856 229 1 hence hence ADV cana-3856 229 2 , , PUNCT cana-3856 229 3 eigenvalues eigenvalue VERB cana-3856 229 4 of of ADP cana-3856 229 5 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 PROPN cana-3856 229 6 ) ) PUNCT cana-3856 229 7 are be AUX cana-3856 229 8 0 0 NUM cana-3856 229 9 with with ADP cana-3856 229 10 multiplicity multiplicity NOUN cana-3856 229 11 𝜎 𝜎 NOUN cana-3856 230 1 βˆ’ βˆ’ PROPN cana-3856 230 2 1 1 NUM cana-3856 230 3 , , PUNCT cana-3856 230 4 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-3856 231 1 βˆ’ βˆ’ PROPN cana-3856 231 2 1)√2 1)√2 NOUN cana-3856 231 3 with with ADP cana-3856 231 4 multiplicity multiplicity NOUN cana-3856 231 5 𝑛 𝑛 PRON cana-3856 231 6 βˆ’ βˆ’ PROPN cana-3856 231 7 𝜎 𝜎 NOUN cana-3856 231 8 βˆ’ βˆ’ PROPN cana-3856 231 9 1 1 NUM cana-3856 231 10 , , PUNCT cana-3856 231 11 √2((π‘›βˆ’1)2 √2((π‘›βˆ’1)2 VERB cana-3856 231 12 + + NOUN cana-3856 231 13 2𝜎)+√2((π‘›βˆ’1)2 2𝜎)+√2((π‘›βˆ’1)2 NUM cana-3856 231 14 + + NOUN cana-3856 231 15 2𝜎)2 2𝜎)2 NUM cana-3856 231 16 + + NOUN cana-3856 231 17 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ NUM cana-3856 231 18 ) ) PUNCT cana-3856 231 19 ) ) PUNCT cana-3856 232 1 2 2 NUM cana-3856 232 2 with with ADP cana-3856 232 3 multiplicity multiplicity NOUN cana-3856 232 4 1 1 NUM cana-3856 232 5 , , PUNCT cana-3856 232 6 and and CCONJ cana-3856 232 7 √2((π‘›βˆ’1)2 √2((π‘›βˆ’1)2 VERB cana-3856 232 8 + + NOUN cana-3856 232 9 2𝜎)βˆ’βˆš2((π‘›βˆ’1)2 2𝜎)βˆ’βˆš2((π‘›βˆ’1)2 ADJ cana-3856 232 10 + + ADJ cana-3856 232 11 2𝜎)2 2𝜎)2 NUM cana-3856 232 12 + + NOUN cana-3856 232 13 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ NUM cana-3856 232 14 ) ) PUNCT cana-3856 232 15 ) ) PUNCT cana-3856 232 16 2 2 NUM cana-3856 232 17 with with ADP cana-3856 232 18 multiplicity multiplicity NOUN cana-3856 232 19 1 1 NUM cana-3856 232 20 . . PUNCT cana-3856 232 21 theorem theorem NOUN cana-3856 232 22 4 4 NUM cana-3856 232 23 . . PUNCT cana-3856 233 1 let let VERB cana-3856 234 1 𝐺 𝐺 PROPN cana-3856 234 2 = = SYM cana-3856 234 3 𝐾𝑛 𝐾𝑛 PROPN cana-3856 234 4 be be AUX cana-3856 234 5 a a DET cana-3856 234 6 complete complete ADJ cana-3856 234 7 graph graph NOUN cana-3856 234 8 with with ADP cana-3856 234 9 n n ADP cana-3856 234 10 vertices vertex NOUN cana-3856 234 11 . . PUNCT cana-3856 235 1 if if SCONJ cana-3856 235 2 𝐺𝑆 𝐺𝑆 PROPN cana-3856 235 3 is be AUX cana-3856 235 4 a a DET cana-3856 235 5 graph graph NOUN cana-3856 235 6 obtained obtain VERB cana-3856 235 7 from from ADP cana-3856 235 8 𝐺 𝐺 PROPN cana-3856 235 9 by by ADP cana-3856 235 10 adding add VERB cana-3856 235 11 𝜎 𝜎 PRON cana-3856 235 12 self self NOUN cana-3856 235 13 loops loop NOUN cana-3856 235 14 , , PUNCT cana-3856 235 15 then then ADV cana-3856 235 16 sombor sombor NOUN cana-3856 235 17 energy energy NOUN cana-3856 235 18 of of ADP cana-3856 235 19 𝐺𝑆 𝐺𝑆 PROPN cana-3856 235 20 is be AUX cana-3856 235 21 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NUM cana-3856 235 22 ) ) PUNCT cana-3856 236 1 = = SYM cana-3856 236 2 √2 √2 X cana-3856 236 3 ( ( PUNCT cana-3856 236 4 𝑛+1)(πœŽβˆ’1)𝜎+(π‘›βˆ’πœŽβˆ’1)(𝑛(π‘›βˆ’1)+(𝑛+1)𝜎 𝑛+1)(πœŽβˆ’1)𝜎+(π‘›βˆ’πœŽβˆ’1)(𝑛(π‘›βˆ’1)+(𝑛+1)𝜎 NOUN cana-3856 236 5 ) ) PUNCT cana-3856 236 6 𝑛 𝑛 PRON cana-3856 236 7 + + ADJ cana-3856 236 8 √2((𝑛 √2((𝑛 NOUN cana-3856 236 9 βˆ’ βˆ’ PROPN cana-3856 236 10 1)2 1)2 NUM cana-3856 236 11 + + CCONJ cana-3856 236 12 2𝜎)2 2𝜎)2 NUM cana-3856 236 13 + + CCONJ cana-3856 236 14 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 236 15 βˆ’ βˆ’ ADP cana-3856 236 16 1 1 NUM cana-3856 236 17 ) ) PUNCT cana-3856 236 18 + + CCONJ cana-3856 236 19 4(𝑛 4(𝑛 NUM cana-3856 236 20 βˆ’ βˆ’ PROPN cana-3856 236 21 𝜎 𝜎 NOUN cana-3856 236 22 ) ) PUNCT cana-3856 236 23 ) ) PUNCT cana-3856 236 24 . . PUNCT cana-3856 237 1 proof proof NOUN cana-3856 237 2 . . PUNCT cana-3856 238 1 by by ADP cana-3856 238 2 definition definition NOUN cana-3856 238 3 , , PUNCT cana-3856 238 4 the the DET cana-3856 238 5 sombor sombor NOUN cana-3856 238 6 energy energy NOUN cana-3856 238 7 of of ADP cana-3856 238 8 matrix matrix NOUN cana-3856 238 9 with with ADP cana-3856 238 10 𝜎 𝜎 PROPN cana-3856 238 11 self self NOUN cana-3856 238 12 loop loop NOUN cana-3856 238 13 is be AUX cana-3856 238 14 , , PUNCT cana-3856 238 15 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 NOUN cana-3856 238 16 ) ) PUNCT cana-3856 238 17 = = NOUN cana-3856 238 18 βˆ‘ βˆ‘ PROPN cana-3856 238 19 𝑛 𝑛 PRON cana-3856 238 20 𝑖=1 𝑖=1 PROPN cana-3856 238 21 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 NUM cana-3856 238 22 ) ) PUNCT cana-3856 238 23 βˆ’ βˆ’ PROPN cana-3856 239 1 √2βˆ‘πœŽπ‘—=1 √2βˆ‘πœŽπ‘—=1 ADV cana-3856 239 2 𝑑𝑗 𝑑𝑗 ADP cana-3856 239 3 𝑛 𝑛 PRON cana-3856 239 4 | | ADJ cana-3856 239 5 communications communication NOUN cana-3856 239 6 on on ADP cana-3856 239 7 applied apply VERB cana-3856 239 8 nonlinear nonlinear ADJ cana-3856 239 9 analysis analysis NOUN cana-3856 239 10 issn issn NOUN cana-3856 239 11 : : PUNCT cana-3856 239 12 1074 1074 NUM cana-3856 239 13 - - PUNCT cana-3856 239 14 133x 133x NUM cana-3856 239 15 vol vol NOUN cana-3856 239 16 32 32 NUM cana-3856 239 17 no no NOUN cana-3856 239 18 . . PUNCT cana-3856 240 1 9s 9s NUM cana-3856 240 2 ( ( PUNCT cana-3856 240 3 2025 2025 NUM cana-3856 240 4 ) ) PUNCT cana-3856 240 5 293 293 NUM cana-3856 240 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 240 7 by by ADP cana-3856 240 8 lemma lemma PROPN cana-3856 240 9 2 2 NUM cana-3856 240 10 , , PUNCT cana-3856 240 11 for for ADP cana-3856 240 12 complete complete ADJ cana-3856 240 13 graph graph NOUN cana-3856 240 14 with with ADP cana-3856 240 15 𝜎 𝜎 PROPN cana-3856 240 16 selfloop selfloop NOUN cana-3856 240 17 has have VERB cana-3856 240 18 βˆ‘πœŽπ‘—=1 βˆ‘πœŽπ‘—=1 ADV cana-3856 240 19 𝑑𝑗 𝑑𝑗 NOUN cana-3856 240 20 = = SYM cana-3856 240 21 ( ( PUNCT cana-3856 240 22 𝑛 𝑛 PROPN cana-3856 240 23 + + NUM cana-3856 240 24 1)𝜎 1)𝜎 NUM cana-3856 240 25 and and CCONJ cana-3856 240 26 eigenvalues eigenvalue VERB cana-3856 240 27 are be AUX cana-3856 240 28 0 0 NUM cana-3856 240 29 with with ADP cana-3856 240 30 multiplicity multiplicity NOUN cana-3856 241 1 𝜎 𝜎 NOUN cana-3856 241 2 βˆ’ βˆ’ PROPN cana-3856 241 3 1 1 NUM cana-3856 241 4 , , PUNCT cana-3856 241 5 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-3856 242 1 βˆ’ βˆ’ PROPN cana-3856 243 1 1)√2 1)√2 NOUN cana-3856 243 2 with with ADP cana-3856 243 3 multiplicity multiplicity NOUN cana-3856 243 4 𝑛 𝑛 PRON cana-3856 243 5 βˆ’ βˆ’ PROPN cana-3856 243 6 𝜎 𝜎 NOUN cana-3856 243 7 βˆ’ βˆ’ PROPN cana-3856 243 8 1 1 NUM cana-3856 243 9 , , PUNCT cana-3856 243 10 √2((π‘›βˆ’1)2 √2((π‘›βˆ’1)2 VERB cana-3856 243 11 + + NOUN cana-3856 243 12 2𝜎)+√2((π‘›βˆ’1)2 2𝜎)+√2((π‘›βˆ’1)2 NUM cana-3856 243 13 + + NOUN cana-3856 243 14 2𝜎)2 2𝜎)2 NUM cana-3856 243 15 + + NOUN cana-3856 243 16 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ NUM cana-3856 243 17 ) ) PUNCT cana-3856 243 18 ) ) PUNCT cana-3856 244 1 2 2 NUM cana-3856 244 2 with with ADP cana-3856 244 3 multiplicity multiplicity NOUN cana-3856 244 4 1 1 NUM cana-3856 244 5 , , PUNCT cana-3856 244 6 √2((π‘›βˆ’1)2 √2((π‘›βˆ’1)2 VERB cana-3856 244 7 + + PRON cana-3856 244 8 2𝜎)βˆ’βˆš2((π‘›βˆ’1)2 2𝜎)βˆ’βˆš2((π‘›βˆ’1)2 ADJ cana-3856 244 9 + + ADJ cana-3856 244 10 2𝜎)2 2𝜎)2 NUM cana-3856 244 11 + + NOUN cana-3856 244 12 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ 4𝜎(2(𝑛2βˆ’1)+4(π‘›βˆ’πœŽ NUM cana-3856 244 13 ) ) PUNCT cana-3856 244 14 ) ) PUNCT cana-3856 245 1 2 2 NUM cana-3856 245 2 with with ADP cana-3856 245 3 multiplicity multiplicity NOUN cana-3856 245 4 1 1 NUM cana-3856 245 5 . . PUNCT cana-3856 246 1 then then ADV cana-3856 246 2 somber somber ADJ cana-3856 246 3 energy energy NOUN cana-3856 246 4 becomes become VERB cana-3856 246 5 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 NOUN cana-3856 246 6 ) ) PUNCT cana-3856 247 1 = = NOUN cana-3856 247 2 βˆ‘ βˆ‘ PROPN cana-3856 247 3 πœŽβˆ’1 πœŽβˆ’1 PROPN cana-3856 247 4 𝑖=1 𝑖=1 PROPN cana-3856 247 5 |0 |0 NUM cana-3856 247 6 βˆ’ βˆ’ NUM cana-3856 247 7 √2(𝑛 √2(𝑛 PROPN cana-3856 247 8 + + CCONJ cana-3856 247 9 1)𝜎 1)𝜎 NUM cana-3856 247 10 𝑛 𝑛 ADP cana-3856 248 1 | | ADV cana-3856 248 2 + + NOUN cana-3856 248 3 βˆ‘ βˆ‘ VERB cana-3856 248 4 π‘›βˆ’πœŽβˆ’1 π‘›βˆ’πœŽβˆ’1 VERB cana-3856 248 5 𝑖=1 𝑖=1 PROPN cana-3856 248 6 | | ADV cana-3856 248 7 βˆ’ βˆ’ PROPN cana-3856 248 8 ( ( PUNCT cana-3856 248 9 𝑛 𝑛 PROPN cana-3856 248 10 βˆ’ βˆ’ NUM cana-3856 248 11 1)√2 1)√2 NUM cana-3856 249 1 βˆ’ βˆ’ PROPN cana-3856 249 2 √2(𝑛 √2(𝑛 PROPN cana-3856 249 3 + + CCONJ cana-3856 249 4 1)𝜎 1)𝜎 NUM cana-3856 249 5 𝑛 𝑛 VERB cana-3856 250 1 | | ADV cana-3856 251 1 + + CCONJ cana-3856 251 2 | | ADV cana-3856 251 3 √2((𝑛 √2((𝑛 VERB cana-3856 251 4 βˆ’ βˆ’ PROPN cana-3856 251 5 1)2 1)2 NUM cana-3856 251 6 + + NUM cana-3856 251 7 2𝜎 2𝜎 NUM cana-3856 251 8 ) ) PUNCT cana-3856 252 1 + + NUM cana-3856 252 2 √2((𝑛 √2((𝑛 NOUN cana-3856 252 3 βˆ’ βˆ’ PROPN cana-3856 253 1 1)2 1)2 NUM cana-3856 253 2 + + CCONJ cana-3856 253 3 2𝜎)2 2𝜎)2 NUM cana-3856 253 4 + + CCONJ cana-3856 253 5 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 253 6 βˆ’ βˆ’ ADP cana-3856 253 7 1 1 NUM cana-3856 253 8 ) ) PUNCT cana-3856 253 9 + + CCONJ cana-3856 254 1 4(𝑛 4(𝑛 NUM cana-3856 254 2 βˆ’ βˆ’ PROPN cana-3856 254 3 𝜎 𝜎 NOUN cana-3856 254 4 ) ) PUNCT cana-3856 254 5 ) ) PUNCT cana-3856 254 6 2 2 NUM cana-3856 254 7 βˆ’ βˆ’ PROPN cana-3856 254 8 √2(𝑛 √2(𝑛 NOUN cana-3856 254 9 + + CCONJ cana-3856 254 10 1))𝜎 1))𝜎 NUM cana-3856 254 11 𝑛 𝑛 VERB cana-3856 255 1 | | ADV cana-3856 256 1 + + CCONJ cana-3856 256 2 | | ADV cana-3856 256 3 √2((𝑛 √2((𝑛 VERB cana-3856 256 4 βˆ’ βˆ’ PROPN cana-3856 257 1 1)2 1)2 NUM cana-3856 257 2 + + NUM cana-3856 257 3 2𝜎 2𝜎 NUM cana-3856 257 4 ) ) PUNCT cana-3856 257 5 βˆ’ βˆ’ PROPN cana-3856 257 6 √2((𝑛 √2((𝑛 SYM cana-3856 257 7 βˆ’ βˆ’ PROPN cana-3856 257 8 1)2 1)2 NUM cana-3856 257 9 + + CCONJ cana-3856 257 10 2𝜎)2 2𝜎)2 NUM cana-3856 257 11 + + CCONJ cana-3856 257 12 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 257 13 βˆ’ βˆ’ ADP cana-3856 257 14 1 1 NUM cana-3856 257 15 ) ) PUNCT cana-3856 257 16 + + CCONJ cana-3856 258 1 4(𝑛 4(𝑛 NUM cana-3856 258 2 βˆ’ βˆ’ PROPN cana-3856 258 3 𝜎 𝜎 NOUN cana-3856 258 4 ) ) PUNCT cana-3856 258 5 ) ) PUNCT cana-3856 258 6 2 2 NUM cana-3856 258 7 βˆ’ βˆ’ PROPN cana-3856 258 8 √2(𝑛 √2(𝑛 NOUN cana-3856 258 9 + + CCONJ cana-3856 258 10 1))𝜎 1))𝜎 NUM cana-3856 258 11 𝑛 𝑛 VERB cana-3856 258 12 | | NOUN cana-3856 258 13 = = SYM cana-3856 258 14 ( ( PUNCT cana-3856 258 15 𝜎 𝜎 INTJ cana-3856 258 16 βˆ’ βˆ’ NOUN cana-3856 258 17 1 1 NUM cana-3856 258 18 ) ) PUNCT cana-3856 258 19 √2(𝑛 √2(𝑛 NOUN cana-3856 258 20 + + CCONJ cana-3856 258 21 1)𝜎 1)𝜎 NUM cana-3856 258 22 𝑛 𝑛 VERB cana-3856 258 23 + + CCONJ cana-3856 258 24 ( ( PUNCT cana-3856 258 25 𝑛 𝑛 DET cana-3856 258 26 βˆ’ βˆ’ NOUN cana-3856 258 27 𝜎 𝜎 NOUN cana-3856 258 28 βˆ’ βˆ’ PROPN cana-3856 258 29 1)((𝑛 1)((𝑛 NOUN cana-3856 258 30 βˆ’ βˆ’ PROPN cana-3856 258 31 1)√2 1)√2 NUM cana-3856 258 32 + + NUM cana-3856 258 33 √2(𝑛 √2(𝑛 NOUN cana-3856 258 34 + + CCONJ cana-3856 258 35 1)𝜎 1)𝜎 NUM cana-3856 258 36 𝑛 𝑛 NOUN cana-3856 258 37 ) ) PUNCT cana-3856 259 1 + + NUM cana-3856 259 2 √2((𝑛 √2((𝑛 NOUN cana-3856 259 3 βˆ’ βˆ’ PROPN cana-3856 260 1 1)2 1)2 NUM cana-3856 260 2 + + NUM cana-3856 260 3 2𝜎 2𝜎 NUM cana-3856 260 4 ) ) PUNCT cana-3856 261 1 + + NUM cana-3856 261 2 √2((𝑛 √2((𝑛 NOUN cana-3856 261 3 βˆ’ βˆ’ PROPN cana-3856 262 1 1)2 1)2 NUM cana-3856 262 2 + + CCONJ cana-3856 262 3 2𝜎)2 2𝜎)2 NUM cana-3856 262 4 + + CCONJ cana-3856 262 5 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 262 6 βˆ’ βˆ’ ADP cana-3856 262 7 1 1 NUM cana-3856 262 8 ) ) PUNCT cana-3856 262 9 + + CCONJ cana-3856 263 1 4(𝑛 4(𝑛 NUM cana-3856 263 2 βˆ’ βˆ’ PROPN cana-3856 263 3 𝜎 𝜎 NOUN cana-3856 263 4 ) ) PUNCT cana-3856 263 5 ) ) PUNCT cana-3856 263 6 2 2 NUM cana-3856 263 7 βˆ’ βˆ’ PROPN cana-3856 263 8 √2(𝑛 √2(𝑛 NOUN cana-3856 263 9 + + CCONJ cana-3856 263 10 1))𝜎 1))𝜎 NUM cana-3856 263 11 𝑛 𝑛 DET cana-3856 263 12 βˆ’ βˆ’ PROPN cana-3856 263 13 √2((𝑛 √2((𝑛 NOUN cana-3856 263 14 βˆ’ βˆ’ PROPN cana-3856 263 15 1)2 1)2 NUM cana-3856 263 16 + + NUM cana-3856 263 17 2𝜎 2𝜎 NUM cana-3856 263 18 ) ) PUNCT cana-3856 264 1 βˆ’ βˆ’ PROPN cana-3856 264 2 √2((𝑛 √2((𝑛 SYM cana-3856 264 3 βˆ’ βˆ’ PROPN cana-3856 265 1 1)2 1)2 NUM cana-3856 265 2 + + CCONJ cana-3856 265 3 2𝜎)2 2𝜎)2 NUM cana-3856 265 4 + + CCONJ cana-3856 265 5 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 265 6 βˆ’ βˆ’ ADP cana-3856 265 7 1 1 NUM cana-3856 265 8 ) ) PUNCT cana-3856 265 9 + + CCONJ cana-3856 266 1 4(𝑛 4(𝑛 NUM cana-3856 266 2 βˆ’ βˆ’ PROPN cana-3856 266 3 𝜎 𝜎 NOUN cana-3856 266 4 ) ) PUNCT cana-3856 266 5 ) ) PUNCT cana-3856 266 6 2 2 NUM cana-3856 267 1 + + NUM cana-3856 267 2 √2(𝑛 √2(𝑛 NOUN cana-3856 267 3 + + CCONJ cana-3856 267 4 1))𝜎 1))𝜎 NUM cana-3856 267 5 𝑛 𝑛 NOUN cana-3856 267 6 = = PUNCT cana-3856 267 7 ( ( PUNCT cana-3856 267 8 𝜎 𝜎 NOUN cana-3856 267 9 βˆ’ βˆ’ NOUN cana-3856 267 10 1 1 NUM cana-3856 267 11 ) ) PUNCT cana-3856 267 12 √2(𝑛 √2(𝑛 NOUN cana-3856 267 13 + + CCONJ cana-3856 267 14 1)𝜎 1)𝜎 NUM cana-3856 267 15 𝑛 𝑛 VERB cana-3856 267 16 + + CCONJ cana-3856 267 17 ( ( PUNCT cana-3856 267 18 𝑛 𝑛 DET cana-3856 267 19 βˆ’ βˆ’ NOUN cana-3856 267 20 𝜎 𝜎 NOUN cana-3856 267 21 βˆ’ βˆ’ PROPN cana-3856 267 22 1)((𝑛 1)((𝑛 NOUN cana-3856 267 23 βˆ’ βˆ’ PROPN cana-3856 267 24 1)√2 1)√2 NUM cana-3856 267 25 + + NUM cana-3856 267 26 √2(𝑛 √2(𝑛 NOUN cana-3856 267 27 + + CCONJ cana-3856 267 28 1)𝜎 1)𝜎 NUM cana-3856 267 29 𝑛 𝑛 NOUN cana-3856 267 30 ) ) PUNCT cana-3856 268 1 + + NUM cana-3856 268 2 √2((𝑛 √2((𝑛 NOUN cana-3856 268 3 βˆ’ βˆ’ PROPN cana-3856 269 1 1)2 1)2 NUM cana-3856 269 2 + + NUM cana-3856 269 3 2𝜎 2𝜎 NUM cana-3856 269 4 ) ) PUNCT cana-3856 270 1 + + NUM cana-3856 270 2 √2((𝑛 √2((𝑛 NOUN cana-3856 270 3 βˆ’ βˆ’ PROPN cana-3856 271 1 1)2 1)2 NUM cana-3856 271 2 + + CCONJ cana-3856 271 3 2𝜎)2 2𝜎)2 NUM cana-3856 271 4 + + CCONJ cana-3856 271 5 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 271 6 βˆ’ βˆ’ ADP cana-3856 271 7 1 1 NUM cana-3856 271 8 ) ) PUNCT cana-3856 271 9 + + CCONJ cana-3856 272 1 4(𝑛 4(𝑛 NUM cana-3856 272 2 βˆ’ βˆ’ PROPN cana-3856 272 3 𝜎 𝜎 NOUN cana-3856 272 4 ) ) PUNCT cana-3856 272 5 ) ) PUNCT cana-3856 272 6 2 2 NUM cana-3856 272 7 βˆ’ βˆ’ NOUN cana-3856 272 8 √2((𝑛 √2((𝑛 NOUN cana-3856 272 9 βˆ’ βˆ’ PROPN cana-3856 272 10 1)2 1)2 NUM cana-3856 272 11 + + NUM cana-3856 272 12 2𝜎 2𝜎 NUM cana-3856 272 13 ) ) PUNCT cana-3856 273 1 βˆ’ βˆ’ PROPN cana-3856 273 2 √2((𝑛 √2((𝑛 SYM cana-3856 273 3 βˆ’ βˆ’ PROPN cana-3856 274 1 1)2 1)2 NUM cana-3856 274 2 + + CCONJ cana-3856 274 3 2𝜎)2 2𝜎)2 NUM cana-3856 274 4 + + CCONJ cana-3856 274 5 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 274 6 βˆ’ βˆ’ ADP cana-3856 274 7 1 1 NUM cana-3856 274 8 ) ) PUNCT cana-3856 274 9 + + CCONJ cana-3856 275 1 4(𝑛 4(𝑛 NUM cana-3856 275 2 βˆ’ βˆ’ PROPN cana-3856 275 3 𝜎 𝜎 NOUN cana-3856 275 4 ) ) PUNCT cana-3856 275 5 ) ) PUNCT cana-3856 275 6 2 2 NUM cana-3856 276 1 = = SYM cana-3856 276 2 √2 √2 X cana-3856 276 3 ( ( PUNCT cana-3856 276 4 𝑛+1)(πœŽβˆ’1)𝜎+(π‘›βˆ’πœŽβˆ’1)(𝑛(π‘›βˆ’1)+(𝑛+1)𝜎 𝑛+1)(πœŽβˆ’1)𝜎+(π‘›βˆ’πœŽβˆ’1)(𝑛(π‘›βˆ’1)+(𝑛+1)𝜎 NOUN cana-3856 276 5 ) ) PUNCT cana-3856 276 6 𝑛 𝑛 PRON cana-3856 276 7 + + ADJ cana-3856 276 8 √2((𝑛 √2((𝑛 NOUN cana-3856 276 9 βˆ’ βˆ’ PROPN cana-3856 276 10 1)2 1)2 NUM cana-3856 276 11 + + CCONJ cana-3856 276 12 2𝜎)2 2𝜎)2 NUM cana-3856 276 13 + + CCONJ cana-3856 276 14 4𝜎(2(𝑛2 4𝜎(2(𝑛2 NUM cana-3856 276 15 βˆ’ βˆ’ ADP cana-3856 276 16 1 1 NUM cana-3856 276 17 ) ) PUNCT cana-3856 276 18 + + CCONJ cana-3856 276 19 4(𝑛 4(𝑛 NUM cana-3856 276 20 βˆ’ βˆ’ PROPN cana-3856 276 21 𝜎 𝜎 NOUN cana-3856 276 22 ) ) PUNCT cana-3856 276 23 ) ) PUNCT cana-3856 276 24 . . PUNCT cana-3856 277 1 communications communication NOUN cana-3856 277 2 on on ADP cana-3856 277 3 applied apply VERB cana-3856 277 4 nonlinear nonlinear ADJ cana-3856 277 5 analysis analysis NOUN cana-3856 277 6 issn issn NOUN cana-3856 277 7 : : PUNCT cana-3856 277 8 1074 1074 NUM cana-3856 277 9 - - PUNCT cana-3856 277 10 133x 133x NUM cana-3856 277 11 vol vol NOUN cana-3856 277 12 32 32 NUM cana-3856 277 13 no no NOUN cana-3856 277 14 . . PUNCT cana-3856 278 1 9s 9s NUM cana-3856 278 2 ( ( PUNCT cana-3856 278 3 2025 2025 NUM cana-3856 278 4 ) ) PUNCT cana-3856 278 5 294 294 NUM cana-3856 278 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 278 7 example example NOUN cana-3856 278 8 2 2 NUM cana-3856 278 9 . . X cana-3856 279 1 consider consider VERB cana-3856 279 2 𝐺 𝐺 NOUN cana-3856 279 3 = = PRON cana-3856 279 4 𝐾3 𝐾3 VERB cana-3856 279 5 is be AUX cana-3856 279 6 complete complete ADJ cana-3856 279 7 graph graph NOUN cana-3856 279 8 with with ADP cana-3856 279 9 3 3 NUM cana-3856 279 10 vertices vertex NOUN cana-3856 279 11 and and CCONJ cana-3856 279 12 𝐺𝑆 𝐺𝑆 NOUN cana-3856 279 13 is be AUX cana-3856 279 14 the the DET cana-3856 279 15 graph graph NOUN cana-3856 279 16 obtained obtain VERB cana-3856 279 17 by by ADP cana-3856 279 18 adding add VERB cana-3856 279 19 1 1 NUM cana-3856 279 20 loops loop NOUN cana-3856 279 21 to to PART cana-3856 279 22 graph graph VERB cana-3856 279 23 𝐺 𝐺 PROPN cana-3856 279 24 = = PUNCT cana-3856 279 25 𝐾3 𝐾3 PROPN cana-3856 279 26 . . PUNCT cana-3856 280 1 sombor sombor NOUN cana-3856 280 2 matrix matrix NOUN cana-3856 280 3 of of ADP cana-3856 280 4 𝐺𝑆 𝐺𝑆 PROPN cana-3856 280 5 is be AUX cana-3856 280 6 𝐴𝑆𝑂(𝐺𝑆 𝐴𝑆𝑂(𝐺𝑆 NUM cana-3856 280 7 ) ) PUNCT cana-3856 281 1 = = NOUN cana-3856 281 2 [ [ PUNCT cana-3856 281 3 4√2 4√2 PROPN cana-3856 281 4 2√5 2√5 NUM cana-3856 281 5 2√5 2√5 NUM cana-3856 281 6 2√5 2√5 NUM cana-3856 281 7 0 0 SYM cana-3856 281 8 2√2 2√2 NUM cana-3856 281 9 2√5 2√5 NUM cana-3856 281 10 2√2 2√2 NUM cana-3856 281 11 0 0 NUM cana-3856 281 12 ] ] PUNCT cana-3856 281 13 hereπœ†1 hereπœ†1 X cana-3856 281 14 = = SYM cana-3856 281 15 10.7234 10.7234 NUM cana-3856 281 16 , , PUNCT cana-3856 281 17 πœ†2 πœ†2 NOUN cana-3856 281 18 = = SYM cana-3856 281 19 βˆ’2.82843 βˆ’2.82843 PROPN cana-3856 281 20 , , PUNCT cana-3856 281 21 πœ†3 πœ†3 NOUN cana-3856 281 22 = = PUNCT cana-3856 282 1 βˆ’2.2381 βˆ’2.2381 PROPN cana-3856 282 2 the the DET cana-3856 282 3 sombor sombor NOUN cana-3856 282 4 energy energy NOUN cana-3856 282 5 of of ADP cana-3856 282 6 graph graph NOUN cana-3856 282 7 𝐺𝑆 𝐺𝑆 PROPN cana-3856 282 8 is be AUX cana-3856 282 9 given give VERB cana-3856 282 10 by by ADP cana-3856 282 11 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NOUN cana-3856 282 12 ) ) PUNCT cana-3856 282 13 = = PUNCT cana-3856 283 1 βˆ‘ βˆ‘ PUNCT cana-3856 283 2 3 3 NUM cana-3856 283 3 𝑖=1 𝑖=1 SYM cana-3856 283 4 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 NUM cana-3856 283 5 ) ) PUNCT cana-3856 283 6 βˆ’ βˆ’ PROPN cana-3856 284 1 √2βˆ‘πœŽπ‘—=1𝑑𝑗 √2βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 284 2 3 3 NUM cana-3856 284 3 | | ADV cana-3856 284 4 = = PUNCT cana-3856 284 5 |10.7234 |10.7234 PUNCT cana-3856 284 6 βˆ’ βˆ’ PROPN cana-3856 284 7 4√2 4√2 PROPN cana-3856 284 8 3 3 NUM cana-3856 285 1 | | ADV cana-3856 286 1 + + CCONJ cana-3856 286 2 | | ADV cana-3856 286 3 βˆ’ βˆ’ PROPN cana-3856 286 4 2.82843 2.82843 NUM cana-3856 286 5 βˆ’ βˆ’ NOUN cana-3856 287 1 4√2 4√2 PROPN cana-3856 287 2 3 3 NUM cana-3856 288 1 | | ADV cana-3856 289 1 + + CCONJ cana-3856 290 1 | | ADV cana-3856 290 2 βˆ’ βˆ’ ADP cana-3856 290 3 2.2381 2.2381 NUM cana-3856 290 4 βˆ’ βˆ’ NOUN cana-3856 290 5 4√2 4√2 NUM cana-3856 290 6 3 3 NUM cana-3856 290 7 | | NOUN cana-3856 290 8 = = SYM cana-3856 290 9 17.675548083 17.675548083 NUM cana-3856 290 10 in in ADP cana-3856 290 11 above above ADP cana-3856 290 12 theorem theorem NOUN cana-3856 290 13 putting put VERB cana-3856 290 14 𝜎 𝜎 NOUN cana-3856 290 15 = = SYM cana-3856 290 16 1 1 NUM cana-3856 290 17 and and CCONJ cana-3856 290 18 n=3 n=3 PUNCT cana-3856 290 19 we we PRON cana-3856 290 20 get get VERB cana-3856 290 21 , , PUNCT cana-3856 290 22 𝐸𝑆𝑂(𝐺𝑆 𝐸𝑆𝑂(𝐺𝑆 NUM cana-3856 290 23 ) ) PUNCT cana-3856 291 1 = = SYM cana-3856 291 2 √2 √2 NOUN cana-3856 291 3 ( ( PUNCT cana-3856 291 4 3 3 NUM cana-3856 291 5 + + NUM cana-3856 291 6 1)(1 1)(1 NUM cana-3856 291 7 βˆ’ βˆ’ NOUN cana-3856 291 8 1 1 NUM cana-3856 291 9 ) ) PUNCT cana-3856 291 10 + + CCONJ cana-3856 291 11 ( ( PUNCT cana-3856 291 12 3 3 NUM cana-3856 291 13 βˆ’ βˆ’ PROPN cana-3856 291 14 1 1 NUM cana-3856 291 15 βˆ’ βˆ’ PROPN cana-3856 291 16 1)(3(3 1)(3(3 NUM cana-3856 291 17 βˆ’ βˆ’ NOUN cana-3856 291 18 1 1 NUM cana-3856 291 19 ) ) PUNCT cana-3856 291 20 + + CCONJ cana-3856 291 21 ( ( PUNCT cana-3856 291 22 3 3 NUM cana-3856 291 23 + + SYM cana-3856 291 24 1)1 1)1 NUM cana-3856 291 25 ) ) PUNCT cana-3856 291 26 3 3 NUM cana-3856 291 27 + + CCONJ cana-3856 291 28 √2((3 √2((3 NUM cana-3856 291 29 βˆ’ βˆ’ PROPN cana-3856 291 30 1)2 1)2 NUM cana-3856 291 31 + + CCONJ cana-3856 291 32 2)2 2)2 NUM cana-3856 291 33 + + SYM cana-3856 291 34 4(2(32 4(2(32 NUM cana-3856 291 35 βˆ’ βˆ’ NOUN cana-3856 291 36 1 1 NUM cana-3856 291 37 ) ) PUNCT cana-3856 291 38 + + CCONJ cana-3856 291 39 4(3 4(3 NUM cana-3856 291 40 βˆ’ βˆ’ NOUN cana-3856 291 41 1 1 NUM cana-3856 291 42 ) ) PUNCT cana-3856 291 43 ) ) PUNCT cana-3856 292 1 = = PUNCT cana-3856 292 2 10 10 NUM cana-3856 293 1 √2 √2 NOUN cana-3856 293 2 3 3 NUM cana-3856 293 3 + + CCONJ cana-3856 293 4 √72 √72 NOUN cana-3856 293 5 + + CCONJ cana-3856 293 6 96 96 NUM cana-3856 293 7 = = SYM cana-3856 293 8 17.675526605 17.675526605 NUM cana-3856 293 9 . . PUNCT cana-3856 293 10 theorem theorem NOUN cana-3856 293 11 5 5 NUM cana-3856 293 12 . . PUNCT cana-3856 294 1 let let VERB cana-3856 294 2 g g NOUN cana-3856 294 3 be be AUX cana-3856 294 4 the the DET cana-3856 294 5 complete complete ADJ cana-3856 294 6 graph graph NOUN cana-3856 294 7 of of ADP cana-3856 294 8 order order NOUN cana-3856 294 9 n n NOUN cana-3856 294 10 and and CCONJ cana-3856 294 11 𝐺𝑙 𝐺𝑙 PROPN cana-3856 294 12 be be VERB cana-3856 294 13 the the DET cana-3856 294 14 graph graph NOUN cana-3856 294 15 obtained obtain VERB cana-3856 294 16 from from ADP cana-3856 294 17 g g NOUN cana-3856 294 18 by by ADP cana-3856 294 19 adding add VERB cana-3856 294 20 a a DET cana-3856 294 21 loop loop NOUN cana-3856 294 22 on on ADP cana-3856 294 23 each each DET cana-3856 294 24 vertex vertex NOUN cana-3856 294 25 of of ADP cana-3856 294 26 g g PROPN cana-3856 294 27 then then ADV cana-3856 294 28 sombor sombor VERB cana-3856 294 29 energy energy NOUN cana-3856 294 30 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 294 31 ⋃ ⋃ NOUN cana-3856 294 32 𝐺𝑙 𝐺𝑙 NOUN cana-3856 294 33 ) ) PUNCT cana-3856 295 1 = = SYM cana-3856 295 2 2𝑛 2𝑛 PROPN cana-3856 295 3 π‘›βˆ’1 π‘›βˆ’1 NUM cana-3856 295 4 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 295 5 ) ) PUNCT cana-3856 295 6 proof proof NOUN cana-3856 295 7 . . PUNCT cana-3856 296 1 let let VERB cana-3856 296 2 𝐻𝑛 𝐻𝑛 VERB cana-3856 296 3 = = SYM cana-3856 296 4 ( ( PUNCT cana-3856 296 5 𝐺⋃ 𝐺⋃ PROPN cana-3856 296 6 𝐺𝑙 𝐺𝑙 PROPN cana-3856 296 7 ) ) PUNCT cana-3856 296 8 . . PUNCT cana-3856 297 1 the the DET cana-3856 297 2 graph graph NOUN cana-3856 297 3 𝐻𝑛 𝐻𝑛 PROPN cana-3856 297 4 contains contain VERB cana-3856 297 5 2n 2n NUM cana-3856 297 6 vertices vertex NOUN cana-3856 297 7 and and CCONJ cana-3856 297 8 n n PRON cana-3856 297 9 loops loop NOUN cana-3856 297 10 . . PUNCT cana-3856 298 1 the the DET cana-3856 298 2 sombor sombor NOUN cana-3856 298 3 matrix matrix NOUN cana-3856 298 4 of of ADP cana-3856 298 5 𝐻𝑛 𝐻𝑛 PROPN cana-3856 298 6 is be AUX cana-3856 298 7 given give VERB cana-3856 298 8 by by ADP cana-3856 298 9 : : PUNCT cana-3856 298 10 𝐴𝑆𝑂(𝐻𝑛 𝐴𝑆𝑂(𝐻𝑛 PROPN cana-3856 298 11 ) ) PUNCT cana-3856 298 12 = = PUNCT cana-3856 299 1 [ [ X cana-3856 299 2 ( ( PUNCT cana-3856 299 3 𝑛 𝑛 PRON cana-3856 299 4 βˆ’ βˆ’ PROPN cana-3856 299 5 1)√2(𝐽 1)√2(𝐽 NUM cana-3856 299 6 βˆ’ βˆ’ PROPN cana-3856 299 7 𝐼)𝑛×𝑛 𝐼)𝑛×𝑛 NOUN cana-3856 300 1 [ [ PUNCT cana-3856 300 2 0]𝑛×𝑛 0]𝑛×𝑛 NOUN cana-3856 300 3 𝑛 𝑛 DET cana-3856 300 4 Γ— Γ— NOUN cana-3856 300 5 𝑛|(𝑛 𝑛|(𝑛 NOUN cana-3856 300 6 + + CCONJ cana-3856 300 7 1)√2(𝐽)𝑛×𝑛 1)√2(𝐽)𝑛×𝑛 X cana-3856 300 8 ] ] PUNCT cana-3856 300 9 the the DET cana-3856 300 10 characteristics characteristic NOUN cana-3856 300 11 polynomial polynomial ADJ cana-3856 300 12 of of ADP cana-3856 300 13 above above ADJ cana-3856 300 14 matrix matrix NOUN cana-3856 300 15 is be AUX cana-3856 300 16 given give VERB cana-3856 300 17 by by ADP cana-3856 300 18 : : PUNCT cana-3856 300 19 communications communication NOUN cana-3856 300 20 on on ADP cana-3856 300 21 applied apply VERB cana-3856 300 22 nonlinear nonlinear ADJ cana-3856 300 23 analysis analysis NOUN cana-3856 300 24 issn issn NOUN cana-3856 300 25 : : PUNCT cana-3856 300 26 1074 1074 NUM cana-3856 300 27 - - PUNCT cana-3856 300 28 133x 133x NUM cana-3856 300 29 vol vol NOUN cana-3856 300 30 32 32 NUM cana-3856 300 31 no no NOUN cana-3856 300 32 . . PUNCT cana-3856 301 1 9s 9s NUM cana-3856 301 2 ( ( PUNCT cana-3856 301 3 2025 2025 NUM cana-3856 301 4 ) ) PUNCT cana-3856 301 5 295 295 NUM cana-3856 301 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 301 7 πœ™(𝐻𝑛 πœ™(𝐻𝑛 NOUN cana-3856 301 8 : : PUNCT cana-3856 301 9 π‘₯ π‘₯ X cana-3856 301 10 ) ) PUNCT cana-3856 302 1 = = NOUN cana-3856 303 1 [ [ X cana-3856 303 2 π‘₯𝐼 π‘₯𝐼 X cana-3856 303 3 βˆ’ βˆ’ NOUN cana-3856 303 4 ( ( PUNCT cana-3856 303 5 𝑛 𝑛 PRON cana-3856 303 6 βˆ’ βˆ’ PROPN cana-3856 303 7 1)√2(𝐽 1)√2(𝐽 NUM cana-3856 303 8 βˆ’ βˆ’ PROPN cana-3856 303 9 𝐼)𝑛×𝑛 𝐼)𝑛×𝑛 NOUN cana-3856 304 1 [ [ PUNCT cana-3856 304 2 0]𝑛×𝑛 0]𝑛×𝑛 NOUN cana-3856 304 3 𝑛 𝑛 PRON cana-3856 304 4 Γ— Γ— NOUN cana-3856 304 5 𝑛|π‘₯𝐼 𝑛|π‘₯𝐼 NOUN cana-3856 304 6 βˆ’ βˆ’ PROPN cana-3856 304 7 ( ( PUNCT cana-3856 304 8 𝑛 𝑛 PROPN cana-3856 304 9 + + X cana-3856 304 10 1)√2(𝐽)𝑛×𝑛 1)√2(𝐽)𝑛×𝑛 ADJ cana-3856 304 11 ] ] PUNCT cana-3856 304 12 if if SCONJ cana-3856 304 13 πœ†1 πœ†1 PROPN cana-3856 304 14 , , PUNCT cana-3856 304 15 πœ†2 πœ†2 NOUN cana-3856 304 16 , , PUNCT cana-3856 304 17 . . PUNCT cana-3856 304 18 . . PUNCT cana-3856 304 19 . . PUNCT cana-3856 305 1 , , PUNCT cana-3856 305 2 πœ†π‘› πœ†π‘› PROPN cana-3856 305 3 are be AUX cana-3856 305 4 eigenvalues eigenvalue NOUN cana-3856 305 5 of of ADP cana-3856 305 6 𝐴𝑆𝑂(𝐻𝑛 𝐴𝑆𝑂(𝐻𝑛 NOUN cana-3856 305 7 ) ) PUNCT cana-3856 305 8 , , PUNCT cana-3856 305 9 then then ADV cana-3856 305 10 πœ†1 πœ†1 VERB cana-3856 305 11 = = PUNCT cana-3856 305 12 ( ( PUNCT cana-3856 305 13 𝑛 𝑛 PRON cana-3856 305 14 βˆ’ βˆ’ PROPN cana-3856 305 15 1 1 NUM cana-3856 305 16 ) ) PUNCT cana-3856 305 17 2√2 2√2 NUM cana-3856 305 18 , , PUNCT cana-3856 305 19 πœ†π‘– πœ†π‘– NOUN cana-3856 305 20 = = PUNCT cana-3856 305 21 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-3856 306 1 βˆ’ βˆ’ PROPN cana-3856 306 2 1)√2 1)√2 NOUN cana-3856 306 3 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-3856 306 4 𝑖 𝑖 NOUN cana-3856 306 5 = = SYM cana-3856 306 6 2,3 2,3 NUM cana-3856 306 7 , , PUNCT cana-3856 306 8 . . PUNCT cana-3856 306 9 . . PUNCT cana-3856 307 1 . . PUNCT cana-3856 308 1 , , PUNCT cana-3856 308 2 𝑛 𝑛 PROPN cana-3856 308 3 , , PUNCT cana-3856 308 4 πœ†π‘›+1 πœ†π‘›+1 NUM cana-3856 308 5 = = PUNCT cana-3856 308 6 𝑛(𝑛 𝑛(𝑛 PROPN cana-3856 308 7 + + CCONJ cana-3856 308 8 1)√2 1)√2 NUM cana-3856 308 9 , , PUNCT cana-3856 308 10 πœ†π‘— πœ†π‘— X cana-3856 309 1 = = SYM cana-3856 309 2 0 0 NUM cana-3856 309 3 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN cana-3856 309 4 𝑗 𝑗 X cana-3856 309 5 = = SYM cana-3856 309 6 𝑛 𝑛 PROPN cana-3856 309 7 + + ADJ cana-3856 309 8 2 2 NUM cana-3856 309 9 , , PUNCT cana-3856 309 10 𝑛 𝑛 PRON cana-3856 309 11 + + NOUN cana-3856 309 12 3 3 NUM cana-3856 309 13 , , PUNCT cana-3856 309 14 . . PUNCT cana-3856 309 15 . . PUNCT cana-3856 309 16 . . PUNCT cana-3856 310 1 , , PUNCT cana-3856 310 2 2𝑛 2𝑛 PROPN cana-3856 310 3 sombor sombor NOUN cana-3856 310 4 energy energy NOUN cana-3856 310 5 of of ADP cana-3856 310 6 𝐻𝑛 𝐻𝑛 PROPN cana-3856 310 7 is be AUX cana-3856 310 8 given give VERB cana-3856 310 9 by by ADP cana-3856 310 10 , , PUNCT cana-3856 310 11 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 PROPN cana-3856 310 12 ) ) PUNCT cana-3856 310 13 = = PUNCT cana-3856 311 1 βˆ‘ βˆ‘ PUNCT cana-3856 311 2 2𝑛 2𝑛 PROPN cana-3856 311 3 𝑖=1 𝑖=1 PROPN cana-3856 311 4 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 PROPN cana-3856 311 5 ) ) PUNCT cana-3856 312 1 βˆ’ βˆ’ PROPN cana-3856 312 2 √2βˆ‘πœŽπ‘—=1𝑑𝑗 √2βˆ‘πœŽπ‘—=1𝑑𝑗 X cana-3856 312 3 𝑛 𝑛 VERB cana-3856 312 4 | | ADV cana-3856 312 5 here here ADV cana-3856 312 6 βˆ‘πœŽπ‘—=1 βˆ‘πœŽπ‘—=1 ADV cana-3856 312 7 𝑑𝑗 𝑑𝑗 NOUN cana-3856 312 8 = = PUNCT cana-3856 312 9 𝑛(𝑛 𝑛(𝑛 PROPN cana-3856 312 10 + + CCONJ cana-3856 312 11 1 1 X cana-3856 312 12 ) ) PUNCT cana-3856 312 13 hence hence ADV cana-3856 312 14 , , PUNCT cana-3856 312 15 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 NOUN cana-3856 312 16 ) ) PUNCT cana-3856 312 17 = = PROPN cana-3856 312 18 βˆ‘ βˆ‘ PROPN cana-3856 312 19 2𝑛 2𝑛 PROPN cana-3856 312 20 𝑖=1 𝑖=1 PROPN cana-3856 312 21 |πœ†π‘–(𝐺𝑠 |πœ†π‘–(𝐺𝑠 PROPN cana-3856 312 22 ) ) PUNCT cana-3856 312 23 βˆ’ βˆ’ NOUN cana-3856 312 24 √2𝑛(𝑛 √2𝑛(𝑛 NOUN cana-3856 312 25 + + CCONJ cana-3856 312 26 1 1 NUM cana-3856 312 27 ) ) PUNCT cana-3856 312 28 2𝑛 2𝑛 NOUN cana-3856 313 1 | | NOUN cana-3856 313 2 = = SYM cana-3856 313 3 |(𝑛 |(𝑛 NOUN cana-3856 313 4 βˆ’ βˆ’ NOUN cana-3856 313 5 1)2√2 1)2√2 NUM cana-3856 313 6 βˆ’ βˆ’ PROPN cana-3856 313 7 √2(𝑛 √2(𝑛 NOUN cana-3856 313 8 + + CCONJ cana-3856 313 9 1 1 NUM cana-3856 313 10 ) ) PUNCT cana-3856 313 11 2 2 NUM cana-3856 314 1 | | ADV cana-3856 315 1 + + CCONJ cana-3856 315 2 ( ( PUNCT cana-3856 315 3 𝑛 𝑛 PROPN cana-3856 315 4 βˆ’ βˆ’ PROPN cana-3856 315 5 1)| 1)| NUM cana-3856 315 6 βˆ’ βˆ’ PROPN cana-3856 315 7 ( ( PUNCT cana-3856 315 8 𝑛 𝑛 PROPN cana-3856 315 9 βˆ’ βˆ’ NUM cana-3856 315 10 1)√2 1)√2 NUM cana-3856 315 11 βˆ’ βˆ’ PROPN cana-3856 315 12 √2(𝑛 √2(𝑛 NOUN cana-3856 315 13 + + CCONJ cana-3856 315 14 1 1 NUM cana-3856 315 15 ) ) PUNCT cana-3856 315 16 2 2 NUM cana-3856 316 1 | | ADV cana-3856 316 2 + + CCONJ cana-3856 316 3 |𝑛(𝑛 |𝑛(𝑛 PROPN cana-3856 316 4 + + CCONJ cana-3856 316 5 1)√2 1)√2 NUM cana-3856 316 6 βˆ’ βˆ’ PROPN cana-3856 316 7 √2(𝑛 √2(𝑛 NOUN cana-3856 316 8 + + CCONJ cana-3856 316 9 1 1 NUM cana-3856 316 10 ) ) PUNCT cana-3856 316 11 2 2 NUM cana-3856 317 1 | | ADV cana-3856 317 2 + + CCONJ cana-3856 317 3 ( ( PUNCT cana-3856 317 4 𝑛 𝑛 PRON cana-3856 317 5 βˆ’ βˆ’ PROPN cana-3856 317 6 1)|0 1)|0 NUM cana-3856 317 7 βˆ’ βˆ’ PROPN cana-3856 317 8 √2(𝑛 √2(𝑛 NOUN cana-3856 317 9 + + CCONJ cana-3856 317 10 1 1 NUM cana-3856 317 11 ) ) PUNCT cana-3856 317 12 2 2 NUM cana-3856 318 1 | | NOUN cana-3856 318 2 = = SYM cana-3856 319 1 [ [ X cana-3856 319 2 2(𝑛 2(𝑛 NUM cana-3856 319 3 βˆ’ βˆ’ PROPN cana-3856 319 4 1)2 1)2 NUM cana-3856 319 5 βˆ’ βˆ’ PROPN cana-3856 319 6 ( ( PUNCT cana-3856 319 7 𝑛 𝑛 PROPN cana-3856 319 8 + + NOUN cana-3856 319 9 1 1 NUM cana-3856 319 10 ) ) PUNCT cana-3856 319 11 ] ] PUNCT cana-3856 320 1 √2 √2 NOUN cana-3856 320 2 2 2 NUM cana-3856 320 3 + + CCONJ cana-3856 320 4 ( ( PUNCT cana-3856 320 5 𝑛 𝑛 PRON cana-3856 320 6 βˆ’ βˆ’ PROPN cana-3856 320 7 1)(2(𝑛 1)(2(𝑛 NUM cana-3856 320 8 βˆ’ βˆ’ NOUN cana-3856 320 9 1 1 NUM cana-3856 320 10 ) ) PUNCT cana-3856 320 11 + + CCONJ cana-3856 320 12 ( ( PUNCT cana-3856 320 13 𝑛 𝑛 PROPN cana-3856 320 14 + + NOUN cana-3856 320 15 1 1 NUM cana-3856 320 16 ) ) PUNCT cana-3856 320 17 ) ) PUNCT cana-3856 321 1 √2 √2 NOUN cana-3856 321 2 2 2 NUM cana-3856 321 3 + + CCONJ cana-3856 321 4 [ [ X cana-3856 321 5 2𝑛(𝑛 2𝑛(𝑛 NUM cana-3856 321 6 + + CCONJ cana-3856 321 7 1 1 NUM cana-3856 321 8 ) ) PUNCT cana-3856 321 9 βˆ’ βˆ’ PROPN cana-3856 322 1 ( ( PUNCT cana-3856 322 2 𝑛 𝑛 PROPN cana-3856 322 3 + + NOUN cana-3856 322 4 1 1 NUM cana-3856 322 5 ) ) PUNCT cana-3856 322 6 ] ] PUNCT cana-3856 323 1 √2 √2 NOUN cana-3856 323 2 2 2 NUM cana-3856 323 3 + + CCONJ cana-3856 323 4 ( ( PUNCT cana-3856 323 5 𝑛 𝑛 PRON cana-3856 323 6 βˆ’ βˆ’ PROPN cana-3856 323 7 1)(𝑛 1)(𝑛 NUM cana-3856 324 1 + + NUM cana-3856 324 2 1 1 X cana-3856 324 3 ) ) PUNCT cana-3856 324 4 √2 √2 NOUN cana-3856 324 5 2 2 NUM cana-3856 324 6 = = SYM cana-3856 324 7 1 1 NUM cana-3856 324 8 √2 √2 PROPN cana-3856 324 9 [ [ X cana-3856 324 10 2𝑛2 2𝑛2 NUM cana-3856 324 11 βˆ’ βˆ’ NOUN cana-3856 324 12 4𝑛 4𝑛 NOUN cana-3856 324 13 + + CCONJ cana-3856 324 14 2 2 NUM cana-3856 324 15 βˆ’ βˆ’ NOUN cana-3856 324 16 𝑛 𝑛 PRON cana-3856 324 17 βˆ’ βˆ’ NUM cana-3856 324 18 1 1 NUM cana-3856 325 1 + + CCONJ cana-3856 325 2 ( ( PUNCT cana-3856 325 3 𝑛 𝑛 DET cana-3856 325 4 βˆ’ βˆ’ PROPN cana-3856 325 5 1)(2𝑛 1)(2𝑛 NUM cana-3856 326 1 βˆ’ βˆ’ NOUN cana-3856 326 2 2 2 NUM cana-3856 326 3 + + CCONJ cana-3856 326 4 𝑛 𝑛 VERB cana-3856 326 5 + + NOUN cana-3856 326 6 1 1 NUM cana-3856 326 7 ) ) PUNCT cana-3856 326 8 + + CCONJ cana-3856 326 9 ( ( PUNCT cana-3856 326 10 2𝑛2 2𝑛2 NUM cana-3856 326 11 + + CCONJ cana-3856 326 12 2𝑛 2𝑛 NUM cana-3856 326 13 βˆ’ βˆ’ NOUN cana-3856 326 14 𝑛 𝑛 PRON cana-3856 326 15 βˆ’ βˆ’ PROPN cana-3856 326 16 1 1 NUM cana-3856 326 17 ) ) PUNCT cana-3856 326 18 + + CCONJ cana-3856 326 19 𝑛2 𝑛2 NOUN cana-3856 326 20 βˆ’ βˆ’ PROPN cana-3856 326 21 1 1 NUM cana-3856 326 22 ] ] SYM cana-3856 326 23 = = SYM cana-3856 326 24 1 1 NUM cana-3856 326 25 √2 √2 PROPN cana-3856 326 26 ( ( PUNCT cana-3856 326 27 8𝑛2 8𝑛2 NUM cana-3856 326 28 βˆ’ βˆ’ PROPN cana-3856 326 29 8𝑛 8𝑛 NOUN cana-3856 326 30 ) ) PUNCT cana-3856 326 31 = = SYM cana-3856 326 32 4√2𝑛(𝑛 4√2𝑛(𝑛 NUM cana-3856 326 33 βˆ’ βˆ’ NOUN cana-3856 326 34 1 1 NUM cana-3856 326 35 ) ) PUNCT cana-3856 326 36 therefore therefore ADV cana-3856 326 37 , , PUNCT cana-3856 326 38 𝐸𝑆𝑂(𝐺𝑠 𝐸𝑆𝑂(𝐺𝑠 PROPN cana-3856 326 39 ) ) PUNCT cana-3856 326 40 = = SYM cana-3856 326 41 4√2𝑛(𝑛 4√2𝑛(𝑛 NUM cana-3856 326 42 βˆ’ βˆ’ NOUN cana-3856 326 43 1 1 NUM cana-3856 326 44 ) ) PUNCT cana-3856 326 45 ( ( PUNCT cana-3856 326 46 3 3 X cana-3856 326 47 ) ) PUNCT cana-3856 326 48 the the DET cana-3856 326 49 sombor sombor NOUN cana-3856 326 50 matrix matrix NOUN cana-3856 326 51 of of ADP cana-3856 326 52 graph graph NOUN cana-3856 326 53 𝐺 𝐺 PROPN cana-3856 326 54 = = SYM cana-3856 326 55 𝐾𝑛 𝐾𝑛 PROPN cana-3856 326 56 is be AUX cana-3856 326 57 , , PUNCT cana-3856 326 58 𝐴𝑆𝑂(𝐺 𝐴𝑆𝑂(𝐺 NOUN cana-3856 326 59 ) ) PUNCT cana-3856 326 60 = = PUNCT cana-3856 327 1 [ [ X cana-3856 327 2 ( ( PUNCT cana-3856 327 3 𝑛 𝑛 PRON cana-3856 327 4 βˆ’ βˆ’ PROPN cana-3856 327 5 1)√2(𝐽 1)√2(𝐽 NUM cana-3856 327 6 βˆ’ βˆ’ PROPN cana-3856 327 7 𝐼)𝑛×𝑛 𝐼)𝑛×𝑛 PROPN cana-3856 327 8 ] ] PUNCT cana-3856 327 9 communications communication NOUN cana-3856 327 10 on on ADP cana-3856 327 11 applied apply VERB cana-3856 327 12 nonlinear nonlinear ADJ cana-3856 327 13 analysis analysis NOUN cana-3856 327 14 issn issn NOUN cana-3856 327 15 : : PUNCT cana-3856 327 16 1074 1074 NUM cana-3856 327 17 - - PUNCT cana-3856 327 18 133x 133x NUM cana-3856 327 19 vol vol NOUN cana-3856 327 20 32 32 NUM cana-3856 327 21 no no NOUN cana-3856 327 22 . . PUNCT cana-3856 328 1 9s 9s NUM cana-3856 328 2 ( ( PUNCT cana-3856 328 3 2025 2025 NUM cana-3856 328 4 ) ) PUNCT cana-3856 328 5 296 296 NUM cana-3856 329 1 https://internationalpubls.com https://internationalpubls.com X cana-3856 329 2 eigenvalues eigenvalue VERB cana-3856 329 3 of of ADP cana-3856 329 4 𝐴𝑆𝑂(𝐺 𝐴𝑆𝑂(𝐺 NOUN cana-3856 329 5 ) ) PUNCT cana-3856 330 1 are(𝑛 are(𝑛 PROPN cana-3856 330 2 βˆ’ βˆ’ NOUN cana-3856 330 3 1)2√2 1)2√2 NUM cana-3856 330 4 with with ADP cana-3856 330 5 multiplicity multiplicity NOUN cana-3856 330 6 1 1 NUM cana-3856 330 7 and and CCONJ cana-3856 330 8 βˆ’(𝑛 βˆ’(𝑛 NOUN cana-3856 330 9 βˆ’ βˆ’ PROPN cana-3856 330 10 1)√2 1)√2 NOUN cana-3856 330 11 with with ADP cana-3856 330 12 𝑛 𝑛 PRON cana-3856 330 13 βˆ’ βˆ’ NUM cana-3856 330 14 1 1 NUM cana-3856 330 15 mutilplicity mutilplicity NOUN cana-3856 330 16 . . PUNCT cana-3856 331 1 hence hence ADV cana-3856 331 2 sombor sombor VERB cana-3856 331 3 energy energy NOUN cana-3856 331 4 of of ADP cana-3856 331 5 𝐺 𝐺 PROPN cana-3856 331 6 is be AUX cana-3856 331 7 given give VERB cana-3856 331 8 by by ADP cana-3856 331 9 , , PUNCT cana-3856 331 10 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 331 11 ) ) PUNCT cana-3856 331 12 = = NOUN cana-3856 331 13 βˆ‘ βˆ‘ PUNCT cana-3856 331 14 𝑛 𝑛 DET cana-3856 331 15 𝑖=1 𝑖=1 PROPN cana-3856 331 16 |πœ†π‘–| |πœ†π‘–| NOUN cana-3856 332 1 = = SYM cana-3856 333 1 ( ( PUNCT cana-3856 333 2 𝑛 𝑛 DET cana-3856 333 3 βˆ’ βˆ’ NOUN cana-3856 333 4 1)2√2 1)2√2 NUM cana-3856 333 5 + + CCONJ cana-3856 333 6 ( ( PUNCT cana-3856 333 7 𝑛 𝑛 PRON cana-3856 333 8 βˆ’ βˆ’ NOUN cana-3856 333 9 1)(𝑛 1)(𝑛 NUM cana-3856 333 10 βˆ’ βˆ’ PROPN cana-3856 333 11 1)√2 1)√2 NUM cana-3856 333 12 = = SYM cana-3856 333 13 2(𝑛 2(𝑛 NUM cana-3856 333 14 βˆ’ βˆ’ NUM cana-3856 333 15 1)2√2 1)2√2 NUM cana-3856 333 16 therefore therefore ADV cana-3856 333 17 , , PUNCT cana-3856 333 18 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 333 19 ) ) PUNCT cana-3856 333 20 = = NOUN cana-3856 333 21 2(𝑛 2(𝑛 NUM cana-3856 333 22 βˆ’ βˆ’ NUM cana-3856 333 23 1 1 NUM cana-3856 333 24 ) ) PUNCT cana-3856 333 25 2√2 2√2 NUM cana-3856 333 26 ( ( PUNCT cana-3856 333 27 4 4 NUM cana-3856 333 28 ) ) PUNCT cana-3856 333 29 hence hence ADV cana-3856 333 30 , , PUNCT cana-3856 333 31 by by ADP cana-3856 333 32 equation equation NOUN cana-3856 333 33 ( ( PUNCT cana-3856 333 34 3 3 NUM cana-3856 333 35 ) ) PUNCT cana-3856 333 36 and and CCONJ cana-3856 333 37 ( ( PUNCT cana-3856 333 38 4 4 NUM cana-3856 333 39 ) ) PUNCT cana-3856 333 40 , , PUNCT cana-3856 333 41 𝐸𝑆𝑂(𝐻𝑛 𝐸𝑆𝑂(𝐻𝑛 NOUN cana-3856 333 42 ) ) PUNCT cana-3856 333 43 = = SYM cana-3856 333 44 2𝑛 2𝑛 PROPN cana-3856 333 45 ( ( PUNCT cana-3856 333 46 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-3856 333 47 ) ) PUNCT cana-3856 333 48 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 333 49 ) ) PUNCT cana-3856 333 50 example example NOUN cana-3856 334 1 3 3 X cana-3856 334 2 . . X cana-3856 334 3 consider consider VERB cana-3856 334 4 the the DET cana-3856 334 5 graph graph NOUN cana-3856 334 6 𝐻3 𝐻3 NOUN cana-3856 334 7 = = PRON cana-3856 334 8 𝐾3⋃ 𝐾3⋃ NOUN cana-3856 334 9 𝐾3 𝐾3 VERB cana-3856 334 10 𝑙 𝑙 X cana-3856 334 11 and and CCONJ cana-3856 334 12 𝐺 𝐺 PROPN cana-3856 334 13 = = PUNCT cana-3856 334 14 𝐾3 𝐾3 PROPN cana-3856 334 15 . . PUNCT cana-3856 335 1 the the DET cana-3856 335 2 graph graph NOUN cana-3856 335 3 𝐻3 𝐻3 NOUN cana-3856 335 4 contains contain VERB cana-3856 335 5 6 6 NUM cana-3856 335 6 vertices vertex NOUN cana-3856 335 7 and and CCONJ cana-3856 335 8 three three NUM cana-3856 335 9 loops loop NOUN cana-3856 335 10 . . PUNCT cana-3856 336 1 it it PRON cana-3856 336 2 is be AUX cana-3856 336 3 known know VERB cana-3856 336 4 that that SCONJ cana-3856 336 5 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 336 6 ) ) PUNCT cana-3856 336 7 = = PUNCT cana-3856 337 1 8√2 8√2 ADP cana-3856 337 2 the the DET cana-3856 337 3 sombor sombor NOUN cana-3856 337 4 matrix matrix NOUN cana-3856 337 5 of of ADP cana-3856 337 6 𝐻3 𝐻3 NOUN cana-3856 337 7 is be AUX cana-3856 337 8 𝐴𝑆𝑂(𝐻3 𝐴𝑆𝑂(𝐻3 ADJ cana-3856 337 9 ) ) PUNCT cana-3856 337 10 = = NOUN cana-3856 338 1 [ [ PUNCT cana-3856 338 2 0 0 NUM cana-3856 338 3 2√2 2√2 NUM cana-3856 338 4 2√2 2√2 NUM cana-3856 338 5 0 0 NUM cana-3856 338 6 0 0 NUM cana-3856 338 7 0 0 NUM cana-3856 338 8 2√2 2√2 NUM cana-3856 338 9 0 0 NUM cana-3856 339 1 2√2 2√2 NUM cana-3856 339 2 0 0 NUM cana-3856 339 3 0 0 NUM cana-3856 339 4 0 0 NUM cana-3856 340 1 2√2 2√2 NUM cana-3856 340 2 2√2 2√2 NUM cana-3856 340 3 0 0 NUM cana-3856 340 4 0 0 NUM cana-3856 340 5 0 0 NUM cana-3856 340 6 0 0 NUM cana-3856 340 7 0 0 NUM cana-3856 340 8 0 0 NUM cana-3856 340 9 0 0 NUM cana-3856 341 1 4√2 4√2 X cana-3856 341 2 4√2 4√2 X cana-3856 341 3 4√2 4√2 PROPN cana-3856 341 4 0 0 NUM cana-3856 341 5 0 0 NUM cana-3856 341 6 0 0 NUM cana-3856 342 1 4√2 4√2 X cana-3856 342 2 4√2 4√2 X cana-3856 342 3 4√2 4√2 PROPN cana-3856 342 4 0 0 NUM cana-3856 342 5 0 0 NUM cana-3856 342 6 0 0 NUM cana-3856 343 1 4√2 4√2 PROPN cana-3856 343 2 4√2 4√2 PROPN cana-3856 343 3 4√2 4√2 PROPN cana-3856 343 4 ] ] PUNCT cana-3856 343 5 communications communication NOUN cana-3856 343 6 on on ADP cana-3856 343 7 applied apply VERB cana-3856 343 8 nonlinear nonlinear ADJ cana-3856 343 9 analysis analysis NOUN cana-3856 343 10 issn issn NOUN cana-3856 343 11 : : PUNCT cana-3856 343 12 1074 1074 NUM cana-3856 343 13 - - PUNCT cana-3856 343 14 133x 133x NUM cana-3856 343 15 vol vol NOUN cana-3856 343 16 32 32 NUM cana-3856 343 17 no no NOUN cana-3856 343 18 . . PUNCT cana-3856 344 1 9s 9s NUM cana-3856 344 2 ( ( PUNCT cana-3856 344 3 2025 2025 NUM cana-3856 344 4 ) ) PUNCT cana-3856 344 5 297 297 NUM cana-3856 344 6 https://internationalpubls.com https://internationalpubls.com X cana-3856 344 7 the the DET cana-3856 344 8 eigen eigen PROPN cana-3856 344 9 values value NOUN cana-3856 344 10 of of ADP cana-3856 344 11 𝐻3 𝐻3 NOUN cana-3856 344 12 are be AUX cana-3856 344 13 4√2 4√2 PROPN cana-3856 344 14 , , PUNCT cana-3856 344 15 12√2 12√2 NUM cana-3856 344 16 , , PUNCT cana-3856 344 17 ( ( PUNCT cana-3856 344 18 βˆ’2√2 βˆ’2√2 PROPN cana-3856 344 19 ) ) PUNCT cana-3856 344 20 with with ADP cana-3856 344 21 multiplicity multiplicity NOUN cana-3856 344 22 2 2 NUM cana-3856 344 23 , , PUNCT cana-3856 344 24 and and CCONJ cana-3856 344 25 0 0 NUM cana-3856 344 26 with with ADP cana-3856 344 27 multiplicity multiplicity NOUN cana-3856 344 28 2 2 NUM cana-3856 344 29 . . PUNCT cana-3856 344 30 hence hence ADV cana-3856 344 31 , , PUNCT cana-3856 344 32 𝐸𝑆𝑂(𝐻3 𝐸𝑆𝑂(𝐻3 NUM cana-3856 344 33 ) ) PUNCT cana-3856 345 1 = = SYM cana-3856 345 2 |4√2 |4√2 NOUN cana-3856 345 3 βˆ’ βˆ’ NOUN cana-3856 345 4 2√2| 2√2| NUM cana-3856 346 1 + + CCONJ cana-3856 346 2 12√2 12√2 NUM cana-3856 346 3 βˆ’ βˆ’ NOUN cana-3856 346 4 2√2| 2√2| NUM cana-3856 347 1 + + CCONJ cana-3856 347 2 2| 2| NUM cana-3856 347 3 βˆ’ βˆ’ PROPN cana-3856 348 1 2√2 2√2 NUM cana-3856 348 2 βˆ’ βˆ’ NOUN cana-3856 348 3 2√2| 2√2| NUM cana-3856 349 1 + + CCONJ cana-3856 349 2 2|0 2|0 NUM cana-3856 349 3 βˆ’ βˆ’ NOUN cana-3856 349 4 2√2| 2√2| NUM cana-3856 350 1 = = SYM cana-3856 350 2 24√2 24√2 NUM cana-3856 350 3 . . PUNCT cana-3856 351 1 therefore therefore ADV cana-3856 351 2 , , PUNCT cana-3856 351 3 𝐸𝑆𝑂(𝐻3 𝐸𝑆𝑂(𝐻3 NUM cana-3856 351 4 ) ) PUNCT cana-3856 351 5 = = NOUN cana-3856 351 6 6 6 NUM cana-3856 351 7 2 2 NUM cana-3856 351 8 𝐸𝑆𝑂(𝐺 𝐸𝑆𝑂(𝐺 NOUN cana-3856 351 9 ) ) PUNCT cana-3856 351 10 . . PUNCT cana-3856 352 1 references reference NOUN cana-3856 352 2 [ [ X cana-3856 352 3 1 1 NUM cana-3856 352 4 ] ] X cana-3856 352 5 b. b. PROPN cana-3856 352 6 furtula furtula PROPN cana-3856 352 7 , , PUNCT cana-3856 352 8 i. i. PROPN cana-3856 352 9 gutman gutman PROPN cana-3856 352 10 , , PUNCT cana-3856 352 11 a a DET cana-3856 352 12 forgotten forget VERB cana-3856 352 13 topological topological ADJ cana-3856 352 14 index index NOUN cana-3856 352 15 , , PUNCT cana-3856 352 16 j j PROPN cana-3856 352 17 math math PROPN cana-3856 352 18 chem chem NOUN cana-3856 352 19 , , PUNCT cana-3856 352 20 53 53 NUM cana-3856 352 21 ( ( PUNCT cana-3856 352 22 2015 2015 NUM cana-3856 352 23 ) ) PUNCT cana-3856 352 24 1184 1184 NUM cana-3856 352 25 - - SYM cana-3856 352 26 1190 1190 NUM cana-3856 352 27 . . PUNCT cana-3856 353 1 [ [ X cana-3856 353 2 2 2 NUM cana-3856 353 3 ] ] X cana-3856 353 4 lin lin PROPN cana-3856 353 5 , , PUNCT cana-3856 353 6 zhen zhen PROPN cana-3856 353 7 . . PUNCT cana-3856 354 1 on on ADP cana-3856 354 2 the the DET cana-3856 354 3 spectral spectral ADJ cana-3856 354 4 radius radius NOUN cana-3856 354 5 , , PUNCT cana-3856 354 6 energy energy NOUN cana-3856 354 7 and and CCONJ cana-3856 354 8 estrada estrada PROPN cana-3856 354 9 index index NOUN cana-3856 354 10 of of ADP cana-3856 354 11 the the DET cana-3856 354 12 sombor sombor NOUN cana-3856 354 13 matrix matrix NOUN cana-3856 354 14 of of ADP cana-3856 354 15 graphs graph NOUN cana-3856 354 16 . . PUNCT cana-3856 355 1 arxiv arxiv PROPN cana-3856 355 2 preprint preprint NOUN cana-3856 355 3 arxiv:2102.03960 arxiv:2102.03960 X cana-3856 355 4 ( ( PUNCT cana-3856 355 5 2021 2021 NUM cana-3856 355 6 ) ) PUNCT cana-3856 355 7 . . PUNCT cana-3856 356 1 [ [ X cana-3856 356 2 3 3 NUM cana-3856 356 3 ] ] X cana-3856 356 4 i. i. PROPN cana-3856 356 5 gutman gutman PROPN cana-3856 356 6 , , PUNCT cana-3856 356 7 the the DET cana-3856 356 8 energy energy NOUN cana-3856 356 9 of of ADP cana-3856 356 10 a a DET cana-3856 356 11 graph graph NOUN cana-3856 356 12 , , PUNCT cana-3856 356 13 ber ber PROPN cana-3856 356 14 . . PUNCT cana-3856 356 15 math math NOUN cana-3856 356 16 . . PUNCT cana-3856 357 1 statist statist PROPN cana-3856 357 2 . . PUNCT cana-3856 358 1 sekt sekt PROPN cana-3856 358 2 . . PUNCT cana-3856 359 1 forschungsz forschungsz PROPN cana-3856 359 2 . . PUNCT cana-3856 360 1 graz graz PROPN cana-3856 360 2 . . PROPN cana-3856 360 3 , , PUNCT cana-3856 360 4 103 103 NUM cana-3856 360 5 ( ( PUNCT cana-3856 360 6 1978 1978 NUM cana-3856 360 7 ) ) PUNCT cana-3856 360 8 1 1 NUM cana-3856 360 9 - - SYM cana-3856 360 10 22 22 NUM cana-3856 360 11 . . PUNCT cana-3856 361 1 [ [ X cana-3856 361 2 4 4 NUM cana-3856 361 3 ] ] X cana-3856 361 4 i. i. PROPN cana-3856 361 5 gutman gutman PROPN cana-3856 361 6 , , PUNCT cana-3856 361 7 topological topological ADJ cana-3856 361 8 studies study NOUN cana-3856 361 9 on on ADP cana-3856 361 10 heteroconjugated heteroconjugate VERB cana-3856 361 11 molecules molecule NOUN cana-3856 361 12 . . PUNCT cana-3856 362 1 alternant alternant ADJ cana-3856 362 2 systems system NOUN cana-3856 362 3 with with ADP cana-3856 362 4 one one NUM cana-3856 362 5 heteroatom heteroatom NOUN cana-3856 362 6 , , PUNCT cana-3856 362 7 theor theor PROPN cana-3856 362 8 . . PUNCT cana-3856 363 1 chim chim PROPN cana-3856 363 2 . . PUNCT cana-3856 364 1 acta acta PROPN cana-3856 364 2 . . PUNCT cana-3856 365 1 50 50 NUM cana-3856 365 2 ( ( PUNCT cana-3856 365 3 1979 1979 NUM cana-3856 365 4 ) ) PUNCT cana-3856 365 5 287–297 287–297 NUM cana-3856 365 6 . . PUNCT cana-3856 366 1 [ [ X cana-3856 366 2 5 5 NUM cana-3856 366 3 ] ] PUNCT cana-3856 366 4 i. i. NOUN cana-3856 366 5 gutman gutman PROPN cana-3856 366 6 , , PUNCT cana-3856 366 7 topological topological ADJ cana-3856 366 8 studies study NOUN cana-3856 366 9 on on ADP cana-3856 366 10 heteroconjugated heteroconjugate VERB cana-3856 366 11 molecules molecule NOUN cana-3856 366 12 . . PUNCT cana-3856 367 1 vi vi X cana-3856 367 2 . . PROPN cana-3856 367 3 alternant alternant PROPN cana-3856 367 4 systems system NOUN cana-3856 367 5 with with ADP cana-3856 367 6 two two NUM cana-3856 367 7 heteroatoms heteroatom NOUN cana-3856 367 8 , , PUNCT cana-3856 367 9 z. z. PROPN cana-3856 367 10 naturforsch naturforsch PROPN cana-3856 367 11 . . PUNCT cana-3856 368 1 45a 45a NUM cana-3856 368 2 ( ( PUNCT cana-3856 368 3 1990 1990 NUM cana-3856 368 4 ) ) PUNCT cana-3856 368 5 1085–1089 1085–1089 NUM cana-3856 368 6 . . PUNCT cana-3856 369 1 [ [ X cana-3856 369 2 6 6 NUM cana-3856 369 3 ] ] PUNCT cana-3856 369 4 i. i. PROPN cana-3856 369 5 gutman gutman PROPN cana-3856 369 6 , , PUNCT cana-3856 369 7 b. b. PROPN cana-3856 369 8 zhou zhou PROPN cana-3856 369 9 , , PUNCT cana-3856 369 10 laplacian laplacian ADJ cana-3856 369 11 energy energy NOUN cana-3856 369 12 of of ADP cana-3856 369 13 a a DET cana-3856 369 14 graph graph NOUN cana-3856 369 15 , , PUNCT cana-3856 369 16 lin lin PROPN cana-3856 369 17 . . PUNCT cana-3856 370 1 algebra algebra PROPN cana-3856 370 2 appl appl PROPN cana-3856 370 3 . . PUNCT cana-3856 371 1 414 414 NUM cana-3856 371 2 ( ( PUNCT cana-3856 371 3 2006 2006 NUM cana-3856 371 4 ) ) PUNCT cana-3856 371 5 29–37 29–37 NOUN cana-3856 371 6 . . PUNCT cana-3856 372 1 [ [ X cana-3856 372 2 7 7 NUM cana-3856 372 3 ] ] X cana-3856 372 4 i. i. NOUN cana-3856 372 5 gutman gutman PROPN cana-3856 372 6 , , PUNCT cana-3856 372 7 geometric geometric ADJ cana-3856 372 8 approach approach NOUN cana-3856 372 9 to to ADP cana-3856 372 10 degree degree NOUN cana-3856 372 11 - - PUNCT cana-3856 372 12 based base VERB cana-3856 372 13 topological topological ADJ cana-3856 372 14 indices index NOUN cana-3856 372 15 : : PUNCT cana-3856 372 16 sombor sombor NOUN cana-3856 372 17 indices index NOUN cana-3856 372 18 , , PUNCT cana-3856 372 19 match match NOUN cana-3856 372 20 commun commun PROPN cana-3856 372 21 . . PUNCT cana-3856 373 1 math math PROPN cana-3856 373 2 . . PUNCT cana-3856 374 1 comput comput NOUN cana-3856 374 2 . . PUNCT cana-3856 375 1 chem chem PROPN cana-3856 375 2 . . PUNCT cana-3856 375 3 , , PUNCT cana-3856 375 4 86 86 NUM cana-3856 375 5 ( ( PUNCT cana-3856 375 6 2021 2021 NUM cana-3856 375 7 ) ) PUNCT cana-3856 375 8 11 11 NUM cana-3856 375 9 - - SYM cana-3856 375 10 16 16 NUM cana-3856 375 11 . . PUNCT cana-3856 376 1 [ [ X cana-3856 376 2 8 8 NUM cana-3856 376 3 ] ] X cana-3856 376 4 i. i. PROPN cana-3856 376 5 gutman gutman PROPN cana-3856 376 6 , , PUNCT cana-3856 376 7 spectrum spectrum NOUN cana-3856 376 8 and and CCONJ cana-3856 376 9 energy energy NOUN cana-3856 376 10 of of ADP cana-3856 376 11 sombor sombor NOUN cana-3856 376 12 matrix matrix NOUN cana-3856 376 13 , , PUNCT cana-3856 376 14 military military ADJ cana-3856 376 15 tech tech NOUN cana-3856 376 16 . . PUNCT cana-3856 377 1 courier courier NOUN cana-3856 377 2 69 69 NUM cana-3856 377 3 ( ( PUNCT cana-3856 377 4 2021 2021 NUM cana-3856 377 5 ) ) PUNCT cana-3856 377 6 551561 551561 NUM cana-3856 377 7 . . PUNCT cana-3856 378 1 [ [ X cana-3856 378 2 9 9 NUM cana-3856 378 3 ] ] SYM cana-3856 378 4 i. i. NOUN cana-3856 378 5 gutman gutman PROPN cana-3856 378 6 , , PUNCT cana-3856 378 7 the the DET cana-3856 378 8 energy energy NOUN cana-3856 378 9 of of ADP cana-3856 378 10 a a DET cana-3856 378 11 graph graph NOUN cana-3856 378 12 with with ADP cana-3856 378 13 self self NOUN cana-3856 378 14 loop loop NOUN cana-3856 378 15 , , PUNCT cana-3856 378 16 match match NOUN cana-3856 378 17 commun commun PROPN cana-3856 378 18 . . PUNCT cana-3856 379 1 math math PROPN cana-3856 379 2 . . PUNCT cana-3856 380 1 comput comput NOUN cana-3856 380 2 . . PUNCT cana-3856 381 1 chem chem NOUN cana-3856 381 2 . . PUNCT cana-3856 382 1 87 87 NUM cana-3856 382 2 ( ( PUNCT cana-3856 382 3 2022 2022 NUM cana-3856 382 4 ) ) PUNCT cana-3856 382 5 645 645 NUM cana-3856 382 6 - - SYM cana-3856 382 7 652 652 NUM cana-3856 382 8 . . PUNCT cana-3856 383 1 [ [ X cana-3856 383 2 10 10 NUM cana-3856 383 3 ] ] PUNCT cana-3856 383 4 m. m. NOUN cana-3856 383 5 cavers caver NOUN cana-3856 383 6 , , PUNCT cana-3856 383 7 s. s. PROPN cana-3856 383 8 fallat fallat PROPN cana-3856 383 9 , , PUNCT cana-3856 383 10 s. s. PROPN cana-3856 383 11 kirkland kirkland PROPN cana-3856 383 12 , , PUNCT cana-3856 383 13 on on ADP cana-3856 383 14 the the DET cana-3856 383 15 normalized normalize VERB cana-3856 383 16 laplacian laplacian ADJ cana-3856 383 17 energy energy NOUN cana-3856 383 18 and and CCONJ cana-3856 383 19 general general ADJ cana-3856 383 20 randiΔ‡ randiΔ‡ PROPN cana-3856 383 21 index index NOUN cana-3856 383 22 π‘…βˆ’1 π‘…βˆ’1 NOUN cana-3856 383 23 of of ADP cana-3856 383 24 graphs graph NOUN cana-3856 383 25 , , PUNCT cana-3856 383 26 lin lin PROPN cana-3856 383 27 . . PUNCT cana-3856 384 1 algebra algebra PROPN cana-3856 384 2 appl appl PROPN cana-3856 384 3 . . PUNCT cana-3856 385 1 433 433 NUM cana-3856 385 2 ( ( PUNCT cana-3856 385 3 2010 2010 NUM cana-3856 385 4 ) ) PUNCT cana-3856 385 5 172–190 172–190 NUM cana-3856 385 6 . . PUNCT cana-3856 386 1 [ [ X cana-3856 386 2 11 11 NUM cana-3856 386 3 ] ] PUNCT cana-3856 386 4 r. r. PROPN cana-3856 386 5 b. b. PROPN cana-3856 386 6 mallion mallion PROPN cana-3856 386 7 , , PUNCT cana-3856 386 8 a. a. PROPN cana-3856 386 9 j. j. PROPN cana-3856 386 10 schwenk schwenk PROPN cana-3856 386 11 , , PUNCT cana-3856 386 12 n. n. PROPN cana-3856 386 13 trinajstiΒ΄c trinajstiΒ΄c PROPN cana-3856 386 14 , , PUNCT cana-3856 386 15 a a DET cana-3856 386 16 graphical graphical ADJ cana-3856 386 17 study study NOUN cana-3856 386 18 of of ADP cana-3856 386 19 heteroconjugated heteroconjugate VERB cana-3856 386 20 molecules molecule NOUN cana-3856 386 21 , , PUNCT cana-3856 386 22 croat croat NOUN cana-3856 386 23 . . PUNCT cana-3856 387 1 chem chem PROPN cana-3856 387 2 . . PUNCT cana-3856 388 1 acta acta PROPN cana-3856 388 2 . . PUNCT cana-3856 389 1 46 46 NUM cana-3856 389 2 ( ( PUNCT cana-3856 389 3 1974 1974 NUM cana-3856 389 4 ) ) PUNCT cana-3856 389 5 171–182 171–182 NUM cana-3856 389 6 . . PUNCT cana-3856 390 1 [ [ X cana-3856 390 2 12 12 NUM cana-3856 390 3 ] ] PUNCT cana-3856 390 4 r. r. PROPN cana-3856 390 5 b. b. PROPN cana-3856 390 6 mallion mallion PROPN cana-3856 390 7 , , PUNCT cana-3856 390 8 n. n. PROPN cana-3856 390 9 trinajstiΒ΄c trinajstiΒ΄c PROPN cana-3856 390 10 , , PUNCT cana-3856 390 11 a. a. PROPN cana-3856 390 12 j. j. PROPN cana-3856 390 13 schwenk schwenk PROPN cana-3856 390 14 , , PUNCT cana-3856 390 15 graph graph NOUN cana-3856 390 16 theory theory NOUN cana-3856 390 17 in in ADP cana-3856 390 18 chemistry chemistry NOUN cana-3856 390 19 – – PUNCT cana-3856 390 20 generalization generalization NOUN cana-3856 390 21 of of ADP cana-3856 390 22 sachs sachs PROPN cana-3856 390 23 ’ ' PUNCT cana-3856 390 24 formula formula NOUN cana-3856 390 25 , , PUNCT cana-3856 390 26 z. z. PROPN cana-3856 390 27 naturforsch naturforsch PROPN cana-3856 390 28 . . PUNCT cana-3856 391 1 29a 29a NUM cana-3856 391 2 ( ( PUNCT cana-3856 391 3 1974 1974 NUM cana-3856 391 4 ) ) PUNCT cana-3856 391 5 1481–1484 1481–1484 NOUN cana-3856 391 6 . . PUNCT cana-3856 392 1 [ [ X cana-3856 392 2 13 13 NUM cana-3856 392 3 ] ] X cana-3856 392 4 v. v. PROPN cana-3856 392 5 consonni consonni PROPN cana-3856 392 6 , , PUNCT cana-3856 392 7 r. r. PROPN cana-3856 392 8 todeschini todeschini PROPN cana-3856 392 9 , , PUNCT cana-3856 392 10 new new ADJ cana-3856 392 11 spectral spectral ADJ cana-3856 392 12 index index NOUN cana-3856 392 13 for for ADP cana-3856 392 14 molecule molecule NOUN cana-3856 392 15 description description NOUN cana-3856 392 16 , , PUNCT cana-3856 392 17 match match NOUN cana-3856 392 18 commun commun PROPN cana-3856 392 19 . . PUNCT cana-3856 392 20 math math PROPN cana-3856 392 21 . . PUNCT cana-3856 393 1 comput comput NOUN cana-3856 393 2 . . PUNCT cana-3856 394 1 chem chem NOUN cana-3856 394 2 . . PUNCT cana-3856 395 1 60 60 NUM cana-3856 395 2 ( ( PUNCT cana-3856 395 3 2008 2008 NUM cana-3856 395 4 ) ) PUNCT cana-3856 395 5 3–14 3–14 PROPN cana-3856 395 6 . . PUNCT