id sid tid token lemma pos cana-4840 1 1 communications communication NOUN cana-4840 1 2 on on ADP cana-4840 1 3 applied apply VERB cana-4840 1 4 nonlinear nonlinear ADJ cana-4840 1 5 analysis analysis NOUN cana-4840 1 6 issn issn NOUN cana-4840 1 7 : : PUNCT cana-4840 1 8 1074 1074 NUM cana-4840 1 9 - - PUNCT cana-4840 1 10 133x 133x NUM cana-4840 1 11 vol vol VERB cana-4840 1 12 32 32 NUM cana-4840 1 13 no no NOUN cana-4840 1 14 . . PUNCT cana-4840 2 1 10s 10 NOUN cana-4840 2 2 ( ( PUNCT cana-4840 2 3 2025 2025 NUM cana-4840 2 4 ) ) PUNCT cana-4840 2 5 444 444 NUM cana-4840 2 6 https://internationalpubls.com https://internationalpubls.com NUM cana-4840 2 7 laplacian laplacian ADJ cana-4840 2 8 minimum minimum NOUN cana-4840 2 9 dominating dominating NOUN cana-4840 2 10 quotient quotient NOUN cana-4840 2 11 energy energy NOUN cana-4840 2 12 of of ADP cana-4840 2 13 a a DET cana-4840 2 14 graph graph NOUN cana-4840 2 15 ramesha ramesha PROPN cana-4840 2 16 m m PROPN cana-4840 2 17 s1 s1 NOUN cana-4840 2 18 , , PUNCT cana-4840 2 19 purushothama purushothama NOUN cana-4840 2 20 s2 s2 NOUN cana-4840 2 21 and and CCONJ cana-4840 2 22 puttaswamy3 puttaswamy3 NOUN cana-4840 2 23 1department 1department NUM cana-4840 2 24 of of ADP cana-4840 2 25 mathematics mathematic NOUN cana-4840 2 26 , , PUNCT cana-4840 2 27 government government NOUN cana-4840 2 28 first first ADJ cana-4840 2 29 grade grade NOUN cana-4840 2 30 college college NOUN cana-4840 2 31 , , PUNCT cana-4840 2 32 kuvempunagara kuvempunagara PROPN cana-4840 2 33 mysore mysore PROPN cana-4840 2 34 , , PUNCT cana-4840 2 35 karanataka karanataka PROPN cana-4840 2 36 , , PUNCT cana-4840 2 37 india india PROPN cana-4840 2 38 e e PROPN cana-4840 2 39 - - NOUN cana-4840 2 40 mail mail NOUN cana-4840 2 41 : : PUNCT cana-4840 2 42 profmsr1978@gmail.com profmsr1978@gmail.com X cana-4840 3 1 2department 2department NUM cana-4840 3 2 of of ADP cana-4840 3 3 mathematics mathematic NOUN cana-4840 3 4 , , PUNCT cana-4840 3 5 maharaja maharaja PROPN cana-4840 3 6 institute institute PROPN cana-4840 3 7 of of ADP cana-4840 3 8 technology technology PROPN cana-4840 3 9 mysore mysore PROPN cana-4840 3 10 , , PUNCT cana-4840 3 11 mandya mandya NOUN cana-4840 3 12 , , PUNCT cana-4840 3 13 karanataka karanataka PROPN cana-4840 3 14 , , PUNCT cana-4840 3 15 india india PROPN cana-4840 3 16 corresponding corresponding PROPN cana-4840 3 17 author author NOUN cana-4840 3 18 : : PUNCT cana-4840 3 19 e e NOUN cana-4840 3 20 - - NOUN cana-4840 3 21 mail mail NOUN cana-4840 3 22 : : PUNCT cana-4840 4 1 psmandya@gmail.com psmandya@gmail.com PROPN cana-4840 4 2 3department 3department NUM cana-4840 4 3 of of ADP cana-4840 4 4 mathematics mathematic NOUN cana-4840 4 5 , , PUNCT cana-4840 4 6 p.e.s p.e.s ADJ cana-4840 4 7 . . PUNCT cana-4840 5 1 college college NOUN cana-4840 5 2 of of ADP cana-4840 5 3 engineering engineering PROPN cana-4840 5 4 , , PUNCT cana-4840 5 5 mandya mandya NOUN cana-4840 5 6 , , PUNCT cana-4840 5 7 karanataka karanataka PROPN cana-4840 5 8 , , PUNCT cana-4840 5 9 india india PROPN cana-4840 5 10 . . PUNCT cana-4840 6 1 e e X cana-4840 6 2 - - NOUN cana-4840 6 3 mail mail NOUN cana-4840 6 4 : : PUNCT cana-4840 6 5 prof.puttaswamy@gmail.com prof.puttaswamy@gmail.com X cana-4840 6 6 article article NOUN cana-4840 6 7 history history NOUN cana-4840 6 8 : : PUNCT cana-4840 6 9 received receive VERB cana-4840 6 10 : : PUNCT cana-4840 6 11 12 12 NUM cana-4840 6 12 - - SYM cana-4840 6 13 01 01 NUM cana-4840 6 14 - - PUNCT cana-4840 6 15 2025 2025 NUM cana-4840 6 16 revised revise VERB cana-4840 6 17 : : PUNCT cana-4840 6 18 15 15 NUM cana-4840 6 19 - - NUM cana-4840 6 20 02 02 NUM cana-4840 6 21 - - PUNCT cana-4840 6 22 2025 2025 NUM cana-4840 6 23 accepted accept VERB cana-4840 6 24 : : PUNCT cana-4840 6 25 01 01 NUM cana-4840 6 26 - - SYM cana-4840 6 27 03 03 NUM cana-4840 6 28 - - PUNCT cana-4840 6 29 2025 2025 NUM cana-4840 6 30 abstract abstract NOUN cana-4840 6 31 : : PUNCT cana-4840 6 32 in in ADP cana-4840 6 33 this this DET cana-4840 6 34 paper paper NOUN cana-4840 6 35 , , PUNCT cana-4840 6 36 we we PRON cana-4840 6 37 present present VERB cana-4840 6 38 the the DET cana-4840 6 39 idea idea NOUN cana-4840 6 40 of of ADP cana-4840 6 41 laplacian laplacian ADJ cana-4840 6 42 minimum minimum ADJ cana-4840 6 43 dominating dominating NOUN cana-4840 6 44 quotient quotient NOUN cana-4840 6 45 energy energy NOUN cana-4840 6 46 of of ADP cana-4840 6 47 graph graph NOUN cana-4840 6 48 , , PUNCT cana-4840 6 49 𝐿𝑄𝐷𝐸(𝐺 𝐿𝑄𝐷𝐸(𝐺 NOUN cana-4840 6 50 ) ) PUNCT cana-4840 6 51 and and CCONJ cana-4840 6 52 compute compute VERB cana-4840 6 53 the the DET cana-4840 6 54 laplacian laplacian ADJ cana-4840 6 55 minimum minimum NOUN cana-4840 6 56 dominating dominating NOUN cana-4840 6 57 quotient quotient NOUN cana-4840 6 58 energy energy NOUN cana-4840 6 59 of of ADP cana-4840 6 60 𝐿𝑄𝐷𝐸(𝐺 𝐿𝑄𝐷𝐸(𝐺 NOUN cana-4840 6 61 ) ) PUNCT cana-4840 6 62 of of ADP cana-4840 6 63 few few ADJ cana-4840 6 64 families family NOUN cana-4840 6 65 of of ADP cana-4840 6 66 graphs graph NOUN cana-4840 6 67 . . PUNCT cana-4840 7 1 additionally additionally ADV cana-4840 7 2 , , PUNCT cana-4840 7 3 we we PRON cana-4840 7 4 derive derive VERB cana-4840 7 5 bounds bound NOUN cana-4840 7 6 for for ADP cana-4840 7 7 the the DET cana-4840 7 8 laplacian laplacian ADJ cana-4840 7 9 minimum minimum NOUN cana-4840 7 10 dominating dominating NOUN cana-4840 7 11 quotient quotient NOUN cana-4840 7 12 energy energy NOUN cana-4840 7 13 , , PUNCT cana-4840 7 14 providing provide VERB cana-4840 7 15 a a DET cana-4840 7 16 comprehensive comprehensive ADJ cana-4840 7 17 understanding understanding NOUN cana-4840 7 18 of of ADP cana-4840 7 19 its its PRON cana-4840 7 20 behavior behavior NOUN cana-4840 7 21 and and CCONJ cana-4840 7 22 properties property NOUN cana-4840 7 23 in in ADP cana-4840 7 24 different different ADJ cana-4840 7 25 graph graph NOUN cana-4840 7 26 structures structure NOUN cana-4840 7 27 . . PUNCT cana-4840 8 1 objectives objective NOUN cana-4840 8 2 : : PUNCT cana-4840 8 3 finding find VERB cana-4840 8 4 the the DET cana-4840 8 5 laplacian laplacian ADJ cana-4840 8 6 minimum minimum NOUN cana-4840 8 7 dominating dominating NOUN cana-4840 8 8 quotient quotient NOUN cana-4840 8 9 energy energy NOUN cana-4840 8 10 of of ADP cana-4840 8 11 different different ADJ cana-4840 8 12 graph graph NOUN cana-4840 8 13 methods method NOUN cana-4840 8 14 : : PUNCT cana-4840 8 15 to to PART cana-4840 8 16 establish establish VERB cana-4840 8 17 the the DET cana-4840 8 18 upper upper ADJ cana-4840 8 19 and and CCONJ cana-4840 8 20 lower low ADJ cana-4840 8 21 bounds bound NOUN cana-4840 8 22 for for ADP cana-4840 8 23 the the DET cana-4840 8 24 energy energy NOUN cana-4840 8 25 of of ADP cana-4840 8 26 graphs graph NOUN cana-4840 8 27 we we PRON cana-4840 8 28 employ employ VERB cana-4840 8 29 the the DET cana-4840 8 30 standard standard ADJ cana-4840 8 31 methods method NOUN cana-4840 8 32 of of ADP cana-4840 8 33 proofs proof NOUN cana-4840 8 34 namely namely ADV cana-4840 8 35 direct direct ADJ cana-4840 8 36 methods method NOUN cana-4840 8 37 and and CCONJ cana-4840 8 38 using use VERB cana-4840 8 39 matlab matlab PROPN cana-4840 8 40 to to PART cana-4840 8 41 compute compute VERB cana-4840 8 42 the the DET cana-4840 8 43 minimum minimum ADJ cana-4840 8 44 pendant pendant ADJ cana-4840 8 45 dominating dominating NOUN cana-4840 8 46 partition partition NOUN cana-4840 8 47 eigen eigen PROPN cana-4840 8 48 values value NOUN cana-4840 8 49 of of ADP cana-4840 8 50 a a DET cana-4840 8 51 graph graph NOUN cana-4840 8 52 𝐺. 𝐺. NOUN cana-4840 8 53 results result NOUN cana-4840 8 54 : : PUNCT cana-4840 8 55 we we PRON cana-4840 8 56 obtain obtain VERB cana-4840 8 57 the the DET cana-4840 8 58 laplacian laplacian ADJ cana-4840 8 59 minimum minimum NOUN cana-4840 8 60 dominating dominating NOUN cana-4840 8 61 quotient quotient NOUN cana-4840 8 62 energy energy NOUN cana-4840 8 63 of of ADP cana-4840 8 64 𝐿𝑄𝐷𝐸(𝐺 𝐿𝑄𝐷𝐸(𝐺 NOUN cana-4840 8 65 ) ) PUNCT cana-4840 8 66 of of ADP cana-4840 8 67 well well ADV cana-4840 8 68 - - PUNCT cana-4840 8 69 known know VERB cana-4840 8 70 families family NOUN cana-4840 8 71 of of ADP cana-4840 8 72 graphs graph NOUN cana-4840 8 73 . . PUNCT cana-4840 9 1 additionally additionally ADV cana-4840 9 2 we we PRON cana-4840 9 3 obtain obtain VERB cana-4840 9 4 upper upper ADJ cana-4840 9 5 and and CCONJ cana-4840 9 6 lower low ADJ cana-4840 9 7 bounds bound NOUN cana-4840 9 8 conclusions conclusion NOUN cana-4840 9 9 : : PUNCT cana-4840 9 10 nowadays nowadays ADV cana-4840 9 11 , , PUNCT cana-4840 9 12 the the DET cana-4840 9 13 study study NOUN cana-4840 9 14 of of ADP cana-4840 9 15 theory theory NOUN cana-4840 9 16 of of ADP cana-4840 9 17 domination domination NOUN cana-4840 9 18 and and CCONJ cana-4840 9 19 energy energy NOUN cana-4840 9 20 of of ADP cana-4840 9 21 graph graph NOUN cana-4840 9 22 is be AUX cana-4840 9 23 an an DET cana-4840 9 24 important important ADJ cana-4840 9 25 area area NOUN cana-4840 9 26 in in ADP cana-4840 9 27 graph graph NOUN cana-4840 9 28 theory theory NOUN cana-4840 9 29 and and CCONJ cana-4840 9 30 also also ADV cana-4840 9 31 remarkable remarkable ADJ cana-4840 9 32 research research NOUN cana-4840 9 33 is be AUX cana-4840 9 34 going go VERB cana-4840 9 35 on on ADP cana-4840 9 36 in in ADP cana-4840 9 37 this this DET cana-4840 9 38 area area NOUN cana-4840 9 39 . . PUNCT cana-4840 10 1 in in ADP cana-4840 10 2 recent recent ADJ cana-4840 10 3 years year NOUN cana-4840 10 4 many many ADJ cana-4840 10 5 scholars scholar NOUN cana-4840 10 6 are be AUX cana-4840 10 7 working work VERB cana-4840 10 8 in in ADP cana-4840 10 9 this this DET cana-4840 10 10 area area NOUN cana-4840 10 11 and and CCONJ cana-4840 10 12 also also ADV cana-4840 10 13 they they PRON cana-4840 10 14 are be AUX cana-4840 10 15 introducing introduce VERB cana-4840 10 16 new new ADJ cana-4840 10 17 domination domination NOUN cana-4840 10 18 parameters parameter NOUN cana-4840 10 19 . . PUNCT cana-4840 11 1 in in ADP cana-4840 11 2 this this DET cana-4840 11 3 paper paper NOUN cana-4840 11 4 we we PRON cana-4840 11 5 have have AUX cana-4840 11 6 initiated initiate VERB cana-4840 11 7 the the DET cana-4840 11 8 study study NOUN cana-4840 11 9 of of ADP cana-4840 11 10 laplacian laplacian ADJ cana-4840 11 11 minimum minimum NOUN cana-4840 11 12 dominating dominating NOUN cana-4840 11 13 quotient quotient NOUN cana-4840 11 14 energy energy NOUN cana-4840 11 15 of of ADP cana-4840 11 16 graph graph NOUN cana-4840 11 17 . . PUNCT cana-4840 12 1 we we PRON cana-4840 12 2 have have AUX cana-4840 12 3 calculated calculate VERB cana-4840 12 4 the the DET cana-4840 12 5 energies energy NOUN cana-4840 12 6 for for ADP cana-4840 12 7 some some DET cana-4840 12 8 standard standard ADJ cana-4840 12 9 family family NOUN cana-4840 12 10 graphs graph NOUN cana-4840 12 11 and and CCONJ cana-4840 12 12 we we PRON cana-4840 12 13 have have AUX cana-4840 12 14 established establish VERB cana-4840 12 15 some some DET cana-4840 12 16 bounds bound NOUN cana-4840 12 17 for for ADP cana-4840 12 18 this this DET cana-4840 12 19 parameter parameter NOUN cana-4840 12 20 . . PUNCT cana-4840 13 1 further far ADV cana-4840 13 2 , , PUNCT cana-4840 13 3 we we PRON cana-4840 13 4 have have AUX cana-4840 13 5 studied study VERB cana-4840 13 6 some some DET cana-4840 13 7 important important ADJ cana-4840 13 8 properties property NOUN cana-4840 13 9 of of ADP cana-4840 13 10 laplacian laplacian ADJ cana-4840 13 11 minimum minimum NOUN cana-4840 13 12 quotient quotient NOUN cana-4840 13 13 dominating dominating NOUN cana-4840 13 14 eigenvalues eigenvalue VERB cana-4840 13 15 keywords keyword NOUN cana-4840 13 16 : : PUNCT cana-4840 13 17 laplacian laplacian ADJ cana-4840 13 18 minimum minimum NOUN cana-4840 13 19 dominating dominating NOUN cana-4840 13 20 set set NOUN cana-4840 13 21 , , PUNCT cana-4840 13 22 minimum minimum ADJ cana-4840 13 23 dominating dominating NOUN cana-4840 13 24 set set NOUN cana-4840 13 25 , , PUNCT cana-4840 13 26 quotient quotient NOUN cana-4840 13 27 energy energy NOUN cana-4840 13 28 , , PUNCT cana-4840 13 29 laplacian laplacian ADJ cana-4840 13 30 dominating dominating NOUN cana-4840 13 31 quotient quotient NOUN cana-4840 13 32 matrix matrix NOUN cana-4840 13 33 . . PUNCT cana-4840 14 1 1 1 X cana-4840 14 2 . . X cana-4840 14 3 introduction introduction NOUN cana-4840 14 4 let let VERB cana-4840 14 5 𝐺 𝐺 PROPN cana-4840 14 6 = = SYM cana-4840 14 7 ( ( PUNCT cana-4840 14 8 𝑉 𝑉 PROPN cana-4840 14 9 , , PUNCT cana-4840 14 10 𝐸 𝐸 PROPN cana-4840 14 11 ) ) PUNCT cana-4840 14 12 be be VERB cana-4840 14 13 a a DET cana-4840 14 14 graph graph NOUN cana-4840 14 15 with with ADP cana-4840 14 16 𝑛 𝑛 DET cana-4840 14 17 nodes node NOUN cana-4840 14 18 and and CCONJ cana-4840 14 19 π‘š π‘š ADP cana-4840 14 20 edges edge NOUN cana-4840 14 21 . . PUNCT cana-4840 15 1 the the DET cana-4840 15 2 degree degree NOUN cana-4840 15 3 of of ADP cana-4840 15 4 𝑣𝑖 𝑣𝑖 ADV cana-4840 15 5 written write VERB cana-4840 15 6 by by ADP cana-4840 15 7 𝑑(𝑣𝑖 𝑑(𝑣𝑖 PROPN cana-4840 15 8 ) ) PUNCT cana-4840 15 9 is be AUX cana-4840 15 10 the the DET cana-4840 15 11 number number NOUN cana-4840 15 12 of of ADP cana-4840 15 13 edges edge NOUN cana-4840 15 14 incident incident NOUN cana-4840 15 15 with with ADP cana-4840 15 16 𝑣𝑖. 𝑣𝑖. NOUN cana-4840 15 17 the the DET cana-4840 15 18 maximum maximum ADJ cana-4840 15 19 node node NOUN cana-4840 15 20 of of ADP cana-4840 15 21 degree degree NOUN cana-4840 15 22 is be AUX cana-4840 15 23 denoted denote VERB cana-4840 15 24 by by ADP cana-4840 15 25 Ξ΄(𝐺 Ξ΄(𝐺 NOUN cana-4840 15 26 ) ) PUNCT cana-4840 15 27 and and CCONJ cana-4840 15 28 minimum minimum ADJ cana-4840 15 29 node node NOUN cana-4840 15 30 of of ADP cana-4840 15 31 degree degree NOUN cana-4840 15 32 is be AUX cana-4840 15 33 denoted denote VERB cana-4840 15 34 by by ADP cana-4840 15 35 𝛿(𝐺 𝛿(𝐺 PROPN cana-4840 15 36 ) ) PUNCT cana-4840 15 37 . . PUNCT cana-4840 16 1 the the DET cana-4840 16 2 adjacency adjacency PROPN cana-4840 16 3 matrix matrix NOUN cana-4840 16 4 𝐴𝐷(𝐺 𝐴𝐷(𝐺 NOUN cana-4840 16 5 ) ) PUNCT cana-4840 16 6 of of ADP cana-4840 16 7 𝐺 𝐺 PROPN cana-4840 16 8 is be AUX cana-4840 16 9 defined define VERB cana-4840 16 10 by by ADP cana-4840 16 11 its its PRON cana-4840 16 12 entries entry NOUN cana-4840 16 13 as as ADP cana-4840 16 14 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-4840 16 15 = = SYM cana-4840 16 16 1 1 NUM cana-4840 16 17 if if SCONJ cana-4840 16 18 𝑣𝑖𝑣𝑗 𝑣𝑖𝑣𝑗 NOUN cana-4840 16 19 ∈ ∈ PROPN cana-4840 16 20 𝐸(𝐺 𝐸(𝐺 PROPN cana-4840 16 21 ) ) PUNCT cana-4840 16 22 π‘œπ‘Ÿ π‘œπ‘Ÿ VERB cana-4840 16 23 𝑣𝑖 𝑣𝑖 ADP cana-4840 16 24 ∈ ∈ PROPN cana-4840 16 25 𝐷 𝐷 PROPN cana-4840 16 26 𝑖𝑓 𝑖𝑓 X cana-4840 16 27 ( ( PUNCT cana-4840 16 28 𝑖 𝑖 NOUN cana-4840 16 29 = = PUNCT cana-4840 16 30 𝑗 𝑗 PROPN cana-4840 16 31 ) ) PUNCT cana-4840 16 32 where where SCONJ cana-4840 16 33 𝐷 𝐷 NOUN cana-4840 16 34 is be AUX cana-4840 16 35 a a DET cana-4840 16 36 dominating dominating NOUN cana-4840 16 37 set set NOUN cana-4840 16 38 of of ADP cana-4840 16 39 𝐺 𝐺 PROPN cana-4840 16 40 and and CCONJ cana-4840 16 41 0 0 NUM cana-4840 16 42 otherwise otherwise ADV cana-4840 16 43 . . PUNCT cana-4840 17 1 the the DET cana-4840 17 2 eigen eigen PROPN cana-4840 17 3 values value NOUN cana-4840 17 4 of of ADP cana-4840 17 5 graph graph NOUN cana-4840 17 6 𝐺 𝐺 PROPN cana-4840 17 7 are be AUX cana-4840 17 8 the the DET cana-4840 17 9 eigenvalues eigenvalue NOUN cana-4840 17 10 of of ADP cana-4840 17 11 its its PRON cana-4840 17 12 adjacency adjacency NOUN cana-4840 17 13 matrix𝐴𝐷(𝐺 matrix𝐴𝐷(𝐺 NOUN cana-4840 17 14 ) ) PUNCT cana-4840 17 15 , , PUNCT cana-4840 17 16 denoted denote VERB cana-4840 17 17 byπœ†1 byπœ†1 PROPN cana-4840 17 18 β‰₯ β‰₯ NUM cana-4840 17 19 πœ†2 πœ†2 PROPN cana-4840 17 20 β‰₯ β‰₯ NOUN cana-4840 17 21 β‹― β‹― PROPN cana-4840 17 22 β‰₯ β‰₯ PROPN cana-4840 17 23 πœ†π‘›. πœ†π‘›. NOUN cana-4840 17 24 a a DET cana-4840 17 25 graph graph NOUN cana-4840 17 26 𝐺 𝐺 NOUN cana-4840 17 27 is be AUX cana-4840 17 28 considered consider VERB cana-4840 17 29 singular singular ADJ cana-4840 17 30 if if SCONJ cana-4840 17 31 it it PRON cana-4840 17 32 has have VERB cana-4840 17 33 at at ADV cana-4840 17 34 least least ADV cana-4840 17 35 one one NUM cana-4840 17 36 eigenvalue eigenvalue NOUN cana-4840 17 37 equal equal ADJ cana-4840 17 38 to to ADP cana-4840 17 39 zero zero NUM cana-4840 17 40 . . PUNCT cana-4840 18 1 in in ADP cana-4840 18 2 the the DET cana-4840 18 3 case case NOUN cana-4840 18 4 of of ADP cana-4840 18 5 singular singular ADJ cana-4840 18 6 graphs graph NOUN cana-4840 18 7 , , PUNCT cana-4840 18 8 it it PRON cana-4840 18 9 is be AUX cana-4840 18 10 clear clear ADJ cana-4840 18 11 that that SCONJ cana-4840 18 12 𝑑𝑒𝑑(𝐴 𝑑𝑒𝑑(𝐴 VERB cana-4840 18 13 ) ) PUNCT cana-4840 18 14 = = SYM cana-4840 19 1 0 0 X cana-4840 19 2 . . PUNCT cana-4840 20 1 a a DET cana-4840 20 2 graph graph NOUN cana-4840 20 3 is be AUX cana-4840 20 4 said say VERB cana-4840 20 5 to to PART cana-4840 20 6 be be AUX cana-4840 20 7 nonsingular nonsingular ADJ cana-4840 20 8 if if SCONJ cana-4840 20 9 all all PRON cana-4840 20 10 of of ADP cana-4840 20 11 its its PRON cana-4840 20 12 eigenvalues eigenvalue NOUN cana-4840 20 13 are be AUX cana-4840 20 14 nonzero nonzero NOUN cana-4840 20 15 . . PUNCT cana-4840 21 1 a a DET cana-4840 21 2 graph graph NOUN cana-4840 21 3 𝐺 𝐺 NOUN cana-4840 21 4 is be AUX cana-4840 21 5 referred refer VERB cana-4840 21 6 to to PART cana-4840 21 7 be be AUX cana-4840 21 8 k k NOUN cana-4840 21 9 - - ADJ cana-4840 21 10 regular regular ADJ cana-4840 21 11 if if SCONJ cana-4840 21 12 every every DET cana-4840 21 13 node node NOUN cana-4840 21 14 in in ADP cana-4840 21 15 𝐺 𝐺 PROPN cana-4840 21 16 has have VERB cana-4840 21 17 degree degree NOUN cana-4840 21 18 π‘˜. π‘˜. ADJ cana-4840 21 19 mailto:psmandya@gmail.com mailto:psmandya@gmail.com X cana-4840 21 20 mailto:prof.puttaswamy@gmail.com mailto:prof.puttaswamy@gmail.com X cana-4840 21 21 communications communication NOUN cana-4840 21 22 on on ADP cana-4840 21 23 applied apply VERB cana-4840 21 24 nonlinear nonlinear ADJ cana-4840 21 25 analysis analysis NOUN cana-4840 21 26 issn issn NOUN cana-4840 21 27 : : PUNCT cana-4840 21 28 1074 1074 NUM cana-4840 21 29 - - PUNCT cana-4840 21 30 133x 133x NUM cana-4840 21 31 vol vol VERB cana-4840 21 32 32 32 NUM cana-4840 21 33 no no NOUN cana-4840 21 34 . . PUNCT cana-4840 22 1 10s 10 NOUN cana-4840 22 2 ( ( PUNCT cana-4840 22 3 2025 2025 NUM cana-4840 22 4 ) ) PUNCT cana-4840 22 5 445 445 NUM cana-4840 22 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 23 1 the the DET cana-4840 23 2 energy energy NOUN cana-4840 23 3 of of ADP cana-4840 23 4 a a DET cana-4840 23 5 graph graph NOUN cana-4840 23 6 𝐺 𝐺 NOUN cana-4840 23 7 is be AUX cana-4840 23 8 defined define VERB cana-4840 23 9 as as ADP cana-4840 23 10 e(g e(g NOUN cana-4840 23 11 ) ) PUNCT cana-4840 24 1 = = PUNCT cana-4840 24 2 βˆ‘ βˆ‘ PUNCT cana-4840 24 3 |Ξ»i| |Ξ»i| PROPN cana-4840 24 4 . . PUNCT cana-4840 25 1 n n PRON cana-4840 25 2 i=1 i=1 PROPN cana-4840 26 1 this this DET cana-4840 26 2 concept concept NOUN cana-4840 26 3 was be AUX cana-4840 26 4 introduced introduce VERB cana-4840 26 5 by by ADP cana-4840 26 6 i. i. PROPN cana-4840 26 7 gutman gutman PROPN cana-4840 26 8 in in ADP cana-4840 26 9 1978 1978 NUM cana-4840 26 10 [ [ X cana-4840 26 11 3 3 NUM cana-4840 26 12 ] ] PUNCT cana-4840 26 13 . . PUNCT cana-4840 27 1 i.gutman i.gutman PROPN cana-4840 27 2 and and CCONJ cana-4840 27 3 b.zhou b.zhou PROPN cana-4840 27 4 [ [ X cana-4840 27 5 4 4 X cana-4840 27 6 ] ] PUNCT cana-4840 27 7 defined define VERB cana-4840 27 8 the the DET cana-4840 27 9 laplacian laplacian ADJ cana-4840 27 10 energy energy NOUN cana-4840 27 11 of of ADP cana-4840 27 12 a a DET cana-4840 27 13 graph graph NOUN cana-4840 27 14 𝐺 𝐺 NOUN cana-4840 27 15 in in ADP cana-4840 27 16 the the DET cana-4840 27 17 year year NOUN cana-4840 27 18 2006 2006 NUM cana-4840 27 19 . . PUNCT cana-4840 28 1 let let VERB cana-4840 28 2 𝐺 𝐺 PRON cana-4840 28 3 be be AUX cana-4840 28 4 a a DET cana-4840 28 5 graph graph NOUN cana-4840 28 6 with with ADP cana-4840 28 7 𝑛 𝑛 DET cana-4840 28 8 nodes node NOUN cana-4840 28 9 and and CCONJ cana-4840 28 10 π‘š π‘š ADP cana-4840 28 11 edges edge NOUN cana-4840 28 12 . . PUNCT cana-4840 29 1 the the DET cana-4840 29 2 laplacian laplacian ADJ cana-4840 29 3 matrix matrix NOUN cana-4840 29 4 of of ADP cana-4840 29 5 the the DET cana-4840 29 6 graph graph NOUN cana-4840 29 7 𝐺 𝐺 NOUN cana-4840 29 8 , , PUNCT cana-4840 29 9 denoted denote VERB cana-4840 29 10 by by ADP cana-4840 29 11 𝐿 𝐿 PROPN cana-4840 29 12 = = PUNCT cana-4840 29 13 𝐿𝑖,𝑗 𝐿𝑖,𝑗 PROPN cana-4840 29 14 , , PUNCT cana-4840 29 15 is be AUX cana-4840 29 16 a a DET cana-4840 29 17 square square ADJ cana-4840 29 18 matrix matrix NOUN cana-4840 29 19 of of ADP cana-4840 29 20 order order NOUN cana-4840 29 21 𝑛. 𝑛. NOUN cana-4840 29 22 the the DET cana-4840 29 23 elements element NOUN cana-4840 29 24 of of ADP cana-4840 29 25 the the DET cana-4840 29 26 laplacian laplacian ADJ cana-4840 29 27 matrix matrix NOUN cana-4840 29 28 are be AUX cana-4840 29 29 defined define VERB cana-4840 29 30 as as ADP cana-4840 29 31 𝐿𝑖𝑗 𝐿𝑖𝑗 PROPN cana-4840 29 32 = = PROPN cana-4840 29 33 { { PUNCT cana-4840 29 34 βˆ’1 βˆ’1 NOUN cana-4840 29 35 , , PUNCT cana-4840 29 36 if if SCONJ cana-4840 29 37 𝑣𝑖 𝑣𝑖 NOUN cana-4840 29 38 and and CCONJ cana-4840 29 39 𝑣𝑗 𝑣𝑗 ADP cana-4840 29 40 are be AUX cana-4840 29 41 adjacent adjacent ADJ cana-4840 29 42 , , PUNCT cana-4840 29 43 0 0 NUM cana-4840 29 44 , , PUNCT cana-4840 29 45 if if SCONJ cana-4840 29 46 𝑣𝑖 𝑣𝑖 NOUN cana-4840 29 47 and and CCONJ cana-4840 29 48 𝑣𝑗 𝑣𝑗 ADP cana-4840 29 49 are be AUX cana-4840 29 50 non non X cana-4840 29 51 adjacent adjacent ADJ cana-4840 29 52 , , PUNCT cana-4840 29 53 𝑑𝑖 𝑑𝑖 VERB cana-4840 29 54 𝑖𝑓 𝑖𝑓 ADP cana-4840 29 55 𝑖 𝑖 X cana-4840 30 1 = = PUNCT cana-4840 30 2 𝑗. 𝑗. NOUN cana-4840 30 3 where where SCONJ cana-4840 30 4 𝑑𝑖 𝑑𝑖 PROPN cana-4840 30 5 is be AUX cana-4840 30 6 the the DET cana-4840 30 7 vertex vertex NOUN cana-4840 30 8 's 's PART cana-4840 30 9 𝑣𝑖 𝑣𝑖 NOUN cana-4840 30 10 degree degree NOUN cana-4840 30 11 let let VERB cana-4840 30 12 πœ†1 πœ†1 NOUN cana-4840 30 13 , , PUNCT cana-4840 30 14 πœ†2 πœ†2 PROPN cana-4840 30 15 , , PUNCT cana-4840 30 16 … … PUNCT cana-4840 30 17 , , PUNCT cana-4840 30 18 πœ†π‘› πœ†π‘› AUX cana-4840 30 19 be be AUX cana-4840 30 20 the the DET cana-4840 30 21 eigen eigen PROPN cana-4840 30 22 values value NOUN cana-4840 30 23 of of ADP cana-4840 30 24 laplacian laplacian ADJ cana-4840 30 25 matrix matrix NOUN cana-4840 30 26 𝐺. 𝐺. PROPN cana-4840 30 27 laplacian laplacian ADJ cana-4840 30 28 energy energy NOUN cana-4840 30 29 of of ADP cana-4840 30 30 𝐺 𝐺 PROPN cana-4840 30 31 is be AUX cana-4840 30 32 defined define VERB cana-4840 30 33 as as ADP cana-4840 30 34 𝐿𝐸(𝐺 𝐿𝐸(𝐺 NOUN cana-4840 30 35 ) ) PUNCT cana-4840 31 1 = = NOUN cana-4840 31 2 βˆ‘| βˆ‘| VERB cana-4840 31 3 πœ†π‘– πœ†π‘– NOUN cana-4840 31 4 βˆ’ βˆ’ NOUN cana-4840 31 5 2π‘š 2π‘š NOUN cana-4840 31 6 𝑛 𝑛 DET cana-4840 31 7 | | NOUN cana-4840 31 8 𝑛 𝑛 ADP cana-4840 31 9 𝑖=1 𝑖=1 PROPN cana-4840 31 10 the the DET cana-4840 31 11 key key ADJ cana-4840 31 12 characteristics characteristic NOUN cana-4840 31 13 of of ADP cana-4840 31 14 laplacian laplacian ADJ cana-4840 31 15 energy energy NOUN cana-4840 31 16 , , PUNCT cana-4840 31 17 including include VERB cana-4840 31 18 various various ADJ cana-4840 31 19 upper upper ADJ cana-4840 31 20 and and CCONJ cana-4840 31 21 lower low ADJ cana-4840 31 22 bounds bound NOUN cana-4840 31 23 , , PUNCT cana-4840 31 24 have have AUX cana-4840 31 25 been be AUX cana-4840 31 26 established establish VERB cana-4840 31 27 in in ADP cana-4840 31 28 [ [ X cana-4840 31 29 4 4 NUM cana-4840 31 30 , , PUNCT cana-4840 31 31 5 5 NUM cana-4840 31 32 ] ] PUNCT cana-4840 31 33 it it PRON cana-4840 31 34 has have AUX cana-4840 31 35 been be AUX cana-4840 31 36 found find VERB cana-4840 31 37 that that SCONJ cana-4840 31 38 laplacian laplacian ADJ cana-4840 31 39 graph graph NOUN cana-4840 31 40 energy energy NOUN cana-4840 31 41 has have VERB cana-4840 31 42 notable notable ADJ cana-4840 31 43 applications application NOUN cana-4840 31 44 in in ADP cana-4840 31 45 areas area NOUN cana-4840 31 46 such such ADJ cana-4840 31 47 as as ADP cana-4840 31 48 chemical chemical ADJ cana-4840 31 49 analysis analysis NOUN cana-4840 31 50 , , PUNCT cana-4840 31 51 high high ADJ cana-4840 31 52 - - PUNCT cana-4840 31 53 resolution resolution NOUN cana-4840 31 54 satellite satellite NOUN cana-4840 31 55 image image NOUN cana-4840 31 56 classification classification NOUN cana-4840 31 57 and and CCONJ cana-4840 31 58 segmentation segmentation NOUN cana-4840 31 59 , , PUNCT cana-4840 31 60 as as ADV cana-4840 31 61 well well ADV cana-4840 31 62 as as ADP cana-4840 31 63 identifying identify VERB cana-4840 31 64 semantic semantic ADJ cana-4840 31 65 structures structure NOUN cana-4840 31 66 in in ADP cana-4840 31 67 image image NOUN cana-4840 31 68 hierarchies hierarchy NOUN cana-4840 31 69 . . PUNCT cana-4840 32 1 in in ADP cana-4840 32 2 this this DET cana-4840 32 3 article article NOUN cana-4840 32 4 , , PUNCT cana-4840 32 5 we we PRON cana-4840 32 6 are be AUX cana-4840 32 7 defining define VERB cana-4840 32 8 a a DET cana-4840 32 9 matrix matrix NOUN cana-4840 32 10 , , PUNCT cana-4840 32 11 called call VERB cana-4840 32 12 the the DET cana-4840 32 13 laplacian laplacian ADJ cana-4840 32 14 minimum minimum NOUN cana-4840 32 15 dominating dominating NOUN cana-4840 32 16 quotient quotient NOUN cana-4840 32 17 matrix matrix NOUN cana-4840 32 18 denoted denote VERB cana-4840 32 19 by by ADP cana-4840 32 20 𝐿𝑄𝐷(𝐺 𝐿𝑄𝐷(𝐺 PROPN cana-4840 32 21 ) ) PUNCT cana-4840 32 22 and and CCONJ cana-4840 32 23 we we PRON cana-4840 32 24 study study VERB cana-4840 32 25 its its PRON cana-4840 32 26 eigenvalues eigenvalue NOUN cana-4840 32 27 and and CCONJ cana-4840 32 28 the the DET cana-4840 32 29 energy energy NOUN cana-4840 32 30 . . PUNCT cana-4840 33 1 further far ADV cana-4840 33 2 , , PUNCT cana-4840 33 3 we we PRON cana-4840 33 4 study study VERB cana-4840 33 5 the the DET cana-4840 33 6 mathematical mathematical ADJ cana-4840 33 7 aspects aspect NOUN cana-4840 33 8 of of ADP cana-4840 33 9 the the DET cana-4840 33 10 laplacian laplacian ADJ cana-4840 33 11 minimum minimum NOUN cana-4840 33 12 dominating dominating NOUN cana-4840 33 13 quotient quotient NOUN cana-4840 33 14 energy energy NOUN cana-4840 33 15 of of ADP cana-4840 33 16 a a DET cana-4840 33 17 graph graph NOUN cana-4840 33 18 . . PUNCT cana-4840 34 1 it it PRON cana-4840 34 2 is be AUX cana-4840 34 3 possible possible ADJ cana-4840 34 4 that that SCONJ cana-4840 34 5 the the DET cana-4840 34 6 laplacian laplacian ADJ cana-4840 34 7 minimum minimum ADJ cana-4840 34 8 dominating dominating NOUN cana-4840 34 9 quotient quotient NOUN cana-4840 34 10 energy energy NOUN cana-4840 34 11 discussed discuss VERB cana-4840 34 12 in in ADP cana-4840 34 13 this this DET cana-4840 34 14 article article NOUN cana-4840 34 15 could could AUX cana-4840 34 16 have have VERB cana-4840 34 17 uses use NOUN cana-4840 34 18 in in ADP cana-4840 34 19 other other ADJ cana-4840 34 20 fields field NOUN cana-4840 34 21 of of ADP cana-4840 34 22 science science NOUN cana-4840 34 23 , , PUNCT cana-4840 34 24 such such ADJ cana-4840 34 25 as as ADP cana-4840 34 26 chemistry chemistry NOUN cana-4840 34 27 , , PUNCT cana-4840 34 28 and and CCONJ cana-4840 34 29 beyond beyond ADP cana-4840 34 30 . . PUNCT cana-4840 35 1 the the DET cana-4840 35 2 graphs graph NOUN cana-4840 35 3 under under ADP cana-4840 35 4 consideration consideration NOUN cana-4840 35 5 are be AUX cana-4840 35 6 assumed assume VERB cana-4840 35 7 to to PART cana-4840 35 8 be be AUX cana-4840 35 9 finite finite ADJ cana-4840 35 10 , , PUNCT cana-4840 35 11 simple simple ADJ cana-4840 35 12 , , PUNCT cana-4840 35 13 undirected undirected ADJ cana-4840 35 14 , , PUNCT cana-4840 35 15 with with ADP cana-4840 35 16 no no DET cana-4840 35 17 isolated isolated ADJ cana-4840 35 18 nodes node NOUN cana-4840 35 19 , , PUNCT cana-4840 35 20 and and CCONJ cana-4840 35 21 of of ADP cana-4840 35 22 order order NOUN cana-4840 35 23 at at ADV cana-4840 35 24 least least ADV cana-4840 35 25 two two NUM cana-4840 35 26 . . PUNCT cana-4840 36 1 2 2 X cana-4840 36 2 . . X cana-4840 36 3 quotient quotient NOUN cana-4840 36 4 energy energy NOUN cana-4840 36 5 of of ADP cana-4840 36 6 graphs graph NOUN cana-4840 36 7 for for ADP cana-4840 36 8 a a DET cana-4840 36 9 graph graph NOUN cana-4840 36 10 𝐺 𝐺 NOUN cana-4840 36 11 , , PUNCT cana-4840 36 12 the the DET cana-4840 36 13 quotient quotient NOUN cana-4840 36 14 matrix matrix NOUN cana-4840 36 15 𝑄 𝑄 PROPN cana-4840 36 16 = = PUNCT cana-4840 36 17 𝑄(𝐺 𝑄(𝐺 PROPN cana-4840 36 18 ) ) PUNCT cana-4840 37 1 = = PRON cana-4840 37 2 π‘žπ‘–π‘— π‘žπ‘–π‘— NOUN cana-4840 37 3 is be AUX cana-4840 37 4 a a DET cana-4840 37 5 𝑝 𝑝 PROPN cana-4840 37 6 Γ— Γ— NOUN cana-4840 37 7 𝑝 𝑝 NOUN cana-4840 37 8 matrix matrix NOUN cana-4840 37 9 defined define VERB cana-4840 37 10 as as ADP cana-4840 37 11 π‘žπ‘–π‘— π‘žπ‘–π‘— NOUN cana-4840 37 12 = = SYM cana-4840 37 13 { { PUNCT cana-4840 37 14 𝑑(𝑣𝑖 𝑑(𝑣𝑖 PROPN cana-4840 37 15 ) ) PUNCT cana-4840 37 16 𝑑(𝑣𝑗 𝑑(𝑣𝑗 PROPN cana-4840 37 17 ) ) PUNCT cana-4840 37 18 , , PUNCT cana-4840 37 19 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-4840 37 20 𝑣𝑖𝑣𝑗 𝑣𝑖𝑣𝑗 PROPN cana-4840 37 21 ∈ ∈ PROPN cana-4840 37 22 𝐸 𝐸 PROPN cana-4840 37 23 , , PUNCT cana-4840 37 24 0 0 NUM cana-4840 37 25 , , PUNCT cana-4840 37 26 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’. π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’. VERB cana-4840 37 27 the the DET cana-4840 37 28 characteristic characteristic ADJ cana-4840 37 29 polynomial polynomial NOUN cana-4840 37 30 of of ADP cana-4840 37 31 𝑄(𝐺 𝑄(𝐺 PROPN cana-4840 37 32 ) ) PUNCT cana-4840 38 1 is be AUX cana-4840 38 2 𝑓(𝐺 𝑓(𝐺 ADJ cana-4840 38 3 , , PUNCT cana-4840 38 4 πœ† πœ† X cana-4840 38 5 ) ) PUNCT cana-4840 38 6 = = SYM cana-4840 38 7 𝑑𝑒𝑑 𝑑𝑒𝑑 NOUN cana-4840 38 8 ( ( PUNCT cana-4840 38 9 𝑄 𝑄 PROPN cana-4840 38 10 βˆ’ βˆ’ PROPN cana-4840 38 11 πœ† πœ† DET cana-4840 38 12 𝐼 𝐼 PROPN cana-4840 38 13 ) ) PUNCT cana-4840 38 14 . . PUNCT cana-4840 39 1 the the DET cana-4840 39 2 quotient quotient NOUN cana-4840 39 3 spectrum spectrum NOUN cana-4840 39 4 of of ADP cana-4840 39 5 the the DET cana-4840 39 6 graph graph NOUN cana-4840 39 7 𝐺 𝐺 NOUN cana-4840 39 8 is be AUX cana-4840 39 9 the the DET cana-4840 39 10 eigenvalues eigenvalue NOUN cana-4840 39 11 of of ADP cana-4840 39 12 the the DET cana-4840 39 13 matrix matrix NOUN cana-4840 39 14 𝑄 𝑄 PRON cana-4840 39 15 and and CCONJ cana-4840 39 16 is be AUX cana-4840 39 17 denoted denote VERB cana-4840 39 18 as as ADP cana-4840 39 19 𝑄 𝑄 PROPN cana-4840 39 20 βˆ’ βˆ’ PROPN cana-4840 39 21 𝑆𝑝𝑒𝑐(𝐺 𝑆𝑝𝑒𝑐(𝐺 NOUN cana-4840 39 22 ) ) PUNCT cana-4840 39 23 . . PUNCT cana-4840 40 1 let let VERB cana-4840 40 2 πœ†1 πœ†1 VERB cana-4840 40 3 β‰₯ β‰₯ PROPN cana-4840 40 4 πœ†2 πœ†2 PROPN cana-4840 40 5 β‰₯ β‰₯ NUM cana-4840 40 6 . . PUNCT cana-4840 40 7 . . PUNCT cana-4840 41 1 . . PUNCT cana-4840 42 1 β‰₯ β‰₯ PRON cana-4840 42 2 πœ†π‘› πœ†π‘› AUX cana-4840 42 3 be be AUX cana-4840 42 4 the the DET cana-4840 42 5 spectrum spectrum NOUN cana-4840 42 6 of of ADP cana-4840 42 7 𝑄(𝐺 𝑄(𝐺 PROPN cana-4840 42 8 ) ) PUNCT cana-4840 42 9 . . PUNCT cana-4840 43 1 then then ADV cana-4840 43 2 the the DET cana-4840 43 3 quotient quotient NOUN cana-4840 43 4 energy energy NOUN cana-4840 43 5 is be AUX cana-4840 43 6 defined define VERB cana-4840 43 7 as as ADP cana-4840 43 8 qe(g qe(g NOUN cana-4840 43 9 ) ) PUNCT cana-4840 44 1 = = PUNCT cana-4840 44 2 βˆ‘ βˆ‘ PUNCT cana-4840 44 3 |Ξ»i| |Ξ»i| PROPN cana-4840 44 4 n n CCONJ cana-4840 44 5 i=1 i=1 PROPN cana-4840 44 6 . . PUNCT cana-4840 45 1 for for ADP cana-4840 45 2 additional additional ADJ cana-4840 45 3 details detail NOUN cana-4840 45 4 about about ADP cana-4840 45 5 quotient quotient NOUN cana-4840 45 6 energy energy NOUN cana-4840 45 7 of of ADP cana-4840 45 8 a a DET cana-4840 45 9 graph graph NOUN cana-4840 45 10 refer refer NOUN cana-4840 45 11 [ [ X cana-4840 45 12 6 6 NUM cana-4840 45 13 ] ] SYM cana-4840 45 14 3 3 NUM cana-4840 45 15 . . PUNCT cana-4840 45 16 minimum minimum ADJ cana-4840 45 17 dominating dominating NOUN cana-4840 45 18 quotient quotient NOUN cana-4840 45 19 energy energy NOUN cana-4840 45 20 of of ADP cana-4840 45 21 graph graph NOUN cana-4840 45 22 let let VERB cana-4840 45 23 𝐺 𝐺 PRON cana-4840 45 24 be be AUX cana-4840 45 25 simple simple ADJ cana-4840 45 26 graph graph NOUN cana-4840 45 27 of of ADP cana-4840 45 28 order order NOUN cana-4840 45 29 𝑛 𝑛 NOUN cana-4840 45 30 with with ADP cana-4840 45 31 node node NOUN cana-4840 45 32 set set VERB cana-4840 45 33 𝑉 𝑉 PROPN cana-4840 45 34 = = SYM cana-4840 45 35 { { PUNCT cana-4840 45 36 𝑣1 𝑣1 PROPN cana-4840 45 37 , , PUNCT cana-4840 45 38 𝑣2 𝑣2 PROPN cana-4840 45 39 , , PUNCT cana-4840 45 40 . . PUNCT cana-4840 45 41 . . PUNCT cana-4840 46 1 . . PUNCT cana-4840 47 1 , , PUNCT cana-4840 47 2 𝑣𝑛 𝑣𝑛 NOUN cana-4840 47 3 } } PUNCT cana-4840 47 4 edge edge NOUN cana-4840 47 5 set set VERB cana-4840 47 6 𝐸. 𝐸. PROPN cana-4840 47 7 let let VERB cana-4840 47 8 𝐷 𝐷 NOUN cana-4840 47 9 be be AUX cana-4840 47 10 the the DET cana-4840 47 11 minimum minimum ADJ cana-4840 47 12 dominating dominating NOUN cana-4840 47 13 set set NOUN cana-4840 47 14 of of ADP cana-4840 47 15 a a DET cana-4840 47 16 graph graph NOUN cana-4840 47 17 𝐺. 𝐺. NOUN cana-4840 47 18 the the DET cana-4840 47 19 minimum minimum ADJ cana-4840 47 20 dominating dominating NOUN cana-4840 47 21 quotient quotient NOUN cana-4840 47 22 matrix matrix NOUN cana-4840 47 23 of of ADP cana-4840 47 24 𝐺 𝐺 PROPN cana-4840 47 25 is be AUX cana-4840 47 26 the the DET cana-4840 47 27 𝑛 𝑛 ADJ cana-4840 47 28 Γ— Γ— NOUN cana-4840 47 29 𝑛 𝑛 DET cana-4840 47 30 matrix matrix NOUN cana-4840 47 31 defined define VERB cana-4840 47 32 by by ADP cana-4840 47 33 𝐴𝑄(𝐺 𝐴𝑄(𝐺 NOUN cana-4840 47 34 ) ) PUNCT cana-4840 48 1 = = PUNCT cana-4840 48 2 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-4840 48 3 where where SCONJ cana-4840 48 4 communications communication NOUN cana-4840 48 5 on on ADP cana-4840 48 6 applied apply VERB cana-4840 48 7 nonlinear nonlinear ADJ cana-4840 48 8 analysis analysis NOUN cana-4840 48 9 issn issn NOUN cana-4840 48 10 : : PUNCT cana-4840 48 11 1074 1074 NUM cana-4840 48 12 - - PUNCT cana-4840 48 13 133x 133x NUM cana-4840 48 14 vol vol VERB cana-4840 48 15 32 32 NUM cana-4840 48 16 no no NOUN cana-4840 48 17 . . PUNCT cana-4840 49 1 10s 10 NOUN cana-4840 49 2 ( ( PUNCT cana-4840 49 3 2025 2025 NUM cana-4840 49 4 ) ) PUNCT cana-4840 49 5 446 446 NUM cana-4840 49 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 49 7 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-4840 49 8 = = SYM cana-4840 49 9 { { PUNCT cana-4840 49 10 𝑑(𝑣𝑖 𝑑(𝑣𝑖 PROPN cana-4840 49 11 ) ) PUNCT cana-4840 49 12 𝑑(𝑣𝑗 𝑑(𝑣𝑗 PROPN cana-4840 49 13 ) ) PUNCT cana-4840 49 14 , , PUNCT cana-4840 49 15 𝑖𝑓 𝑖𝑓 CCONJ cana-4840 49 16 𝑣𝑖𝑣𝑗 𝑣𝑖𝑣𝑗 PROPN cana-4840 49 17 ∈ ∈ PROPN cana-4840 49 18 𝐸 𝐸 PROPN cana-4840 49 19 , , PUNCT cana-4840 49 20 1 1 NUM cana-4840 49 21 , , PUNCT cana-4840 49 22 𝑖𝑓 𝑖𝑓 ADP cana-4840 49 23 𝑣𝑖 𝑣𝑖 ADP cana-4840 49 24 = = SYM cana-4840 49 25 𝑣𝑗 𝑣𝑗 ADP cana-4840 49 26 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN cana-4840 49 27 𝑣𝑖 𝑣𝑖 ADP cana-4840 49 28 ∈ ∈ PROPN cana-4840 49 29 𝐷 𝐷 PROPN cana-4840 49 30 , , PUNCT cana-4840 49 31 0 0 NUM cana-4840 49 32 , , PUNCT cana-4840 49 33 𝑖𝑓 𝑖𝑓 VERB cana-4840 49 34 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ NOUN cana-4840 49 35 . . PUNCT cana-4840 50 1 the the DET cana-4840 50 2 characteristic characteristic ADJ cana-4840 50 3 polynomial polynomial NOUN cana-4840 50 4 of of ADP cana-4840 50 5 𝐴𝑄(𝐺 𝐴𝑄(𝐺 NOUN cana-4840 50 6 ) ) PUNCT cana-4840 50 7 is be AUX cana-4840 50 8 indicated indicate VERB cana-4840 50 9 by𝑓(𝐺 by𝑓(𝐺 NUM cana-4840 50 10 , , PUNCT cana-4840 50 11 πœ† πœ† X cana-4840 50 12 ) ) PUNCT cana-4840 50 13 = = SYM cana-4840 50 14 𝑑𝑒𝑑 𝑑𝑒𝑑 NOUN cana-4840 50 15 ( ( PUNCT cana-4840 50 16 πœ† πœ† X cana-4840 50 17 𝐼 𝐼 PROPN cana-4840 50 18 βˆ’ βˆ’ PROPN cana-4840 50 19 𝐴𝑄(𝐺 𝐴𝑄(𝐺 NOUN cana-4840 50 20 ) ) PUNCT cana-4840 50 21 ) ) PUNCT cana-4840 50 22 . . PUNCT cana-4840 51 1 the the DET cana-4840 51 2 minimum minimum ADJ cana-4840 51 3 dominating dominating NOUN cana-4840 51 4 quotient quotient NOUN cana-4840 51 5 eigenvalues eigenvalue NOUN cana-4840 51 6 of of ADP cana-4840 51 7 the the DET cana-4840 51 8 graph graph NOUN cana-4840 51 9 𝐺 𝐺 NOUN cana-4840 51 10 are be AUX cana-4840 51 11 the the DET cana-4840 51 12 eigenvalues eigenvalue NOUN cana-4840 51 13 of of ADP cana-4840 51 14 𝐴𝑄(𝐺 𝐴𝑄(𝐺 NOUN cana-4840 51 15 ) ) PUNCT cana-4840 51 16 . . PUNCT cana-4840 52 1 since since SCONJ cana-4840 52 2 𝐴𝑄(𝐺 𝐴𝑄(𝐺 PROPN cana-4840 52 3 ) ) PUNCT cana-4840 52 4 is be AUX cana-4840 52 5 real real ADJ cana-4840 52 6 and and CCONJ cana-4840 52 7 symmetric symmetric ADJ cana-4840 52 8 , , PUNCT cana-4840 52 9 its its PRON cana-4840 52 10 eigenvalues eigenvalue NOUN cana-4840 52 11 are be AUX cana-4840 52 12 real real ADJ cana-4840 52 13 numbers number NOUN cana-4840 52 14 and and CCONJ cana-4840 52 15 are be AUX cana-4840 52 16 labelled label VERB cana-4840 52 17 in in ADP cana-4840 52 18 non non ADJ cana-4840 52 19 - - ADJ cana-4840 52 20 increasing increase VERB cana-4840 52 21 order order NOUN cana-4840 52 22 πœ†1 πœ†1 VERB cana-4840 52 23 β‰₯ β‰₯ PROPN cana-4840 52 24 πœ†2 πœ†2 PROPN cana-4840 52 25 β‰₯ β‰₯ NUM cana-4840 52 26 . . PUNCT cana-4840 52 27 . . PUNCT cana-4840 53 1 . . PUNCT cana-4840 54 1 β‰₯ β‰₯ PRON cana-4840 54 2 πœ†π‘›. πœ†π‘›. VERB cana-4840 54 3 the the DET cana-4840 54 4 minimum minimum ADJ cana-4840 54 5 dominating dominating NOUN cana-4840 54 6 quotient quotient NOUN cana-4840 54 7 energy energy NOUN cana-4840 54 8 of of ADP cana-4840 54 9 𝐺 𝐺 PROPN cana-4840 54 10 is be AUX cana-4840 54 11 defined define VERB cana-4840 54 12 as as ADP cana-4840 54 13 qde(g qde(g NOUN cana-4840 54 14 ) ) PUNCT cana-4840 55 1 = = NOUN cana-4840 55 2 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 55 3 n n PROPN cana-4840 55 4 i=1 i=1 PROPN cana-4840 55 5 4 4 NUM cana-4840 55 6 . . PUNCT cana-4840 56 1 the the DET cana-4840 56 2 laplacian laplacian ADJ cana-4840 56 3 minimum minimum NOUN cana-4840 56 4 dominating dominating NOUN cana-4840 56 5 quotient quotient NOUN cana-4840 56 6 energy energy NOUN cana-4840 56 7 of of ADP cana-4840 56 8 a a DET cana-4840 56 9 graph graph NOUN cana-4840 56 10 let let VERB cana-4840 56 11 𝐷(𝐺 𝐷(𝐺 PROPN cana-4840 56 12 ) ) PUNCT cana-4840 56 13 represent represent VERB cana-4840 56 14 the the DET cana-4840 56 15 diagonal diagonal ADJ cana-4840 56 16 matrix matrix NOUN cana-4840 56 17 of of ADP cana-4840 56 18 the the DET cana-4840 56 19 node node ADJ cana-4840 56 20 degrees degree NOUN cana-4840 56 21 of of ADP cana-4840 56 22 the the DET cana-4840 56 23 graph graph NOUN cana-4840 56 24 𝐺. 𝐺. NOUN cana-4840 56 25 then then ADV cana-4840 56 26 the the DET cana-4840 56 27 laplacian laplacian ADJ cana-4840 56 28 minimum minimum NOUN cana-4840 56 29 dominating dominating NOUN cana-4840 56 30 quotient quotient NOUN cana-4840 56 31 matrix matrix NOUN cana-4840 56 32 of of ADP cana-4840 56 33 𝐺 𝐺 PROPN cana-4840 56 34 is be AUX cana-4840 56 35 denoted denote VERB cana-4840 56 36 by by ADP cana-4840 56 37 𝐿𝑄𝐷𝐸(𝐺 𝐿𝑄𝐷𝐸(𝐺 NOUN cana-4840 56 38 ) ) PUNCT cana-4840 56 39 and and CCONJ cana-4840 56 40 is be AUX cana-4840 56 41 defined define VERB cana-4840 56 42 as as SCONJ cana-4840 56 43 follows follow VERB cana-4840 56 44 𝐿𝑄𝐷𝐸(𝐺 𝐿𝑄𝐷𝐸(𝐺 NOUN cana-4840 56 45 ) ) PUNCT cana-4840 56 46 = = SYM cana-4840 56 47 𝐷(𝐺 𝐷(𝐺 NOUN cana-4840 56 48 ) ) PUNCT cana-4840 57 1 βˆ’ βˆ’ PROPN cana-4840 57 2 𝐴𝐷(𝐺 𝐴𝐷(𝐺 NOUN cana-4840 57 3 ) ) PUNCT cana-4840 57 4 . . PUNCT cana-4840 58 1 let let VERB cana-4840 58 2 πœ†1 πœ†1 VERB cana-4840 58 3 β‰₯ β‰₯ PROPN cana-4840 58 4 πœ†2 πœ†2 PROPN cana-4840 58 5 β‰₯ β‰₯ NUM cana-4840 58 6 . . PUNCT cana-4840 58 7 . . PUNCT cana-4840 59 1 . . PUNCT cana-4840 60 1 β‰₯ β‰₯ PRON cana-4840 60 2 πœ†π‘› πœ†π‘› AUX cana-4840 60 3 be be AUX cana-4840 60 4 the the DET cana-4840 60 5 eigen eigen PROPN cana-4840 60 6 values value NOUN cana-4840 60 7 𝐿𝑄𝐷𝐸(𝐺 𝐿𝑄𝐷𝐸(𝐺 NOUN cana-4840 60 8 ) ) PUNCT cana-4840 60 9 organized organize VERB cana-4840 60 10 in in ADP cana-4840 60 11 non non ADJ cana-4840 60 12 - - ADJ cana-4840 60 13 increasing increase VERB cana-4840 60 14 order order NOUN cana-4840 60 15 . . PUNCT cana-4840 61 1 these these DET cana-4840 61 2 eigen eigen PROPN cana-4840 61 3 values value NOUN cana-4840 61 4 are be AUX cana-4840 61 5 called call VERB cana-4840 61 6 laplacian laplacian ADJ cana-4840 61 7 minimum minimum NOUN cana-4840 61 8 dominating dominating NOUN cana-4840 61 9 quotient quotient NOUN cana-4840 61 10 eigen eigen PROPN cana-4840 61 11 values value NOUN cana-4840 61 12 of of ADP cana-4840 61 13 𝐺. 𝐺. NOUN cana-4840 61 14 the the DET cana-4840 61 15 laplacian laplacian ADJ cana-4840 61 16 minimum minimum NOUN cana-4840 61 17 dominating dominating NOUN cana-4840 61 18 quotient quotient NOUN cana-4840 61 19 energy energy NOUN cana-4840 61 20 of of ADP cana-4840 61 21 a a DET cana-4840 61 22 graph graph NOUN cana-4840 61 23 𝐺 𝐺 NOUN cana-4840 61 24 is be AUX cana-4840 61 25 defined define VERB cana-4840 61 26 as as ADP cana-4840 61 27 lqde(g lqde(g NOUN cana-4840 61 28 ) ) PUNCT cana-4840 62 1 = = NOUN cana-4840 62 2 βˆ‘|Ξ»i βˆ‘|Ξ»i NOUN cana-4840 62 3 βˆ’ βˆ’ PROPN cana-4840 62 4 2 2 NUM cana-4840 62 5 m m NOUN cana-4840 62 6 n n NUM cana-4840 62 7 | | ADV cana-4840 62 8 n n ADV cana-4840 62 9 i=1 i=1 PROPN cana-4840 62 10 where where SCONJ cana-4840 62 11 π‘š π‘š PROPN cana-4840 62 12 is be AUX cana-4840 62 13 the the DET cana-4840 62 14 number number NOUN cana-4840 62 15 of of ADP cana-4840 62 16 edges edge NOUN cana-4840 62 17 of of ADP cana-4840 62 18 𝐺 𝐺 NOUN cana-4840 62 19 and and CCONJ cana-4840 62 20 2π‘š 2π‘š NOUN cana-4840 62 21 𝑛 𝑛 PRON cana-4840 62 22 is be AUX cana-4840 62 23 the the DET cana-4840 62 24 average average ADJ cana-4840 62 25 degree degree NOUN cana-4840 62 26 of of ADP cana-4840 62 27 𝐺. 𝐺. NOUN cana-4840 62 28 example.4.1 example.4.1 PROPN cana-4840 62 29 let let VERB cana-4840 62 30 𝐺 𝐺 PROPN cana-4840 62 31 be be AUX cana-4840 62 32 a a DET cana-4840 62 33 graph graph NOUN cana-4840 62 34 with with ADP cana-4840 62 35 6 6 NUM cana-4840 62 36 nodes node NOUN cana-4840 62 37 , , PUNCT cana-4840 62 38 as as SCONJ cana-4840 62 39 illustrated illustrate VERB cana-4840 62 40 in in ADP cana-4840 62 41 figure figure NOUN cana-4840 62 42 4.1 4.1 NUM cana-4840 62 43 . . PUNCT cana-4840 63 1 the the DET cana-4840 63 2 possible possible ADJ cana-4840 63 3 𝛾 𝛾 ADP cana-4840 63 4 βˆ’sets βˆ’set NOUN cana-4840 63 5 are be AUX cana-4840 63 6 ( ( PUNCT cana-4840 63 7 𝑖)𝐷1 𝑖)𝐷1 X cana-4840 63 8 = = SYM cana-4840 63 9 { { PUNCT cana-4840 63 10 𝑣2 𝑣2 NOUN cana-4840 63 11 , , PUNCT cana-4840 63 12 𝑣4 𝑣4 NOUN cana-4840 63 13 } } PUNCT cana-4840 63 14 ( ( PUNCT cana-4840 63 15 𝑖𝑖)𝐷2 𝑖𝑖)𝐷2 PROPN cana-4840 63 16 = = SYM cana-4840 63 17 { { PUNCT cana-4840 63 18 𝑣1𝑣4 𝑣1𝑣4 NOUN cana-4840 63 19 } } PUNCT cana-4840 63 20 figure figure NOUN cana-4840 63 21 . . PUNCT cana-4840 64 1 4.1 4.1 NUM cana-4840 64 2 ( ( PUNCT cana-4840 64 3 i i NOUN cana-4840 64 4 ) ) PUNCT cana-4840 65 1 if if SCONJ cana-4840 65 2 the the DET cana-4840 65 3 𝛾 𝛾 NOUN cana-4840 65 4 βˆ’ βˆ’ PROPN cana-4840 65 5 set set NOUN cana-4840 65 6 is be AUX cana-4840 65 7 𝐷1 𝐷1 NOUN cana-4840 65 8 = = SYM cana-4840 65 9 { { PUNCT cana-4840 65 10 𝑣2 𝑣2 NOUN cana-4840 65 11 , , PUNCT cana-4840 65 12 𝑣4 𝑣4 NOUN cana-4840 65 13 } } PUNCT cana-4840 65 14 then then ADV cana-4840 65 15 communications communication NOUN cana-4840 65 16 on on ADP cana-4840 65 17 applied apply VERB cana-4840 65 18 nonlinear nonlinear ADJ cana-4840 65 19 analysis analysis NOUN cana-4840 65 20 issn issn NOUN cana-4840 65 21 : : PUNCT cana-4840 65 22 1074 1074 NUM cana-4840 65 23 - - PUNCT cana-4840 65 24 133x 133x NUM cana-4840 65 25 vol vol VERB cana-4840 65 26 32 32 NUM cana-4840 65 27 no no NOUN cana-4840 65 28 . . PUNCT cana-4840 66 1 10s 10 NOUN cana-4840 66 2 ( ( PUNCT cana-4840 66 3 2025 2025 NUM cana-4840 66 4 ) ) PUNCT cana-4840 66 5 447 447 NUM cana-4840 66 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 67 1 the the DET cana-4840 67 2 characteristic characteristic ADJ cana-4840 67 3 polynomial polynomial NOUN cana-4840 67 4 is be AUX cana-4840 67 5 expressed express VERB cana-4840 67 6 as as ADP cana-4840 67 7 𝑓{𝑛}(𝐺 𝑓{𝑛}(𝐺 NOUN cana-4840 67 8 , , PUNCT cana-4840 67 9 πœ† πœ† NOUN cana-4840 67 10 ) ) PUNCT cana-4840 67 11 = = SYM cana-4840 67 12 πœ† πœ† ADP cana-4840 67 13 7 7 NUM cana-4840 67 14 βˆ’ βˆ’ NUM cana-4840 67 15 14πœ†6 14πœ†6 NUM cana-4840 68 1 + + NUM cana-4840 68 2 72πœ†5 72πœ†5 NUM cana-4840 68 3 βˆ’ βˆ’ NOUN cana-4840 68 4 166πœ†4 166πœ†4 NUM cana-4840 68 5 + + NUM cana-4840 68 6 162πœ†3 162πœ†3 NUM cana-4840 68 7 βˆ’ βˆ’ NOUN cana-4840 68 8 30πœ†2 30πœ†2 NUM cana-4840 68 9 βˆ’ βˆ’ NOUN cana-4840 68 10 30πœ† 30πœ† NUM cana-4840 68 11 + + CCONJ cana-4840 68 12 4 4 NUM cana-4840 68 13 = = SYM cana-4840 68 14 0 0 NUM cana-4840 68 15 . . PUNCT cana-4840 69 1 the the DET cana-4840 69 2 laplacian laplacian ADJ cana-4840 69 3 minimum minimum NOUN cana-4840 69 4 dominating dominating NOUN cana-4840 69 5 quotient quotient NOUN cana-4840 69 6 eigen eigen PROPN cana-4840 69 7 values value NOUN cana-4840 69 8 are be AUX cana-4840 69 9 πœ†1 πœ†1 ADJ cana-4840 69 10 = = PUNCT cana-4840 69 11 βˆ’0.5597 βˆ’0.5597 PROPN cana-4840 69 12 , , PUNCT cana-4840 69 13 πœ†2 πœ†2 NOUN cana-4840 69 14 = = SYM cana-4840 69 15 βˆ’0.1842 βˆ’0.1842 PROPN cana-4840 69 16 , , PUNCT cana-4840 69 17 πœ†3 πœ†3 NOUN cana-4840 69 18 = = NOUN cana-4840 69 19 0.9583 0.9583 NUM cana-4840 69 20 , , PUNCT cana-4840 69 21 πœ†4 πœ†4 NOUN cana-4840 69 22 = = SYM cana-4840 69 23 2 2 NUM cana-4840 69 24 , , PUNCT cana-4840 69 25 πœ†5 πœ†5 ADV cana-4840 69 26 = = PUNCT cana-4840 69 27 2.1960 2.1960 NUM cana-4840 69 28 , , PUNCT cana-4840 69 29 πœ†6 πœ†6 PROPN cana-4840 69 30 = = PROPN cana-4840 69 31 4.1923 4.1923 NUM cana-4840 69 32 , , PUNCT cana-4840 69 33 πœ†_7 πœ†_7 PROPN cana-4840 70 1 = = SYM cana-4840 70 2 4.3973 4.3973 NUM cana-4840 70 3 . . PUNCT cana-4840 71 1 the the DET cana-4840 71 2 mean mean ADJ cana-4840 71 3 degree degree NOUN cana-4840 71 4 of of ADP cana-4840 71 5 the the DET cana-4840 71 6 graph graph NOUN cana-4840 71 7 2π‘š 2π‘š NOUN cana-4840 71 8 𝑛 𝑛 NOUN cana-4840 71 9 = = SYM cana-4840 71 10 2Γ—8 2Γ—8 PROPN cana-4840 71 11 7 7 NUM cana-4840 71 12 = = SYM cana-4840 71 13 16 16 NUM cana-4840 71 14 7 7 NUM cana-4840 71 15 hence hence ADV cana-4840 71 16 , , PUNCT cana-4840 71 17 laplacian laplacian ADJ cana-4840 71 18 minimum minimum NOUN cana-4840 71 19 dominating dominating NOUN cana-4840 71 20 quotient quotient NOUN cana-4840 71 21 energy energy NOUN cana-4840 71 22 , , PUNCT cana-4840 71 23 𝐿𝑄𝐷1𝐸(𝐺 𝐿𝑄𝐷1𝐸(𝐺 ADJ cana-4840 71 24 ) ) PUNCT cana-4840 72 1 β‰ˆ β‰ˆ PROPN cana-4840 72 2 12.43638 12.43638 NUM cana-4840 72 3 ( ( PUNCT cana-4840 72 4 ii ii NOUN cana-4840 72 5 ) ) PUNCT cana-4840 72 6 if if SCONJ cana-4840 72 7 the the DET cana-4840 72 8 𝛾 𝛾 NOUN cana-4840 72 9 βˆ’ βˆ’ PROPN cana-4840 72 10 set set NOUN cana-4840 72 11 is be AUX cana-4840 72 12 𝐷2 𝐷2 NOUN cana-4840 72 13 = = SYM cana-4840 72 14 { { PUNCT cana-4840 72 15 𝑣1 𝑣1 PROPN cana-4840 72 16 , , PUNCT cana-4840 72 17 𝑣4 𝑣4 NOUN cana-4840 72 18 } } PUNCT cana-4840 72 19 then then ADV cana-4840 72 20 the the DET cana-4840 72 21 characteristic characteristic ADJ cana-4840 72 22 polynomial polynomial NOUN cana-4840 72 23 is be AUX cana-4840 72 24 expressed express VERB cana-4840 72 25 as as ADP cana-4840 72 26 𝑓{𝑛}(𝐺 𝑓{𝑛}(𝐺 NOUN cana-4840 72 27 , , PUNCT cana-4840 72 28 πœ† πœ† NOUN cana-4840 72 29 ) ) PUNCT cana-4840 72 30 = = SYM cana-4840 72 31 πœ† πœ† ADP cana-4840 72 32 7 7 NUM cana-4840 72 33 βˆ’ βˆ’ NUM cana-4840 72 34 14πœ†6 14πœ†6 NUM cana-4840 72 35 + + NUM cana-4840 72 36 71πœ†5 71πœ†5 NUM cana-4840 72 37 βˆ’ βˆ’ NUM cana-4840 72 38 155πœ†4 155πœ†4 NUM cana-4840 72 39 + + SYM cana-4840 72 40 121πœ†3 121πœ†3 NUM cana-4840 72 41 + + SYM cana-4840 72 42 27πœ†2 27πœ†2 NUM cana-4840 72 43 βˆ’ βˆ’ NOUN cana-4840 72 44 48πœ† 48πœ† NOUN cana-4840 72 45 βˆ’ βˆ’ PROPN cana-4840 72 46 4 4 NUM cana-4840 72 47 = = SYM cana-4840 72 48 0 0 NUM cana-4840 72 49 . . PUNCT cana-4840 73 1 the the DET cana-4840 73 2 laplacian laplacian ADJ cana-4840 73 3 minimum minimum NOUN cana-4840 73 4 dominating dominating NOUN cana-4840 73 5 quotient quotient NOUN cana-4840 73 6 eigen eigen PROPN cana-4840 73 7 values value NOUN cana-4840 73 8 areπœ†1 areπœ†1 PUNCT cana-4840 74 1 = = PUNCT cana-4840 74 2 βˆ’0.4956 βˆ’0.4956 PROPN cana-4840 74 3 , , PUNCT cana-4840 74 4 πœ†2 πœ†2 PROPN cana-4840 74 5 = = SYM cana-4840 74 6 βˆ’0.0811 βˆ’0.0811 PROPN cana-4840 74 7 , , PUNCT cana-4840 74 8 πœ†3 πœ†3 NOUN cana-4840 74 9 = = SYM cana-4840 74 10 1.0381 1.0381 NUM cana-4840 74 11 , , PUNCT cana-4840 74 12 πœ†4 πœ†4 NOUN cana-4840 74 13 = = SYM cana-4840 74 14 2.2342 2.2342 NUM cana-4840 74 15 , , PUNCT cana-4840 74 16 πœ†5 πœ†5 NOUN cana-4840 74 17 = = PUNCT cana-4840 74 18 5.09551 5.09551 NUM cana-4840 74 19 , , PUNCT cana-4840 74 20 πœ†6 πœ†6 PROPN cana-4840 74 21 = = SYM cana-4840 74 22 4.2094 4.2094 NUM cana-4840 74 23 , , PUNCT cana-4840 74 24 πœ†7 πœ†7 NOUN cana-4840 74 25 = = NOUN cana-4840 74 26 2 2 X cana-4840 74 27 . . X cana-4840 74 28 average average ADJ cana-4840 74 29 degree degree NOUN cana-4840 74 30 of of ADP cana-4840 74 31 the the DET cana-4840 74 32 graph graph NOUN cana-4840 74 33 2π‘š 2π‘š NOUN cana-4840 74 34 𝑛 𝑛 NOUN cana-4840 74 35 = = SYM cana-4840 74 36 2Γ—8 2Γ—8 PROPN cana-4840 74 37 7 7 NUM cana-4840 74 38 = = SYM cana-4840 74 39 16 16 NUM cana-4840 74 40 7 7 NUM cana-4840 74 41 communications communication NOUN cana-4840 74 42 on on ADP cana-4840 74 43 applied apply VERB cana-4840 74 44 nonlinear nonlinear ADJ cana-4840 74 45 analysis analysis NOUN cana-4840 74 46 issn issn NOUN cana-4840 74 47 : : PUNCT cana-4840 74 48 1074 1074 NUM cana-4840 74 49 - - PUNCT cana-4840 74 50 133x 133x NUM cana-4840 74 51 vol vol VERB cana-4840 74 52 32 32 NUM cana-4840 74 53 no no NOUN cana-4840 74 54 . . PUNCT cana-4840 74 55 10s 10 NOUN cana-4840 74 56 ( ( PUNCT cana-4840 74 57 2025 2025 NUM cana-4840 74 58 ) ) PUNCT cana-4840 74 59 448 448 NUM cana-4840 74 60 https://internationalpubls.com https://internationalpubls.com X cana-4840 74 61 thus thus ADV cana-4840 74 62 , , PUNCT cana-4840 74 63 the the DET cana-4840 74 64 laplacian laplacian ADJ cana-4840 74 65 minimum minimum NOUN cana-4840 74 66 dominating dominating NOUN cana-4840 74 67 quotient quotient NOUN cana-4840 74 68 energy energy NOUN cana-4840 74 69 , , PUNCT cana-4840 74 70 𝐿𝑄𝐷2𝐸(𝐺 𝐿𝑄𝐷2𝐸(𝐺 NUM cana-4840 74 71 ) ) PUNCT cana-4840 75 1 β‰ˆ β‰ˆ PROPN cana-4840 75 2 12.8677 12.8677 NUM cana-4840 75 3 therefore therefore ADV cana-4840 75 4 , , PUNCT cana-4840 75 5 based base VERB cana-4840 75 6 on on ADP cana-4840 75 7 the the DET cana-4840 75 8 above above ADJ cana-4840 75 9 example example NOUN cana-4840 75 10 , , PUNCT cana-4840 75 11 it it PRON cana-4840 75 12 is be AUX cana-4840 75 13 evident evident ADJ cana-4840 75 14 that that SCONJ cana-4840 75 15 the the DET cana-4840 75 16 laplacian laplacian ADJ cana-4840 75 17 minimum minimum ADJ cana-4840 75 18 dominating dominating NOUN cana-4840 75 19 quotient quotient NOUN cana-4840 75 20 energy energy NOUN cana-4840 75 21 of of ADP cana-4840 75 22 a a DET cana-4840 75 23 graph graph NOUN cana-4840 75 24 𝐺 𝐺 NOUN cana-4840 75 25 is be AUX cana-4840 75 26 influenced influence VERB cana-4840 75 27 by by ADP cana-4840 75 28 the the DET cana-4840 75 29 minimum minimum ADJ cana-4840 75 30 dominating dominating NOUN cana-4840 75 31 set set NOUN cana-4840 75 32 of of ADP cana-4840 75 33 𝐺 𝐺 PROPN cana-4840 75 34 5 5 NUM cana-4840 75 35 . . PUNCT cana-4840 76 1 laplacian laplacian ADJ cana-4840 76 2 minimum minimum NOUN cana-4840 76 3 dominating dominating NOUN cana-4840 76 4 quotient quotient NOUN cana-4840 76 5 energy energy NOUN cana-4840 76 6 of of ADP cana-4840 76 7 some some DET cana-4840 76 8 standard standard ADJ cana-4840 76 9 graphs graph NOUN cana-4840 76 10 theorem theorem VERB cana-4840 76 11 5.1 5.1 NUM cana-4840 76 12 . . PUNCT cana-4840 77 1 if if SCONJ cana-4840 77 2 𝐾𝑛 𝐾𝑛 PROPN cana-4840 77 3 is be AUX cana-4840 77 4 the the DET cana-4840 77 5 complete complete ADJ cana-4840 77 6 graph graph NOUN cana-4840 77 7 with with ADP cana-4840 77 8 𝑛 𝑛 DET cana-4840 77 9 nodes node NOUN cana-4840 77 10 , , PUNCT cana-4840 77 11 then then ADV cana-4840 77 12 𝐿𝑄𝐷𝐸(𝐾𝑛 𝐿𝑄𝐷𝐸(𝐾𝑛 PROPN cana-4840 77 13 ) ) PUNCT cana-4840 78 1 = = PUNCT cana-4840 78 2 ( ( PUNCT cana-4840 78 3 𝑛 𝑛 PRON cana-4840 78 4 βˆ’ βˆ’ NOUN cana-4840 78 5 2 2 NUM cana-4840 78 6 ) ) PUNCT cana-4840 78 7 + + CCONJ cana-4840 78 8 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 79 1 βˆ’ βˆ’ PRON cana-4840 79 2 2𝑛 2𝑛 NOUN cana-4840 79 3 + + CCONJ cana-4840 79 4 5 5 X cana-4840 79 5 . . X cana-4840 79 6 proof proof NOUN cana-4840 79 7 : : PUNCT cana-4840 79 8 let let VERB cana-4840 79 9 𝐾𝑛 𝐾𝑛 PROPN cana-4840 79 10 be be AUX cana-4840 79 11 the the DET cana-4840 79 12 complete complete ADJ cana-4840 79 13 graph graph NOUN cana-4840 79 14 with with ADP cana-4840 79 15 node node NOUN cana-4840 79 16 set set VERB cana-4840 79 17 𝑉 𝑉 PROPN cana-4840 79 18 = = SYM cana-4840 79 19 { { PUNCT cana-4840 79 20 𝑣1 𝑣1 PROPN cana-4840 79 21 , , PUNCT cana-4840 79 22 𝑣2 𝑣2 PROPN cana-4840 79 23 , , PUNCT cana-4840 79 24 … … PUNCT cana-4840 79 25 , , PUNCT cana-4840 79 26 𝑣𝑛 𝑣𝑛 NOUN cana-4840 79 27 } } PUNCT cana-4840 79 28 . . PUNCT cana-4840 80 1 the the DET cana-4840 80 2 𝛾 𝛾 PROPN cana-4840 80 3 βˆ’set βˆ’set NOUN cana-4840 80 4 𝐷 𝐷 NOUN cana-4840 80 5 = = SYM cana-4840 80 6 { { PUNCT cana-4840 80 7 𝑣1 𝑣1 NOUN cana-4840 80 8 } } PUNCT cana-4840 80 9 . . PUNCT cana-4840 81 1 and and CCONJ cana-4840 81 2 its its PRON cana-4840 81 3 characteristic characteristic ADJ cana-4840 81 4 polynomial polynomial NOUN cana-4840 81 5 is be AUX cana-4840 81 6 [ [ X cana-4840 81 7 πœ† πœ† X cana-4840 81 8 βˆ’ βˆ’ PROPN cana-4840 81 9 𝑛](π‘›βˆ’2)[πœ†2 𝑛](π‘›βˆ’2)[πœ†2 NOUN cana-4840 81 10 βˆ’ βˆ’ PROPN cana-4840 81 11 ( ( PUNCT cana-4840 81 12 𝑛 𝑛 PRON cana-4840 81 13 βˆ’ βˆ’ PROPN cana-4840 81 14 1)πœ† 1)πœ† NUM cana-4840 81 15 βˆ’ βˆ’ PROPN cana-4840 81 16 1 1 NUM cana-4840 81 17 ] ] PUNCT cana-4840 81 18 the the DET cana-4840 81 19 laplacian laplacian ADJ cana-4840 81 20 minimum minimum NOUN cana-4840 81 21 dominating dominating NOUN cana-4840 81 22 quotient quotient NOUN cana-4840 81 23 eigen eigen PROPN cana-4840 81 24 values value NOUN cana-4840 81 25 are be AUX cana-4840 81 26 : : PUNCT cana-4840 81 27 πœ† πœ† X cana-4840 81 28 = = PUNCT cana-4840 81 29 𝑛[(𝑛 𝑛[(𝑛 VERB cana-4840 81 30 βˆ’ βˆ’ NOUN cana-4840 81 31 2)π‘‘π‘–π‘šπ‘’ 2)π‘‘π‘–π‘šπ‘’ NUM cana-4840 81 32 ] ] PUNCT cana-4840 81 33 , , PUNCT cana-4840 81 34 πœ† πœ† X cana-4840 81 35 = = PUNCT cana-4840 81 36 ( ( PUNCT cana-4840 81 37 𝑛 𝑛 PRON cana-4840 81 38 βˆ’ βˆ’ PROPN cana-4840 81 39 1 1 NUM cana-4840 81 40 ) ) PUNCT cana-4840 81 41 Β± Β± NOUN cana-4840 81 42 βˆšπ‘›2 βˆšπ‘›2 PROPN cana-4840 81 43 βˆ’ βˆ’ PROPN cana-4840 81 44 2𝑛 2𝑛 NOUN cana-4840 82 1 + + CCONJ cana-4840 82 2 5 5 NUM cana-4840 82 3 2 2 NUM cana-4840 82 4 [ [ PUNCT cana-4840 82 5 π‘œπ‘›π‘’ π‘œπ‘›π‘’ NOUN cana-4840 82 6 π‘‘π‘–π‘šπ‘’ π‘‘π‘–π‘šπ‘’ VERB cana-4840 82 7 π‘’π‘Žπ‘β„Ž π‘’π‘Žπ‘β„Ž PROPN cana-4840 82 8 ] ] PUNCT cana-4840 82 9 average average ADJ cana-4840 82 10 degree degree NOUN cana-4840 82 11 of of ADP cana-4840 82 12 𝐾𝑛 𝐾𝑛 NOUN cana-4840 82 13 = = SYM cana-4840 82 14 2π‘š 2π‘š NUM cana-4840 82 15 𝑛 𝑛 NOUN cana-4840 82 16 = = SYM cana-4840 82 17 2 2 NUM cana-4840 82 18 𝑛(π‘›βˆ’1 𝑛(π‘›βˆ’1 NOUN cana-4840 82 19 ) ) PUNCT cana-4840 82 20 2 2 NUM cana-4840 82 21 𝑛 𝑛 NOUN cana-4840 82 22 = = SYM cana-4840 82 23 𝑛 𝑛 PRON cana-4840 82 24 βˆ’ βˆ’ NOUN cana-4840 82 25 1 1 NUM cana-4840 82 26 hence hence ADV cana-4840 82 27 , , PUNCT cana-4840 82 28 the the DET cana-4840 82 29 laplacian laplacian ADJ cana-4840 82 30 minimum minimum NOUN cana-4840 82 31 dominating dominating NOUN cana-4840 82 32 quotient quotient NOUN cana-4840 82 33 energy energy NOUN cana-4840 82 34 of of ADP cana-4840 82 35 𝐾𝑛 𝐾𝑛 PROPN cana-4840 82 36 is be AUX cana-4840 82 37 communications communication NOUN cana-4840 82 38 on on ADP cana-4840 82 39 applied apply VERB cana-4840 82 40 nonlinear nonlinear ADJ cana-4840 82 41 analysis analysis NOUN cana-4840 82 42 issn issn NOUN cana-4840 82 43 : : PUNCT cana-4840 82 44 1074 1074 NUM cana-4840 82 45 - - PUNCT cana-4840 82 46 133x 133x NUM cana-4840 82 47 vol vol VERB cana-4840 82 48 32 32 NUM cana-4840 82 49 no no NOUN cana-4840 82 50 . . PUNCT cana-4840 83 1 10s 10 NOUN cana-4840 83 2 ( ( PUNCT cana-4840 83 3 2025 2025 NUM cana-4840 83 4 ) ) PUNCT cana-4840 83 5 449 449 NUM cana-4840 83 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 83 7 𝐿𝑄𝐷𝐸(𝐾𝑛 𝐿𝑄𝐷𝐸(𝐾𝑛 PROPN cana-4840 83 8 ) ) PUNCT cana-4840 83 9 = = SYM cana-4840 83 10 |𝑛 |𝑛 PROPN cana-4840 83 11 βˆ’ βˆ’ X cana-4840 83 12 ( ( PUNCT cana-4840 83 13 𝑛 𝑛 PROPN cana-4840 83 14 βˆ’ βˆ’ PROPN cana-4840 83 15 1)| 1)| NUM cana-4840 83 16 ( ( PUNCT cana-4840 83 17 𝑛 𝑛 PROPN cana-4840 83 18 βˆ’ βˆ’ PROPN cana-4840 83 19 2 2 NUM cana-4840 83 20 ) ) PUNCT cana-4840 84 1 + + CCONJ cana-4840 84 2 | | ADV cana-4840 84 3 ( ( PUNCT cana-4840 84 4 𝑛 𝑛 PROPN cana-4840 84 5 βˆ’ βˆ’ NOUN cana-4840 84 6 1 1 NUM cana-4840 84 7 ) ) PUNCT cana-4840 84 8 + + CCONJ cana-4840 84 9 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 85 1 βˆ’ βˆ’ PRON cana-4840 85 2 2𝑛 2𝑛 NOUN cana-4840 85 3 + + CCONJ cana-4840 85 4 5 5 NUM cana-4840 85 5 2 2 NUM cana-4840 85 6 βˆ’ βˆ’ NOUN cana-4840 85 7 ( ( PUNCT cana-4840 85 8 𝑛 𝑛 PROPN cana-4840 85 9 βˆ’ βˆ’ PROPN cana-4840 85 10 1)| 1)| NUM cana-4840 86 1 + + CCONJ cana-4840 87 1 | | ADV cana-4840 87 2 ( ( PUNCT cana-4840 87 3 𝑛 𝑛 PROPN cana-4840 87 4 βˆ’ βˆ’ PROPN cana-4840 87 5 1 1 NUM cana-4840 87 6 ) ) PUNCT cana-4840 87 7 βˆ’ βˆ’ PROPN cana-4840 87 8 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 88 1 βˆ’ βˆ’ PRON cana-4840 88 2 2𝑛 2𝑛 NOUN cana-4840 88 3 + + CCONJ cana-4840 88 4 5 5 NUM cana-4840 88 5 2 2 NUM cana-4840 88 6 βˆ’ βˆ’ NOUN cana-4840 88 7 ( ( PUNCT cana-4840 88 8 𝑛 𝑛 PROPN cana-4840 88 9 βˆ’ βˆ’ PROPN cana-4840 88 10 1)| 1)| NUM cana-4840 88 11 𝐿𝑄𝐷𝐸(𝐾𝑛 𝐿𝑄𝐷𝐸(𝐾𝑛 PROPN cana-4840 88 12 ) ) PUNCT cana-4840 88 13 = = PUNCT cana-4840 88 14 ( ( PUNCT cana-4840 88 15 𝑛 𝑛 PRON cana-4840 88 16 βˆ’ βˆ’ NOUN cana-4840 88 17 2 2 NUM cana-4840 88 18 ) ) PUNCT cana-4840 88 19 + + CCONJ cana-4840 88 20 | | ADV cana-4840 88 21 βˆ’π‘› βˆ’π‘› VERB cana-4840 88 22 + + CCONJ cana-4840 88 23 1 1 NUM cana-4840 88 24 + + CCONJ cana-4840 88 25 βˆšπ‘›2 βˆšπ‘›2 PROPN cana-4840 89 1 βˆ’ βˆ’ PROPN cana-4840 89 2 2𝑛 2𝑛 NOUN cana-4840 89 3 + + CCONJ cana-4840 89 4 5 5 NUM cana-4840 89 5 2 2 NUM cana-4840 90 1 | | ADV cana-4840 90 2 + + CCONJ cana-4840 90 3 | | ADV cana-4840 90 4 βˆ’π‘› βˆ’π‘› VERB cana-4840 90 5 + + CCONJ cana-4840 91 1 1 1 NUM cana-4840 91 2 βˆ’ βˆ’ NOUN cana-4840 91 3 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 91 4 βˆ’ βˆ’ PROPN cana-4840 91 5 2𝑛 2𝑛 NOUN cana-4840 91 6 + + CCONJ cana-4840 91 7 5 5 NUM cana-4840 91 8 2 2 NUM cana-4840 91 9 | | ADV cana-4840 91 10 therefore therefore ADV cana-4840 91 11 , , PUNCT cana-4840 91 12 𝐿𝑄𝐷𝐸(𝐾𝑛 𝐿𝑄𝐷𝐸(𝐾𝑛 PROPN cana-4840 91 13 ) ) PUNCT cana-4840 91 14 = = SYM cana-4840 91 15 ( ( PUNCT cana-4840 91 16 𝑛 𝑛 PRON cana-4840 91 17 βˆ’ βˆ’ NOUN cana-4840 91 18 2 2 NUM cana-4840 91 19 ) ) PUNCT cana-4840 91 20 + + CCONJ cana-4840 91 21 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 92 1 βˆ’ βˆ’ PRON cana-4840 92 2 2𝑛 2𝑛 PROPN cana-4840 92 3 + + CCONJ cana-4840 92 4 5 5 NUM cana-4840 92 5 theorem theorem NOUN cana-4840 92 6 . . PUNCT cana-4840 93 1 5.2 5.2 NUM cana-4840 93 2 . . PUNCT cana-4840 94 1 if if SCONJ cana-4840 94 2 𝐾1,π‘›βˆ’1 𝐾1,π‘›βˆ’1 PROPN cana-4840 94 3 is be AUX cana-4840 94 4 the the DET cana-4840 94 5 star star NOUN cana-4840 94 6 graph graph NOUN cana-4840 94 7 with with ADP cana-4840 94 8 𝑛 𝑛 DET cana-4840 94 9 node node NOUN cana-4840 94 10 , , PUNCT cana-4840 94 11 then then ADV cana-4840 94 12 𝐿𝑄𝐷𝐸(𝐺 𝐿𝑄𝐷𝐸(𝐺 NOUN cana-4840 94 13 ) ) PUNCT cana-4840 95 1 = = PUNCT cana-4840 95 2 ( ( PUNCT cana-4840 95 3 π‘›βˆ’2)2 π‘›βˆ’2)2 VERB cana-4840 95 4 𝑛 𝑛 VERB cana-4840 95 5 + + NUM cana-4840 95 6 βˆšπ‘›2 βˆšπ‘›2 PROPN cana-4840 95 7 βˆ’ βˆ’ PROPN cana-4840 95 8 2𝑛 2𝑛 NOUN cana-4840 95 9 + + CCONJ cana-4840 95 10 5 5 NUM cana-4840 95 11 proof proof NOUN cana-4840 95 12 : : PUNCT cana-4840 95 13 let let VERB cana-4840 95 14 𝐾1,π‘›βˆ’1 𝐾1,π‘›βˆ’1 PROPN cana-4840 95 15 be be AUX cana-4840 95 16 the the DET cana-4840 95 17 star star NOUN cana-4840 95 18 graph graph NOUN cana-4840 95 19 with with ADP cana-4840 95 20 node node NOUN cana-4840 95 21 set set VERB cana-4840 95 22 𝑉 𝑉 PROPN cana-4840 95 23 = = SYM cana-4840 95 24 { { PUNCT cana-4840 95 25 𝑣1 𝑣1 PROPN cana-4840 95 26 , , PUNCT cana-4840 95 27 𝑣2 𝑣2 PROPN cana-4840 95 28 … … PUNCT cana-4840 95 29 . . PUNCT cana-4840 96 1 , , PUNCT cana-4840 96 2 π‘£π‘›βˆ’1 π‘£π‘›βˆ’1 PROPN cana-4840 96 3 } } PUNCT cana-4840 96 4 . . PUNCT cana-4840 97 1 the the DET cana-4840 97 2 minimum minimum ADJ cana-4840 97 3 dominating dominating NOUN cana-4840 97 4 set set VERB cana-4840 97 5 𝐷 𝐷 NOUN cana-4840 97 6 = = SYM cana-4840 97 7 { { PUNCT cana-4840 97 8 𝑣1 𝑣1 NOUN cana-4840 97 9 } } PUNCT cana-4840 97 10 . . PUNCT cana-4840 98 1 and and CCONJ cana-4840 98 2 its its PRON cana-4840 98 3 characteristic characteristic ADJ cana-4840 98 4 polynomial polynomial NOUN cana-4840 98 5 is be AUX cana-4840 98 6 [ [ X cana-4840 98 7 πœ† πœ† X cana-4840 98 8 βˆ’ βˆ’ PROPN cana-4840 98 9 1](π‘›βˆ’2)[πœ†2 1](π‘›βˆ’2)[πœ†2 NOUN cana-4840 98 10 βˆ’ βˆ’ PROPN cana-4840 99 1 ( ( PUNCT cana-4840 99 2 𝑛 𝑛 DET cana-4840 99 3 βˆ’ βˆ’ PROPN cana-4840 99 4 1)πœ† 1)πœ† NUM cana-4840 99 5 βˆ’ βˆ’ PROPN cana-4840 99 6 1 1 NUM cana-4840 99 7 ] ] PUNCT cana-4840 99 8 the the DET cana-4840 99 9 laplacian laplacian ADJ cana-4840 99 10 minimum minimum NOUN cana-4840 99 11 dominating dominating NOUN cana-4840 99 12 quotient quotient NOUN cana-4840 99 13 eigen eigen PROPN cana-4840 99 14 values value NOUN cana-4840 99 15 are be AUX cana-4840 99 16 : : PUNCT cana-4840 99 17 communications communication NOUN cana-4840 99 18 on on ADP cana-4840 99 19 applied apply VERB cana-4840 99 20 nonlinear nonlinear ADJ cana-4840 99 21 analysis analysis NOUN cana-4840 99 22 issn issn NOUN cana-4840 99 23 : : PUNCT cana-4840 99 24 1074 1074 NUM cana-4840 99 25 - - PUNCT cana-4840 99 26 133x 133x NUM cana-4840 99 27 vol vol VERB cana-4840 99 28 32 32 NUM cana-4840 99 29 no no NOUN cana-4840 99 30 . . PUNCT cana-4840 100 1 10s 10 NOUN cana-4840 100 2 ( ( PUNCT cana-4840 100 3 2025 2025 NUM cana-4840 100 4 ) ) PUNCT cana-4840 100 5 450 450 NUM cana-4840 100 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 100 7 πœ† πœ† PUNCT cana-4840 100 8 = = SYM cana-4840 100 9 1[(𝑛 1[(𝑛 NUM cana-4840 100 10 βˆ’ βˆ’ NOUN cana-4840 100 11 2)π‘‘π‘–π‘šπ‘’ 2)π‘‘π‘–π‘šπ‘’ NUM cana-4840 100 12 ] ] PUNCT cana-4840 100 13 , , PUNCT cana-4840 100 14 πœ† πœ† X cana-4840 100 15 = = PUNCT cana-4840 100 16 ( ( PUNCT cana-4840 100 17 𝑛 𝑛 PRON cana-4840 100 18 βˆ’ βˆ’ PROPN cana-4840 100 19 1 1 NUM cana-4840 100 20 ) ) PUNCT cana-4840 100 21 Β± Β± NOUN cana-4840 100 22 βˆšπ‘›2 βˆšπ‘›2 PROPN cana-4840 100 23 βˆ’ βˆ’ PROPN cana-4840 100 24 2𝑛 2𝑛 NOUN cana-4840 101 1 + + CCONJ cana-4840 101 2 5 5 NUM cana-4840 101 3 2 2 NUM cana-4840 101 4 [ [ PUNCT cana-4840 101 5 π‘œπ‘›π‘’ π‘œπ‘›π‘’ NOUN cana-4840 101 6 π‘‘π‘–π‘šπ‘’ π‘‘π‘–π‘šπ‘’ VERB cana-4840 101 7 π‘’π‘Žπ‘β„Ž π‘’π‘Žπ‘β„Ž PROPN cana-4840 101 8 ] ] PUNCT cana-4840 101 9 average average ADJ cana-4840 101 10 degree degree NOUN cana-4840 101 11 of of ADP cana-4840 101 12 𝐾1,π‘›βˆ’1 𝐾1,π‘›βˆ’1 NOUN cana-4840 101 13 = = SYM cana-4840 101 14 2π‘š 2π‘š NOUN cana-4840 101 15 𝑛 𝑛 NOUN cana-4840 101 16 = = SYM cana-4840 101 17 2(π‘›βˆ’1 2(π‘›βˆ’1 NOUN cana-4840 101 18 ) ) PUNCT cana-4840 102 1 𝑛 𝑛 PRON cana-4840 102 2 hence hence ADV cana-4840 102 3 , , PUNCT cana-4840 102 4 the the DET cana-4840 102 5 laplacian laplacian ADJ cana-4840 102 6 minimum minimum NOUN cana-4840 102 7 dominating dominating NOUN cana-4840 102 8 quotient quotient NOUN cana-4840 102 9 energy energy NOUN cana-4840 102 10 of of ADP cana-4840 102 11 𝐾1,π‘›βˆ’1 𝐾1,π‘›βˆ’1 PROPN cana-4840 102 12 is be AUX cana-4840 102 13 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1 PROPN cana-4840 102 14 ) ) PUNCT cana-4840 103 1 = = PRON cana-4840 103 2 |𝑛 |𝑛 X cana-4840 104 1 βˆ’ βˆ’ NOUN cana-4840 104 2 2(𝑛 2(𝑛 NUM cana-4840 104 3 βˆ’ βˆ’ PROPN cana-4840 104 4 1 1 NUM cana-4840 104 5 ) ) PUNCT cana-4840 104 6 𝑛 𝑛 NOUN cana-4840 105 1 | | ADV cana-4840 105 2 ( ( PUNCT cana-4840 105 3 𝑛 𝑛 PROPN cana-4840 105 4 βˆ’ βˆ’ PROPN cana-4840 105 5 2 2 NUM cana-4840 105 6 ) ) PUNCT cana-4840 105 7 + + CCONJ cana-4840 106 1 | | ADV cana-4840 106 2 ( ( PUNCT cana-4840 106 3 𝑛 𝑛 PROPN cana-4840 106 4 βˆ’ βˆ’ NOUN cana-4840 106 5 1 1 NUM cana-4840 106 6 ) ) PUNCT cana-4840 106 7 + + CCONJ cana-4840 106 8 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 107 1 βˆ’ βˆ’ PRON cana-4840 107 2 2𝑛 2𝑛 NOUN cana-4840 107 3 + + CCONJ cana-4840 107 4 5 5 NUM cana-4840 107 5 2 2 NUM cana-4840 107 6 βˆ’ βˆ’ NOUN cana-4840 107 7 2(𝑛 2(𝑛 NUM cana-4840 107 8 βˆ’ βˆ’ NOUN cana-4840 107 9 1 1 NUM cana-4840 107 10 ) ) PUNCT cana-4840 107 11 𝑛 𝑛 NOUN cana-4840 108 1 | | ADV cana-4840 109 1 + + CCONJ cana-4840 109 2 | | ADV cana-4840 109 3 ( ( PUNCT cana-4840 109 4 𝑛 𝑛 PROPN cana-4840 109 5 βˆ’ βˆ’ PROPN cana-4840 109 6 1 1 NUM cana-4840 109 7 ) ) PUNCT cana-4840 109 8 βˆ’ βˆ’ PROPN cana-4840 109 9 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 110 1 βˆ’ βˆ’ PRON cana-4840 110 2 2𝑛 2𝑛 NOUN cana-4840 110 3 + + CCONJ cana-4840 110 4 5 5 NUM cana-4840 110 5 2 2 NUM cana-4840 110 6 βˆ’ βˆ’ NOUN cana-4840 110 7 2(𝑛 2(𝑛 NUM cana-4840 110 8 βˆ’ βˆ’ NOUN cana-4840 110 9 1 1 NUM cana-4840 110 10 ) ) PUNCT cana-4840 110 11 𝑛 𝑛 DET cana-4840 110 12 | | PROPN cana-4840 110 13 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1 PROPN cana-4840 110 14 ) ) PUNCT cana-4840 111 1 = = PUNCT cana-4840 111 2 ( ( PUNCT cana-4840 111 3 𝑛 𝑛 PRON cana-4840 111 4 βˆ’ βˆ’ PROPN cana-4840 111 5 2)2 2)2 NUM cana-4840 111 6 𝑛 𝑛 NOUN cana-4840 112 1 + + CCONJ cana-4840 112 2 | | ADV cana-4840 112 3 βˆ’π‘› βˆ’π‘› VERB cana-4840 112 4 + + CCONJ cana-4840 112 5 1 1 NUM cana-4840 112 6 + + CCONJ cana-4840 112 7 βˆšπ‘›2 βˆšπ‘›2 PROPN cana-4840 113 1 βˆ’ βˆ’ PROPN cana-4840 113 2 2𝑛 2𝑛 NOUN cana-4840 113 3 + + CCONJ cana-4840 113 4 5 5 NUM cana-4840 113 5 2 2 NUM cana-4840 114 1 | | ADV cana-4840 114 2 + + CCONJ cana-4840 114 3 | | ADV cana-4840 114 4 βˆ’π‘› βˆ’π‘› VERB cana-4840 114 5 + + CCONJ cana-4840 115 1 1 1 NUM cana-4840 115 2 βˆ’ βˆ’ NOUN cana-4840 115 3 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-4840 115 4 βˆ’ βˆ’ PROPN cana-4840 115 5 2𝑛 2𝑛 NOUN cana-4840 115 6 + + CCONJ cana-4840 115 7 5 5 NUM cana-4840 115 8 2 2 NUM cana-4840 116 1 | | ADV cana-4840 116 2 therefore therefore ADV cana-4840 116 3 , , PUNCT cana-4840 116 4 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1 PROPN cana-4840 116 5 ) ) PUNCT cana-4840 117 1 = = PUNCT cana-4840 117 2 ( ( PUNCT cana-4840 117 3 π‘›βˆ’2)2 π‘›βˆ’2)2 VERB cana-4840 117 4 𝑛 𝑛 VERB cana-4840 117 5 + + NUM cana-4840 117 6 βˆšπ‘›2 βˆšπ‘›2 PROPN cana-4840 118 1 βˆ’ βˆ’ PROPN cana-4840 118 2 2𝑛 2𝑛 NOUN cana-4840 118 3 + + CCONJ cana-4840 118 4 5 5 NUM cana-4840 118 5 6 6 NUM cana-4840 118 6 . . PUNCT cana-4840 118 7 bounds bound NOUN cana-4840 118 8 on on ADP cana-4840 118 9 laplacian laplacian ADJ cana-4840 118 10 minimum minimum ADJ cana-4840 118 11 dominating dominating NOUN cana-4840 118 12 quotient quotient NOUN cana-4840 118 13 energy energy NOUN cana-4840 118 14 of of ADP cana-4840 118 15 graphs graph NOUN cana-4840 118 16 theorem theorem VERB cana-4840 118 17 6.1 6.1 NUM cana-4840 118 18 . . PUNCT cana-4840 119 1 let let VERB cana-4840 119 2 𝐷 𝐷 NOUN cana-4840 119 3 be be AUX cana-4840 119 4 a a DET cana-4840 119 5 minimum minimum ADJ cana-4840 119 6 dominating dominating NOUN cana-4840 119 7 set set NOUN cana-4840 119 8 of of ADP cana-4840 119 9 a a DET cana-4840 119 10 graph graph NOUN cana-4840 119 11 𝐺 𝐺 NOUN cana-4840 119 12 and and CCONJ cana-4840 119 13 Ξ»1 Ξ»1 ADJ cana-4840 119 14 , , PUNCT cana-4840 119 15 Ξ»2 Ξ»2 NOUN cana-4840 119 16 , , PUNCT cana-4840 119 17 . . PUNCT cana-4840 119 18 . . PUNCT cana-4840 120 1 . . PUNCT cana-4840 121 1 , , PUNCT cana-4840 121 2 Ξ»n Ξ»n PROPN cana-4840 121 3 are be AUX cana-4840 121 4 the the DET cana-4840 121 5 eigenvalues eigenvalue NOUN cana-4840 121 6 of of ADP cana-4840 121 7 lq𝐷(g lq𝐷(g PROPN cana-4840 121 8 ) ) PUNCT cana-4840 121 9 then then ADV cana-4840 121 10 ( ( PUNCT cana-4840 121 11 𝑖 𝑖 X cana-4840 121 12 ) ) PUNCT cana-4840 121 13 βˆ‘πœ†π‘– βˆ‘πœ†π‘– PUNCT cana-4840 122 1 = = PUNCT cana-4840 123 1 2|𝐸| 2|𝐸| NUM cana-4840 123 2 βˆ’ βˆ’ NOUN cana-4840 123 3 |𝐷| |𝐷| NOUN cana-4840 123 4 𝑛 𝑛 X cana-4840 123 5 𝑖=1 𝑖=1 PROPN cana-4840 123 6 ( ( PUNCT cana-4840 123 7 𝑖𝑖 𝑖𝑖 NOUN cana-4840 123 8 ) ) PUNCT cana-4840 123 9 βˆ‘πœ†π‘– βˆ‘πœ†π‘– PUNCT cana-4840 123 10 2 2 NUM cana-4840 123 11 = = SYM cana-4840 123 12 2|𝐸| 2|𝐸| NUM cana-4840 123 13 + + NOUN cana-4840 123 14 βˆ‘(𝑑𝑖 βˆ‘(𝑑𝑖 PROPN cana-4840 123 15 βˆ’ βˆ’ PROPN cana-4840 123 16 β„Žπ‘– β„Žπ‘– NOUN cana-4840 123 17 ) ) PUNCT cana-4840 123 18 2 2 NUM cana-4840 123 19 𝑛 𝑛 NOUN cana-4840 123 20 𝑖=1 𝑖=1 PUNCT cana-4840 123 21 𝑛 𝑛 PRON cana-4840 123 22 𝑖=1 𝑖=1 PROPN cana-4840 123 23 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN cana-4840 123 24 β„Žπ‘– β„Žπ‘– NOUN cana-4840 123 25 = = PUNCT cana-4840 123 26 { { PUNCT cana-4840 123 27 1 1 NUM cana-4840 123 28 , , PUNCT cana-4840 123 29 𝑖𝑓 𝑖𝑓 NUM cana-4840 123 30 𝑣𝑖 𝑣𝑖 ADP cana-4840 123 31 ∈ ∈ PROPN cana-4840 123 32 𝐷 𝐷 PROPN cana-4840 123 33 0 0 NUM cana-4840 123 34 , , PUNCT cana-4840 123 35 𝑖𝑓 𝑖𝑓 NUM cana-4840 123 36 𝑣𝑖 𝑣𝑖 ADP cana-4840 123 37 βˆ‰ βˆ‰ PROPN cana-4840 123 38 𝐷 𝐷 PROPN cana-4840 123 39 proof proof NOUN cana-4840 123 40 : : PUNCT cana-4840 123 41 ( ( PUNCT cana-4840 123 42 i i NOUN cana-4840 123 43 ) ) PUNCT cana-4840 123 44 by by ADP cana-4840 123 45 definition definition NOUN cana-4840 123 46 , , PUNCT cana-4840 123 47 the the DET cana-4840 123 48 sum sum NOUN cana-4840 123 49 of of ADP cana-4840 123 50 the the DET cana-4840 123 51 principal principal ADJ cana-4840 123 52 diagonal diagonal ADJ cana-4840 123 53 elements element NOUN cana-4840 123 54 of of ADP cana-4840 123 55 lq𝐷(g lq𝐷(g PROPN cana-4840 123 56 ) ) PUNCT cana-4840 123 57 is be AUX cana-4840 123 58 equal equal ADJ cana-4840 123 59 to to ADP cana-4840 123 60 βˆ‘di βˆ‘di NOUN cana-4840 123 61 βˆ’ βˆ’ PROPN cana-4840 123 62 |d| |d| PROPN cana-4840 123 63 = = PUNCT cana-4840 124 1 2|e| 2|e| NUM cana-4840 124 2 βˆ’ βˆ’ NUM cana-4840 124 3 |d| |d| PROPN cana-4840 124 4 n n CCONJ cana-4840 124 5 i=1 i=1 PROPN cana-4840 124 6 also also ADV cana-4840 124 7 , , PUNCT cana-4840 124 8 the the DET cana-4840 124 9 sum sum NOUN cana-4840 124 10 of of ADP cana-4840 124 11 the the DET cana-4840 124 12 eigenvalues eigenvalue NOUN cana-4840 124 13 of of ADP cana-4840 124 14 the the DET cana-4840 124 15 matrix matrix NOUN cana-4840 124 16 lq𝐷(g lq𝐷(g PROPN cana-4840 124 17 ) ) PUNCT cana-4840 124 18 is be AUX cana-4840 124 19 equal equal ADJ cana-4840 124 20 to to ADP cana-4840 124 21 the the DET cana-4840 124 22 tracelq𝐷(g tracelq𝐷(g PROPN cana-4840 124 23 ) ) PUNCT cana-4840 124 24 . . PUNCT cana-4840 125 1 ( ( PUNCT cana-4840 125 2 ii ii X cana-4840 125 3 ) ) PUNCT cana-4840 125 4 the the DET cana-4840 125 5 result result NOUN cana-4840 125 6 that that SCONJ cana-4840 125 7 the the DET cana-4840 125 8 sum sum NOUN cana-4840 125 9 of of ADP cana-4840 125 10 the the DET cana-4840 125 11 squares square NOUN cana-4840 125 12 of of ADP cana-4840 125 13 the the DET cana-4840 125 14 eigenvalues eigenvalue NOUN cana-4840 125 15 of of ADP cana-4840 125 16 lq𝐷(g lq𝐷(g PROPN cana-4840 125 17 ) ) PUNCT cana-4840 125 18 is be AUX cana-4840 125 19 equal equal ADJ cana-4840 125 20 to to ADP cana-4840 125 21 the the DET cana-4840 125 22 trace trace NOUN cana-4840 125 23 of of ADP cana-4840 125 24 lq𝐷(g lq𝐷(g PROPN cana-4840 125 25 ) ) PUNCT cana-4840 125 26 2 2 NUM cana-4840 125 27 is be AUX cana-4840 125 28 a a DET cana-4840 125 29 direct direct ADJ cana-4840 125 30 application application NOUN cana-4840 125 31 of of ADP cana-4840 125 32 a a DET cana-4840 125 33 general general ADJ cana-4840 125 34 property property NOUN cana-4840 125 35 of of ADP cana-4840 125 36 matrices matrix NOUN cana-4840 125 37 . . PUNCT cana-4840 126 1 therefore therefore ADV cana-4840 126 2 , , PUNCT cana-4840 126 3 βˆ‘Ξ»i βˆ‘Ξ»i ADV cana-4840 126 4 2 2 NUM cana-4840 126 5 = = NOUN cana-4840 126 6 βˆ‘βˆ‘lijlji βˆ‘βˆ‘lijlji PUNCT cana-4840 126 7 n n NOUN cana-4840 126 8 j=1 j=1 NOUN cana-4840 126 9 = = SYM cana-4840 126 10 βˆ‘(lij βˆ‘(lij PROPN cana-4840 126 11 ) ) PUNCT cana-4840 126 12 2 2 NUM cana-4840 126 13 n n CCONJ cana-4840 126 14 i=1 i=1 X cana-4840 127 1 + + ADJ cana-4840 127 2 βˆ‘(lji βˆ‘(lji PROPN cana-4840 127 3 ) ) PUNCT cana-4840 127 4 2 2 NUM cana-4840 127 5 n n NOUN cana-4840 127 6 j=1 j=1 PROPN cana-4840 127 7 n n CCONJ cana-4840 127 8 i=1 i=1 PROPN cana-4840 127 9 n n CCONJ cana-4840 127 10 i=1 i=1 PROPN cana-4840 127 11 = = SYM cana-4840 127 12 2βˆ‘(lij 2βˆ‘(lij NUM cana-4840 127 13 ) ) PUNCT cana-4840 127 14 2 2 NUM cana-4840 127 15 n n NOUN cana-4840 128 1 i i X cana-4840 128 2 < < X cana-4840 128 3 j j PROPN cana-4840 128 4 + + ADJ cana-4840 128 5 βˆ‘(lji βˆ‘(lji PROPN cana-4840 128 6 ) ) PUNCT cana-4840 128 7 2 2 NUM cana-4840 128 8 n n NOUN cana-4840 128 9 j=1 j=1 NOUN cana-4840 128 10 βˆ‘Ξ»i βˆ‘Ξ»i PUNCT cana-4840 129 1 2 2 NUM cana-4840 129 2 = = SYM cana-4840 129 3 2|e| 2|e| NUM cana-4840 130 1 + + NOUN cana-4840 130 2 βˆ‘(di βˆ‘(di ADJ cana-4840 130 3 βˆ’ βˆ’ PROPN cana-4840 130 4 hi hi NOUN cana-4840 130 5 ) ) PUNCT cana-4840 130 6 2 2 NUM cana-4840 130 7 n n NOUN cana-4840 130 8 i=1 i=1 PROPN cana-4840 130 9 n n X cana-4840 130 10 i=1 i=1 PROPN cana-4840 130 11 where where SCONJ cana-4840 130 12 hi hi INTJ cana-4840 130 13 = = PUNCT cana-4840 130 14 { { PUNCT cana-4840 130 15 1 1 NUM cana-4840 130 16 , , PUNCT cana-4840 130 17 if if SCONJ cana-4840 130 18 vi vi ADP cana-4840 130 19 ∈ ∈ PROPN cana-4840 130 20 d d NOUN cana-4840 130 21 0 0 PROPN cana-4840 130 22 , , PUNCT cana-4840 130 23 if if SCONJ cana-4840 130 24 vi vi PROPN cana-4840 130 25 βˆ‰ βˆ‰ PROPN cana-4840 130 26 d d PROPN cana-4840 130 27 communications communication NOUN cana-4840 130 28 on on ADP cana-4840 130 29 applied apply VERB cana-4840 130 30 nonlinear nonlinear ADJ cana-4840 130 31 analysis analysis NOUN cana-4840 130 32 issn issn NOUN cana-4840 130 33 : : PUNCT cana-4840 130 34 1074 1074 NUM cana-4840 130 35 - - PUNCT cana-4840 130 36 133x 133x NUM cana-4840 130 37 vol vol VERB cana-4840 130 38 32 32 NUM cana-4840 130 39 no no NOUN cana-4840 130 40 . . PUNCT cana-4840 131 1 10s 10 NOUN cana-4840 131 2 ( ( PUNCT cana-4840 131 3 2025 2025 NUM cana-4840 131 4 ) ) PUNCT cana-4840 131 5 451 451 NUM cana-4840 131 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 131 7 βˆ‘Ξ»i βˆ‘Ξ»i SYM cana-4840 131 8 2 2 NUM cana-4840 131 9 = = SYM cana-4840 131 10 2n 2n NUM cana-4840 132 1 where where SCONJ cana-4840 132 2 n n AUX cana-4840 132 3 = = SYM cana-4840 132 4 |e| |e| PRON cana-4840 132 5 + + CCONJ cana-4840 132 6 1 1 NUM cana-4840 132 7 2 2 NUM cana-4840 132 8 βˆ‘(di βˆ‘(di NOUN cana-4840 132 9 βˆ’ βˆ’ NOUN cana-4840 132 10 hi hi INTJ cana-4840 132 11 ) ) PUNCT cana-4840 132 12 2 2 NUM cana-4840 132 13 n n NOUN cana-4840 132 14 i=1 i=1 PROPN cana-4840 132 15 n n CCONJ cana-4840 132 16 i=1 i=1 PROPN cana-4840 132 17 theorem theorem ADJ cana-4840 132 18 6.2 6.2 NUM cana-4840 132 19 . . PUNCT cana-4840 133 1 given give VERB cana-4840 133 2 the the DET cana-4840 133 3 graph graph NOUN cana-4840 133 4 g g NOUN cana-4840 133 5 with with ADP cana-4840 133 6 n n ADP cana-4840 133 7 vertices vertex NOUN cana-4840 133 8 and and CCONJ cana-4840 133 9 m m NOUN cana-4840 133 10 edges edge NOUN cana-4840 133 11 , , PUNCT cana-4840 133 12 and and CCONJ cana-4840 133 13 d d NOUN cana-4840 133 14 is be AUX cana-4840 133 15 a a DET cana-4840 133 16 minimum minimum ADJ cana-4840 133 17 dominating dominating NOUN cana-4840 133 18 set set NOUN cana-4840 133 19 of of ADP cana-4840 133 20 g g PROPN cana-4840 133 21 then then ADV cana-4840 133 22 lq𝐷(g lq𝐷(g PROPN cana-4840 133 23 ) ) PUNCT cana-4840 133 24 ≀ ≀ NOUN cana-4840 133 25 √2nn √2nn VERB cana-4840 133 26 + + CCONJ cana-4840 133 27 2 2 NUM cana-4840 133 28 m m NOUN cana-4840 133 29 proof proof NOUN cana-4840 133 30 : : PUNCT cana-4840 133 31 given give VERB cana-4840 133 32 a a DET cana-4840 133 33 graph graph NOUN cana-4840 133 34 𝐺 𝐺 NOUN cana-4840 133 35 with with ADP cana-4840 133 36 𝑛 𝑛 PROPN cana-4840 133 37 vertices vertex NOUN cana-4840 133 38 and and CCONJ cana-4840 133 39 π‘š π‘š ADP cana-4840 133 40 edges edge NOUN cana-4840 133 41 , , PUNCT cana-4840 133 42 and and CCONJ cana-4840 133 43 the the DET cana-4840 133 44 eigenvalues eigenvalue NOUN cana-4840 133 45 Ξ»1 Ξ»1 ADJ cana-4840 133 46 , , PUNCT cana-4840 133 47 Ξ»2 Ξ»2 NOUN cana-4840 133 48 , , PUNCT cana-4840 133 49 . . PUNCT cana-4840 133 50 . . PUNCT cana-4840 133 51 . . PUNCT cana-4840 133 52 . . PUNCT cana-4840 134 1 . . PUNCT cana-4840 135 1 , , PUNCT cana-4840 135 2 Ξ»n Ξ»n NOUN cana-4840 135 3 of of ADP cana-4840 135 4 the the DET cana-4840 135 5 laplacian laplacian ADJ cana-4840 135 6 matrix matrix NOUN cana-4840 135 7 lq𝐷(g lq𝐷(g PROPN cana-4840 135 8 ) ) PUNCT cana-4840 135 9 . . PUNCT cana-4840 136 1 by by ADP cana-4840 136 2 using use VERB cana-4840 136 3 cauchy cauchy PROPN cana-4840 136 4 ’s ’s PART cana-4840 136 5 schwarz schwarz PROPN cana-4840 136 6 inequality inequality NOUN cana-4840 136 7 we we PRON cana-4840 136 8 have have VERB cana-4840 136 9 , , PUNCT cana-4840 136 10 ( ( PUNCT cana-4840 136 11 βˆ‘aibi βˆ‘aibi PROPN cana-4840 136 12 n n CCONJ cana-4840 136 13 i=1 i=1 PROPN cana-4840 136 14 ) ) PUNCT cana-4840 136 15 2 2 NUM cana-4840 136 16 ≀ ≀ NOUN cana-4840 136 17 ( ( PUNCT cana-4840 136 18 βˆ‘ai βˆ‘ai NUM cana-4840 136 19 2 2 NUM cana-4840 136 20 n n NOUN cana-4840 136 21 i=1 i=1 PRON cana-4840 136 22 ) ) PUNCT cana-4840 136 23 ( ( PUNCT cana-4840 136 24 βˆ‘bi βˆ‘bi NOUN cana-4840 136 25 2 2 NUM cana-4840 136 26 n n NOUN cana-4840 136 27 i=1 i=1 PRON cana-4840 136 28 ) ) PUNCT cana-4840 136 29 put put VERB cana-4840 136 30 ai ai NOUN cana-4840 136 31 = = NOUN cana-4840 136 32 1 1 NUM cana-4840 136 33 , , PUNCT cana-4840 136 34 bi bi NOUN cana-4840 137 1 = = NOUN cana-4840 137 2 Ξ»i Ξ»i X cana-4840 137 3 then then ADV cana-4840 137 4 , , PUNCT cana-4840 137 5 ( ( PUNCT cana-4840 137 6 βˆ‘|Ξ»i| βˆ‘|Ξ»i| NOUN cana-4840 137 7 n n X cana-4840 137 8 i=1 i=1 PROPN cana-4840 137 9 ) ) PUNCT cana-4840 137 10 2 2 NUM cana-4840 137 11 ≀ ≀ NOUN cana-4840 137 12 ( ( PUNCT cana-4840 137 13 βˆ‘1 βˆ‘1 NOUN cana-4840 137 14 n n PRON cana-4840 137 15 i=1 i=1 PROPN cana-4840 137 16 ) ) PUNCT cana-4840 138 1 ( ( PUNCT cana-4840 138 2 βˆ‘|Ξ»i| βˆ‘|Ξ»i| NOUN cana-4840 138 3 2 2 NUM cana-4840 138 4 n n CCONJ cana-4840 138 5 i=1 i=1 PROPN cana-4840 138 6 ) ) PUNCT cana-4840 139 1 ( ( PUNCT cana-4840 139 2 βˆ‘|Ξ»i| βˆ‘|Ξ»i| NOUN cana-4840 139 3 n n X cana-4840 139 4 i=1 i=1 PROPN cana-4840 139 5 ) ) PUNCT cana-4840 139 6 2 2 NUM cana-4840 139 7 ≀ ≀ NOUN cana-4840 139 8 ( ( PUNCT cana-4840 139 9 n)(2n n)(2n NOUN cana-4840 139 10 ) ) PUNCT cana-4840 139 11 ∴ ∴ NOUN cana-4840 139 12 ( ( PUNCT cana-4840 139 13 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 139 14 n n CCONJ cana-4840 139 15 i=1 i=1 PROPN cana-4840 139 16 ) ) PUNCT cana-4840 139 17 ≀ ≀ PUNCT cana-4840 140 1 √2nn √2nn VERB cana-4840 140 2 by by ADP cana-4840 140 3 triangle triangle NOUN cana-4840 140 4 inequality inequality NOUN cana-4840 140 5 , , PUNCT cana-4840 140 6 |Ξ»i |Ξ»i PROPN cana-4840 140 7 βˆ’ βˆ’ PROPN cana-4840 140 8 2 2 NUM cana-4840 140 9 m m VERB cana-4840 140 10 n n NUM cana-4840 140 11 | | ADV cana-4840 140 12 ≀ ≀ NUM cana-4840 140 13 |Ξ»i| |Ξ»i| NOUN cana-4840 141 1 + + CCONJ cana-4840 142 1 | | ADV cana-4840 142 2 2 2 NUM cana-4840 142 3 m m NOUN cana-4840 142 4 n n PRON cana-4840 142 5 |βˆ€ |βˆ€ NOUN cana-4840 142 6 i i NOUN cana-4840 142 7 = = SYM cana-4840 142 8 1,2 1,2 NUM cana-4840 142 9 , , PUNCT cana-4840 142 10 . . PUNCT cana-4840 142 11 . . PUNCT cana-4840 143 1 . . PUNCT cana-4840 144 1 , , PUNCT cana-4840 144 2 n n CCONJ cana-4840 144 3 𝑖. 𝑖. ADV cana-4840 144 4 𝑒. 𝑒. NOUN cana-4840 144 5 , , PUNCT cana-4840 144 6 |Ξ»i |Ξ»i PROPN cana-4840 144 7 βˆ’ βˆ’ PROPN cana-4840 144 8 2 2 NUM cana-4840 144 9 m m VERB cana-4840 144 10 n n NUM cana-4840 144 11 | | ADV cana-4840 144 12 ≀ ≀ NUM cana-4840 144 13 |Ξ»i| |Ξ»i| NOUN cana-4840 144 14 + + CCONJ cana-4840 144 15 2 2 NUM cana-4840 144 16 m m NOUN cana-4840 144 17 n n NUM cana-4840 144 18 βˆ€ βˆ€ NOUN cana-4840 145 1 i i NOUN cana-4840 145 2 ( ( PUNCT cana-4840 145 3 βˆ‘ βˆ‘ PUNCT cana-4840 145 4 |Ξ»i |Ξ»i PROPN cana-4840 145 5 βˆ’ βˆ’ PROPN cana-4840 145 6 2 2 NUM cana-4840 145 7 m m NOUN cana-4840 145 8 n n NUM cana-4840 145 9 | | ADV cana-4840 145 10 n n CCONJ cana-4840 145 11 i=1 i=1 PROPN cana-4840 145 12 ) ) PUNCT cana-4840 145 13 ≀ ≀ NOUN cana-4840 145 14 ( ( PUNCT cana-4840 145 15 βˆ‘Ξ»i βˆ‘Ξ»i CCONJ cana-4840 145 16 n n CCONJ cana-4840 145 17 i=1 i=1 PROPN cana-4840 145 18 ) ) PUNCT cana-4840 145 19 ( ( PUNCT cana-4840 145 20 βˆ‘ βˆ‘ PROPN cana-4840 145 21 2 2 NUM cana-4840 145 22 m m NOUN cana-4840 145 23 n n ADP cana-4840 145 24 n n NOUN cana-4840 145 25 i=1 i=1 PROPN cana-4840 145 26 ) ) PUNCT cana-4840 145 27 ≀ ≀ PROPN cana-4840 145 28 √2𝑁𝑛 √2𝑁𝑛 NOUN cana-4840 145 29 + + CCONJ cana-4840 145 30 2 2 NUM cana-4840 145 31 m m NOUN cana-4840 145 32 lq𝐷(g lq𝐷(g PROPN cana-4840 145 33 ) ) PUNCT cana-4840 145 34 ≀ ≀ PUNCT cana-4840 145 35 √2𝑁𝑛 √2𝑁𝑛 NOUN cana-4840 146 1 + + X cana-4840 146 2 2 2 NUM cana-4840 146 3 m m NOUN cana-4840 146 4 theorem theorem VERB cana-4840 146 5 6.3 6.3 NUM cana-4840 146 6 . . PUNCT cana-4840 147 1 given give VERB cana-4840 147 2 the the DET cana-4840 147 3 graph graph NOUN cana-4840 147 4 g g NOUN cana-4840 147 5 with with ADP cana-4840 147 6 n n ADP cana-4840 147 7 vertices vertex NOUN cana-4840 147 8 and and CCONJ cana-4840 147 9 m m NOUN cana-4840 147 10 edges edge NOUN cana-4840 147 11 , , PUNCT cana-4840 147 12 and and CCONJ cana-4840 147 13 d d NOUN cana-4840 147 14 is be AUX cana-4840 147 15 a a DET cana-4840 147 16 minimum minimum ADJ cana-4840 147 17 dominating dominating NOUN cana-4840 147 18 set set NOUN cana-4840 147 19 of of ADP cana-4840 147 20 g g PROPN cana-4840 147 21 and and CCONJ cana-4840 147 22 if if SCONJ cana-4840 147 23 d d PROPN cana-4840 147 24 = = SYM cana-4840 147 25 |det |det NOUN cana-4840 147 26 lepe(g lepe(g NOUN cana-4840 147 27 ) ) PUNCT cana-4840 147 28 | | ADV cana-4840 147 29 then then ADV cana-4840 147 30 lq𝐷(g lq𝐷(g PROPN cana-4840 147 31 ) ) PUNCT cana-4840 147 32 β‰₯ β‰₯ NOUN cana-4840 147 33 √2n √2n NUM cana-4840 147 34 + + PUNCT cana-4840 147 35 n(n n(n PROPN cana-4840 147 36 βˆ’ βˆ’ NOUN cana-4840 147 37 1)d 1)d NUM cana-4840 147 38 2 2 NUM cana-4840 147 39 𝑛 𝑛 PRON cana-4840 147 40 βˆ’ βˆ’ PROPN cana-4840 147 41 2 2 NUM cana-4840 147 42 m. m. NOUN cana-4840 147 43 proof proof NOUN cana-4840 147 44 : : PUNCT cana-4840 147 45 consider consider VERB cana-4840 147 46 ( ( PUNCT cana-4840 147 47 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 147 48 n n X cana-4840 147 49 i=1 i=1 PROPN cana-4840 147 50 ) ) PUNCT cana-4840 147 51 2 2 NUM cana-4840 147 52 = = SYM cana-4840 147 53 ( ( PUNCT cana-4840 147 54 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 147 55 n n X cana-4840 147 56 i=1 i=1 PROPN cana-4840 147 57 ) ) PUNCT cana-4840 148 1 βˆ™ βˆ™ PROPN cana-4840 148 2 ( ( PUNCT cana-4840 148 3 βˆ‘|Ξ»j| βˆ‘|Ξ»j| NOUN cana-4840 148 4 n n CCONJ cana-4840 148 5 j=1 j=1 ADJ cana-4840 148 6 ) ) PUNCT cana-4840 148 7 communications communication NOUN cana-4840 148 8 on on ADP cana-4840 148 9 applied apply VERB cana-4840 148 10 nonlinear nonlinear ADJ cana-4840 148 11 analysis analysis NOUN cana-4840 148 12 issn issn NOUN cana-4840 148 13 : : PUNCT cana-4840 148 14 1074 1074 NUM cana-4840 148 15 - - PUNCT cana-4840 148 16 133x 133x NUM cana-4840 148 17 vol vol VERB cana-4840 148 18 32 32 NUM cana-4840 148 19 no no NOUN cana-4840 148 20 . . PUNCT cana-4840 149 1 10s 10 NOUN cana-4840 149 2 ( ( PUNCT cana-4840 149 3 2025 2025 NUM cana-4840 149 4 ) ) PUNCT cana-4840 149 5 452 452 NUM cana-4840 149 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 149 7 = = SYM cana-4840 149 8 βˆ‘|Ξ»i| βˆ‘|Ξ»i| NOUN cana-4840 149 9 2 2 NUM cana-4840 149 10 n n NOUN cana-4840 149 11 i=1 i=1 X cana-4840 150 1 + + ADJ cana-4840 150 2 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 150 3 iβ‰ j iβ‰ j PROPN cana-4840 150 4 |Ξ»j| |Ξ»j| PROPN cana-4840 150 5 βˆ΄βˆ‘|Ξ»i| βˆ΄βˆ‘|Ξ»i| PROPN cana-4840 150 6 iβ‰ j iβ‰ j NOUN cana-4840 150 7 |Ξ»j| |Ξ»j| NOUN cana-4840 150 8 = = PUNCT cana-4840 151 1 ( ( PUNCT cana-4840 151 2 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 151 3 n n X cana-4840 151 4 i=1 i=1 PROPN cana-4840 151 5 ) ) PUNCT cana-4840 151 6 2 2 NUM cana-4840 151 7 βˆ’βˆ‘|Ξ»i| βˆ’βˆ‘|Ξ»i| NOUN cana-4840 151 8 2 2 NUM cana-4840 151 9 n n NOUN cana-4840 151 10 i=1 i=1 PROPN cana-4840 151 11 ( ( PUNCT cana-4840 151 12 theorem theorem VERB cana-4840 151 13 6.1 6.1 NUM cana-4840 151 14 . . PUNCT cana-4840 151 15 ) ) PUNCT cana-4840 152 1 using use VERB cana-4840 152 2 am be AUX cana-4840 152 3 - - PUNCT cana-4840 152 4 gm gm NOUN cana-4840 152 5 inequality inequality NOUN cana-4840 152 6 for for ADP cana-4840 152 7 𝑛(𝑛 𝑛(𝑛 PROPN cana-4840 152 8 βˆ’ βˆ’ PROPN cana-4840 152 9 1 1 NUM cana-4840 152 10 ) ) PUNCT cana-4840 152 11 non non ADJ cana-4840 152 12 - - ADJ cana-4840 152 13 negative negative ADJ cana-4840 152 14 terms term NOUN cana-4840 152 15 shows show VERB cana-4840 152 16 that that SCONJ cana-4840 152 17 the the DET cana-4840 152 18 arithmetic arithmetic ADJ cana-4840 152 19 mean mean NOUN cana-4840 152 20 of of ADP cana-4840 152 21 these these DET cana-4840 152 22 terms term NOUN cana-4840 152 23 is be AUX cana-4840 152 24 atleast atleast VERB cana-4840 152 25 as as ADV cana-4840 152 26 large large ADJ cana-4840 152 27 as as ADP cana-4840 152 28 their their PRON cana-4840 152 29 geometric geometric ADJ cana-4840 152 30 mean mean NOUN cana-4840 152 31 and and CCONJ cana-4840 152 32 thus thus ADV cana-4840 152 33 it it PRON cana-4840 152 34 follows follow VERB cana-4840 152 35 that that SCONJ cana-4840 152 36 : : PUNCT cana-4840 152 37 βˆ‘ βˆ‘ PUNCT cana-4840 152 38 |Ξ»i| |Ξ»i| NOUN cana-4840 152 39 iβ‰ j iβ‰ j NOUN cana-4840 152 40 |Ξ»j| |Ξ»j| NOUN cana-4840 152 41 n(n n(n NOUN cana-4840 152 42 βˆ’ βˆ’ PROPN cana-4840 152 43 1 1 NUM cana-4840 152 44 ) ) PUNCT cana-4840 152 45 β‰₯ β‰₯ NOUN cana-4840 153 1 [ [ X cana-4840 153 2 ∏|Ξ»i| ∏|Ξ»i| PROPN cana-4840 153 3 |Ξ»j| |Ξ»j| NOUN cana-4840 153 4 iβ‰ j iβ‰ j NOUN cana-4840 153 5 ] ] PUNCT cana-4840 153 6 1 1 NUM cana-4840 153 7 n(nβˆ’1 n(nβˆ’1 NUM cana-4840 153 8 ) ) PUNCT cana-4840 153 9 𝑖. 𝑖. ADV cana-4840 153 10 𝑒. 𝑒. ADJ cana-4840 153 11 , , PUNCT cana-4840 153 12 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 153 13 iβ‰ j iβ‰ j PROPN cana-4840 153 14 |Ξ»j| |Ξ»j| PROPN cana-4840 153 15 β‰₯ β‰₯ NOUN cana-4840 153 16 n(n n(n NOUN cana-4840 153 17 βˆ’ βˆ’ PROPN cana-4840 153 18 1 1 NUM cana-4840 153 19 ) ) PUNCT cana-4840 154 1 [ [ X cana-4840 154 2 ∏|Ξ»i| ∏|Ξ»i| PROPN cana-4840 154 3 |Ξ»j| |Ξ»j| NOUN cana-4840 154 4 iβ‰ j iβ‰ j NOUN cana-4840 154 5 ] ] PUNCT cana-4840 154 6 1 1 NUM cana-4840 154 7 n(nβˆ’1 n(nβˆ’1 NUM cana-4840 154 8 ) ) PUNCT cana-4840 154 9 using use VERB cana-4840 154 10 theorem.6.1 theorem.6.1 PROPN cana-4840 154 11 we we PRON cana-4840 154 12 get get VERB cana-4840 154 13 , , PUNCT cana-4840 154 14 ( ( PUNCT cana-4840 154 15 βˆ‘|Ξ»i| βˆ‘|Ξ»i| NOUN cana-4840 154 16 n n X cana-4840 154 17 i=1 i=1 PROPN cana-4840 154 18 ) ) PUNCT cana-4840 154 19 2 2 NUM cana-4840 154 20 βˆ’βˆ‘|Ξ»i| βˆ’βˆ‘|Ξ»i| NOUN cana-4840 154 21 2 2 NUM cana-4840 154 22 β‰₯ β‰₯ NOUN cana-4840 154 23 n(n n(n NOUN cana-4840 154 24 βˆ’ βˆ’ PROPN cana-4840 154 25 1 1 NUM cana-4840 154 26 ) ) PUNCT cana-4840 154 27 n n NOUN cana-4840 154 28 i=1 i=1 X cana-4840 155 1 [ [ X cana-4840 155 2 ∏|Ξ»i| ∏|Ξ»i| PROPN cana-4840 155 3 2(nβˆ’1 2(nβˆ’1 NOUN cana-4840 155 4 ) ) PUNCT cana-4840 156 1 n n CCONJ cana-4840 156 2 i=1 i=1 X cana-4840 156 3 ] ] PUNCT cana-4840 156 4 1 1 NUM cana-4840 156 5 n(nβˆ’1 n(nβˆ’1 NUM cana-4840 156 6 ) ) PUNCT cana-4840 157 1 ( ( PUNCT cana-4840 157 2 βˆ‘|Ξ»i| βˆ‘|Ξ»i| NOUN cana-4840 157 3 n n X cana-4840 157 4 i=1 i=1 PROPN cana-4840 157 5 ) ) PUNCT cana-4840 157 6 2 2 NUM cana-4840 157 7 βˆ’ βˆ’ PROPN cana-4840 157 8 2n 2n NUM cana-4840 157 9 β‰₯ β‰₯ NUM cana-4840 157 10 n(n n(n NOUN cana-4840 157 11 βˆ’ βˆ’ PROPN cana-4840 157 12 1 1 NUM cana-4840 157 13 ) ) PUNCT cana-4840 158 1 [ [ X cana-4840 158 2 ∏|Ξ»i| ∏|Ξ»i| PROPN cana-4840 158 3 n n CCONJ cana-4840 158 4 i=1 i=1 X cana-4840 158 5 ] ] PUNCT cana-4840 158 6 2 2 NUM cana-4840 158 7 n n NOUN cana-4840 158 8 ( ( PUNCT cana-4840 158 9 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 158 10 n n X cana-4840 158 11 i=1 i=1 PROPN cana-4840 158 12 ) ) PUNCT cana-4840 158 13 2 2 NUM cana-4840 158 14 β‰₯ β‰₯ NOUN cana-4840 158 15 2n 2n NUM cana-4840 159 1 + + CCONJ cana-4840 159 2 n(n n(n NOUN cana-4840 159 3 βˆ’ βˆ’ NOUN cana-4840 159 4 1 1 NUM cana-4840 159 5 ) ) PUNCT cana-4840 160 1 [ [ X cana-4840 160 2 ∏|Ξ»i| ∏|Ξ»i| PROPN cana-4840 160 3 n n CCONJ cana-4840 160 4 i=1 i=1 X cana-4840 160 5 ] ] PUNCT cana-4840 160 6 2 2 NUM cana-4840 160 7 n n NOUN cana-4840 160 8 βˆ΄βˆ‘|Ξ»i| βˆ΄βˆ‘|Ξ»i| NOUN cana-4840 160 9 n n CCONJ cana-4840 160 10 i=1 i=1 PROPN cana-4840 160 11 β‰₯ β‰₯ VERB cana-4840 160 12 √2n √2n NUM cana-4840 160 13 + + PUNCT cana-4840 160 14 n(n n(n PROPN cana-4840 160 15 βˆ’ βˆ’ NOUN cana-4840 160 16 1)d 1)d NUM cana-4840 160 17 2 2 NUM cana-4840 160 18 𝑛 𝑛 PRON cana-4840 160 19 w.k.t w.k.t ADJ cana-4840 160 20 |Ξ»i| |Ξ»i| NOUN cana-4840 160 21 βˆ’ βˆ’ NOUN cana-4840 161 1 | | ADV cana-4840 161 2 2 2 NUM cana-4840 161 3 m m VERB cana-4840 161 4 n n NUM cana-4840 161 5 | | ADV cana-4840 161 6 ≀ ≀ CCONJ cana-4840 161 7 |Ξ»i |Ξ»i PROPN cana-4840 161 8 βˆ’ βˆ’ PROPN cana-4840 161 9 2 2 NUM cana-4840 161 10 m m NOUN cana-4840 161 11 n n PRON cana-4840 161 12 |βˆ€ |βˆ€ NOUN cana-4840 161 13 i i PRON cana-4840 161 14 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 161 15 n n PROPN cana-4840 161 16 i=1 i=1 PROPN cana-4840 161 17 βˆ’βˆ‘ βˆ’βˆ‘ PROPN cana-4840 161 18 2 2 NUM cana-4840 161 19 m m PROPN cana-4840 161 20 n n VERB cana-4840 161 21 n n NOUN cana-4840 161 22 i=1 i=1 NOUN cana-4840 162 1 β‰€βˆ‘|Ξ»i β‰€βˆ‘|Ξ»i NOUN cana-4840 162 2 βˆ’ βˆ’ PROPN cana-4840 162 3 2 2 NUM cana-4840 162 4 m m NOUN cana-4840 162 5 n n NUM cana-4840 162 6 | | ADV cana-4840 162 7 n n ADV cana-4840 162 8 i=1 i=1 PROPN cana-4840 162 9 βˆ‘|Ξ»i| βˆ‘|Ξ»i| PROPN cana-4840 163 1 n n PRON cana-4840 163 2 i=1 i=1 PROPN cana-4840 164 1 βˆ’ βˆ’ PROPN cana-4840 164 2 2 2 NUM cana-4840 164 3 m m PROPN cana-4840 164 4 ≀ ≀ NUM cana-4840 164 5 lq𝐷(g lq𝐷(g PROPN cana-4840 164 6 ) ) PUNCT cana-4840 164 7 lq𝐷(g lq𝐷(g PROPN cana-4840 164 8 ) ) PUNCT cana-4840 164 9 β‰₯βˆ‘|Ξ»i| β‰₯βˆ‘|Ξ»i| PROPN cana-4840 164 10 n n NOUN cana-4840 164 11 i=1 i=1 PROPN cana-4840 164 12 βˆ’ βˆ’ PROPN cana-4840 164 13 2 2 NUM cana-4840 164 14 m m PROPN cana-4840 164 15 communications communication NOUN cana-4840 164 16 on on ADP cana-4840 164 17 applied apply VERB cana-4840 164 18 nonlinear nonlinear ADJ cana-4840 164 19 analysis analysis NOUN cana-4840 164 20 issn issn NOUN cana-4840 164 21 : : PUNCT cana-4840 164 22 1074 1074 NUM cana-4840 164 23 - - PUNCT cana-4840 164 24 133x 133x NUM cana-4840 164 25 vol vol VERB cana-4840 164 26 32 32 NUM cana-4840 164 27 no no NOUN cana-4840 164 28 . . PUNCT cana-4840 165 1 10s 10 NOUN cana-4840 165 2 ( ( PUNCT cana-4840 165 3 2025 2025 NUM cana-4840 165 4 ) ) PUNCT cana-4840 165 5 453 453 NUM cana-4840 165 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 165 7 β‰₯ β‰₯ NOUN cana-4840 165 8 √2n √2n NUM cana-4840 165 9 + + PUNCT cana-4840 165 10 n(n n(n PROPN cana-4840 165 11 βˆ’ βˆ’ NOUN cana-4840 165 12 1)d 1)d NUM cana-4840 165 13 2 2 NUM cana-4840 165 14 𝑛 𝑛 DET cana-4840 165 15 βˆ’ βˆ’ PROPN cana-4840 165 16 2 2 NUM cana-4840 165 17 m m NOUN cana-4840 165 18 ∴ ∴ PROPN cana-4840 165 19 lq𝐷(g lq𝐷(g PROPN cana-4840 165 20 ) ) PUNCT cana-4840 165 21 β‰₯ β‰₯ NOUN cana-4840 165 22 √2n √2n NUM cana-4840 165 23 + + PUNCT cana-4840 165 24 n(n n(n PROPN cana-4840 165 25 βˆ’ βˆ’ NOUN cana-4840 165 26 1)d 1)d NUM cana-4840 165 27 2 2 NUM cana-4840 165 28 𝑛 𝑛 PRON cana-4840 165 29 βˆ’ βˆ’ PROPN cana-4840 165 30 2 2 NUM cana-4840 165 31 m m NOUN cana-4840 165 32 theorem theorem VERB cana-4840 165 33 6.4 6.4 NUM cana-4840 165 34 . . PUNCT cana-4840 166 1 given give VERB cana-4840 166 2 the the DET cana-4840 166 3 graph graph NOUN cana-4840 166 4 g g NOUN cana-4840 166 5 with with ADP cana-4840 166 6 n n ADP cana-4840 166 7 vertices vertex NOUN cana-4840 166 8 and and CCONJ cana-4840 166 9 m m NOUN cana-4840 166 10 edges edge NOUN cana-4840 166 11 , , PUNCT cana-4840 166 12 and and CCONJ cana-4840 166 13 d d NOUN cana-4840 166 14 is be AUX cana-4840 166 15 a a DET cana-4840 166 16 minimum minimum ADJ cana-4840 166 17 dominating dominating NOUN cana-4840 166 18 set set NOUN cana-4840 166 19 of of ADP cana-4840 166 20 g g PROPN cana-4840 166 21 then then ADV cana-4840 166 22 lq𝐷(g lq𝐷(g PROPN cana-4840 166 23 ) ) PUNCT cana-4840 166 24 ≀ ≀ PUNCT cana-4840 166 25 √2nn √2nn VERB cana-4840 166 26 + + CCONJ cana-4840 166 27 4m(|d| 4m(|d| PROPN cana-4840 166 28 βˆ’ βˆ’ PROPN cana-4840 166 29 m m NOUN cana-4840 166 30 ) ) PUNCT cana-4840 166 31 proof proof NOUN cana-4840 166 32 : : PUNCT cana-4840 166 33 by by ADP cana-4840 166 34 using use VERB cana-4840 166 35 cauchy cauchy PROPN cana-4840 166 36 ’s ’s PART cana-4840 166 37 schwarz schwarz PROPN cana-4840 166 38 inequality inequality NOUN cana-4840 166 39 we we PRON cana-4840 166 40 have have VERB cana-4840 166 41 , , PUNCT cana-4840 166 42 ( ( PUNCT cana-4840 166 43 βˆ‘aibi βˆ‘aibi PROPN cana-4840 166 44 n n CCONJ cana-4840 166 45 i=1 i=1 PROPN cana-4840 166 46 ) ) PUNCT cana-4840 166 47 2 2 NUM cana-4840 166 48 ≀ ≀ NOUN cana-4840 166 49 ( ( PUNCT cana-4840 166 50 βˆ‘ai βˆ‘ai NUM cana-4840 166 51 2 2 NUM cana-4840 166 52 n n NOUN cana-4840 166 53 i=1 i=1 PRON cana-4840 166 54 ) ) PUNCT cana-4840 166 55 ( ( PUNCT cana-4840 166 56 βˆ‘bi βˆ‘bi NOUN cana-4840 166 57 2 2 NUM cana-4840 166 58 n n NOUN cana-4840 166 59 i=1 i=1 PRON cana-4840 166 60 ) ) PUNCT cana-4840 166 61 put put VERB cana-4840 166 62 ai ai NOUN cana-4840 166 63 = = NOUN cana-4840 166 64 1 1 NUM cana-4840 166 65 , , PUNCT cana-4840 166 66 bi bi NOUN cana-4840 166 67 = = NOUN cana-4840 166 68 |Ξ»i |Ξ»i PROPN cana-4840 166 69 βˆ’ βˆ’ NUM cana-4840 166 70 2 2 NUM cana-4840 166 71 m m NOUN cana-4840 166 72 n n NUM cana-4840 167 1 | | ADV cana-4840 167 2 then then ADV cana-4840 167 3 , , PUNCT cana-4840 167 4 ( ( PUNCT cana-4840 167 5 βˆ‘|Ξ»i βˆ‘|Ξ»i ADJ cana-4840 167 6 βˆ’ βˆ’ PROPN cana-4840 167 7 2 2 NUM cana-4840 167 8 m m NOUN cana-4840 167 9 n n NUM cana-4840 167 10 | | ADV cana-4840 167 11 n n CCONJ cana-4840 167 12 i=1 i=1 PROPN cana-4840 167 13 ) ) PUNCT cana-4840 167 14 2 2 NUM cana-4840 167 15 ≀ ≀ NOUN cana-4840 167 16 ( ( PUNCT cana-4840 167 17 βˆ‘1 βˆ‘1 NOUN cana-4840 167 18 n n PRON cana-4840 167 19 i=1 i=1 PROPN cana-4840 167 20 ) ) PUNCT cana-4840 168 1 ( ( PUNCT cana-4840 168 2 βˆ‘|Ξ»i βˆ‘|Ξ»i ADJ cana-4840 168 3 βˆ’ βˆ’ PROPN cana-4840 168 4 2 2 NUM cana-4840 168 5 m m NOUN cana-4840 168 6 n n NOUN cana-4840 168 7 | | ADV cana-4840 168 8 2 2 NUM cana-4840 168 9 n n NOUN cana-4840 168 10 i=1 i=1 PRON cana-4840 168 11 ) ) PUNCT cana-4840 169 1 [ [ X cana-4840 169 2 lq𝐷(g lq𝐷(g PROPN cana-4840 169 3 ) ) PUNCT cana-4840 169 4 ] ] PUNCT cana-4840 169 5 2 2 NUM cana-4840 169 6 = = SYM cana-4840 169 7 n n PRON cana-4840 170 1 [ [ X cana-4840 170 2 βˆ‘Ξ»i βˆ‘Ξ»i ADJ cana-4840 170 3 2 2 NUM cana-4840 170 4 n n NOUN cana-4840 170 5 i=1 i=1 X cana-4840 171 1 + + NOUN cana-4840 171 2 βˆ‘ βˆ‘ PROPN cana-4840 171 3 n n PROPN cana-4840 171 4 i=1 i=1 PROPN cana-4840 171 5 4m2 4m2 NUM cana-4840 171 6 n2 n2 NOUN cana-4840 171 7 βˆ’ βˆ’ PROPN cana-4840 171 8 4 4 NUM cana-4840 171 9 m m NOUN cana-4840 171 10 n n NUM cana-4840 171 11 βˆ‘Ξ»i βˆ‘Ξ»i PUNCT cana-4840 172 1 n n CCONJ cana-4840 172 2 i=1 i=1 PRON cana-4840 172 3 ] ] PUNCT cana-4840 173 1 = = PUNCT cana-4840 173 2 n n PRON cana-4840 173 3 [ [ X cana-4840 173 4 2n 2n X cana-4840 173 5 + + CCONJ cana-4840 173 6 4m2 4m2 NUM cana-4840 173 7 n2 n2 NOUN cana-4840 173 8 βˆ™ βˆ™ PROPN cana-4840 173 9 n n CCONJ cana-4840 173 10 βˆ’ βˆ’ PROPN cana-4840 173 11 4 4 NUM cana-4840 173 12 m m NOUN cana-4840 173 13 n n PRON cana-4840 173 14 ( ( PUNCT cana-4840 173 15 2 2 NUM cana-4840 173 16 m m NOUN cana-4840 173 17 βˆ’ βˆ’ PROPN cana-4840 173 18 |d| |d| PROPN cana-4840 173 19 ) ) PUNCT cana-4840 173 20 ] ] PUNCT cana-4840 174 1 = = PUNCT cana-4840 174 2 n n NUM cana-4840 174 3 [ [ X cana-4840 174 4 2n 2n X cana-4840 174 5 + + CCONJ cana-4840 174 6 4m2 4m2 NUM cana-4840 174 7 n n CCONJ cana-4840 174 8 βˆ’ βˆ’ PROPN cana-4840 174 9 8m2 8m2 NUM cana-4840 174 10 n n NOUN cana-4840 174 11 + + CCONJ cana-4840 174 12 4m|d| 4m|d| NUM cana-4840 174 13 n n NOUN cana-4840 174 14 ] ] PUNCT cana-4840 174 15 = = PUNCT cana-4840 175 1 2nn 2nn NOUN cana-4840 176 1 + + CCONJ cana-4840 176 2 4m(|d| 4m(|d| NUM cana-4840 176 3 βˆ’ βˆ’ PROPN cana-4840 176 4 m m NOUN cana-4840 176 5 ) ) PUNCT cana-4840 176 6 ∴ ∴ PROPN cana-4840 176 7 lq𝐷(g lq𝐷(g PROPN cana-4840 176 8 ) ) PUNCT cana-4840 176 9 ≀ ≀ PUNCT cana-4840 176 10 √2nn √2nn VERB cana-4840 176 11 + + CCONJ cana-4840 177 1 4m(|d| 4m(|d| PROPN cana-4840 178 1 βˆ’ βˆ’ PROPN cana-4840 179 1 m m PROPN cana-4840 179 2 ) ) PUNCT cana-4840 180 1 . . PUNCT cana-4840 181 1 refrences refrence VERB cana-4840 182 1 [ [ X cana-4840 182 2 1 1 X cana-4840 182 3 ] ] PUNCT cana-4840 182 4 c.adiga c.adiga ADJ cana-4840 182 5 , , PUNCT cana-4840 182 6 a.bayad a.bayad PROPN cana-4840 182 7 , , PUNCT cana-4840 182 8 i.gutman i.gutman ADJ cana-4840 182 9 , , PUNCT cana-4840 182 10 s.a.srinivas s.a.sriniva NOUN cana-4840 182 11 , , PUNCT cana-4840 182 12 the the DET cana-4840 182 13 minimum minimum NOUN cana-4840 182 14 covering cover VERB cana-4840 182 15 energy energy NOUN cana-4840 182 16 of of ADP cana-4840 182 17 a a DET cana-4840 182 18 graph graph NOUN cana-4840 182 19 , , PUNCT cana-4840 182 20 kragujevac kragujevac PROPN cana-4840 182 21 j. j. PROPN cana-4840 182 22 sci sci PROPN cana-4840 182 23 . . PROPN cana-4840 183 1 34 34 NUM cana-4840 183 2 ( ( PUNCT cana-4840 183 3 2012 2012 NUM cana-4840 183 4 ) ) PUNCT cana-4840 183 5 39 39 NUM cana-4840 183 6 - - SYM cana-4840 183 7 56 56 NUM cana-4840 183 8 . . PUNCT cana-4840 184 1 [ [ X cana-4840 184 2 2 2 NUM cana-4840 184 3 ] ] X cana-4840 184 4 diaz diaz PROPN cana-4840 184 5 , , PUNCT cana-4840 184 6 jb jb PROPN cana-4840 184 7 and and CCONJ cana-4840 184 8 metcalf metcalf PROPN cana-4840 184 9 , , PUNCT cana-4840 184 10 ft ft PROPN cana-4840 184 11 1963 1963 NUM cana-4840 184 12 , , PUNCT cana-4840 184 13 ' ' PUNCT cana-4840 184 14 stroger strog ADJ cana-4840 184 15 forms form NOUN cana-4840 184 16 of of ADP cana-4840 184 17 a a DET cana-4840 184 18 class class NOUN cana-4840 184 19 of of ADP cana-4840 184 20 inequalities inequality NOUN cana-4840 184 21 of of ADP cana-4840 184 22 g. g. PROPN cana-4840 184 23 polya polya PROPN cana-4840 184 24 - - PUNCT cana-4840 184 25 g g PROPN cana-4840 184 26 - - PUNCT cana-4840 184 27 szeg"o szeg"o NOUN cana-4840 184 28 and and CCONJ cana-4840 184 29 lv lv PROPN cana-4840 184 30 kantorovich kantorovich PROPN cana-4840 184 31 ' ' PUNCT cana-4840 184 32 , , PUNCT cana-4840 184 33 bulletin bulletin NOUN cana-4840 184 34 of of ADP cana-4840 184 35 the the DET cana-4840 184 36 amsamerican amsamerican PROPN cana-4840 184 37 mathematical mathematical PROPN cana-4840 184 38 society society NOUN cana-4840 184 39 , , PUNCT cana-4840 184 40 vol vol NOUN cana-4840 184 41 69 69 NUM cana-4840 184 42 , , PUNCT cana-4840 184 43 pp.415418 pp.415418 PROPN cana-4840 184 44 . . PUNCT cana-4840 185 1 [ [ X cana-4840 185 2 3 3 X cana-4840 185 3 ] ] X cana-4840 185 4 s.t s.t PROPN cana-4840 185 5 . . PROPN cana-4840 185 6 hedetniemi hedetniemi PROPN cana-4840 185 7 and and CCONJ cana-4840 185 8 r.c r.c PROPN cana-4840 185 9 . . PROPN cana-4840 185 10 laskar laskar PROPN cana-4840 185 11 , , PUNCT cana-4840 185 12 topics topic NOUN cana-4840 185 13 on on ADP cana-4840 185 14 domination domination NOUN cana-4840 185 15 , , PUNCT cana-4840 185 16 discrete discrete ADJ cana-4840 185 17 math math NOUN cana-4840 185 18 . . PUNCT cana-4840 186 1 86 86 NUM cana-4840 186 2 ( ( PUNCT cana-4840 186 3 1990 1990 NUM cana-4840 186 4 ) ) PUNCT cana-4840 186 5 . . PUNCT cana-4840 187 1 [ [ X cana-4840 187 2 4 4 X cana-4840 187 3 ] ] PUNCT cana-4840 187 4 i.gutman i.gutman NOUN cana-4840 187 5 , , PUNCT cana-4840 187 6 the the DET cana-4840 187 7 energy energy NOUN cana-4840 187 8 of of ADP cana-4840 187 9 a a DET cana-4840 187 10 graph graph NOUN cana-4840 187 11 . . PUNCT cana-4840 187 12 ber ber NOUN cana-4840 187 13 . . PUNCT cana-4840 188 1 math math NOUN cana-4840 188 2 - - PUNCT cana-4840 188 3 statist statist NOUN cana-4840 188 4 . . PUNCT cana-4840 189 1 sekt sekt PROPN cana-4840 189 2 . . PUNCT cana-4840 190 1 forschungsz.graz forschungsz.graz PROPN cana-4840 190 2 103 103 NUM cana-4840 190 3 ( ( PUNCT cana-4840 190 4 1978 1978 NUM cana-4840 190 5 ) ) PUNCT cana-4840 190 6 1 1 NUM cana-4840 190 7 - - SYM cana-4840 190 8 22 22 NUM cana-4840 190 9 . . PUNCT cana-4840 191 1 [ [ X cana-4840 191 2 5 5 X cana-4840 191 3 ] ] PUNCT cana-4840 191 4 i.gutman i.gutman NOUN cana-4840 191 5 , , PUNCT cana-4840 191 6 b.zhou b.zhou PROPN cana-4840 191 7 laplacian laplacian ADJ cana-4840 191 8 energy energy NOUN cana-4840 191 9 of of ADP cana-4840 191 10 a a DET cana-4840 191 11 graph graph NOUN cana-4840 191 12 . . PUNCT cana-4840 192 1 lin lin PROPN cana-4840 192 2 . . PUNCT cana-4840 193 1 algebra algebra PROPN cana-4840 193 2 appl appl PROPN cana-4840 193 3 . . PUNCT cana-4840 194 1 414 414 NUM cana-4840 194 2 , , PUNCT cana-4840 194 3 29 29 NUM cana-4840 194 4 - - SYM cana-4840 194 5 37(2006 37(2006 NUM cana-4840 194 6 ) ) PUNCT cana-4840 194 7 . . PUNCT cana-4840 195 1 [ [ X cana-4840 195 2 6 6 NUM cana-4840 195 3 ] ] PUNCT cana-4840 195 4 m. m. NOUN cana-4840 195 5 lalitha lalitha PROPN cana-4840 195 6 kumari kumari PROPN cana-4840 195 7 l l PROPN cana-4840 195 8 , , PUNCT cana-4840 195 9 pandiselvi pandiselvi ADJ cana-4840 195 10 and and CCONJ cana-4840 195 11 k.palani1 k.palani1 NOUN cana-4840 195 12 , , PUNCT cana-4840 195 13 quotient quotient VERB cana-4840 195 14 energy energy NOUN cana-4840 195 15 of of ADP cana-4840 195 16 zero zero NUM cana-4840 195 17 divisor divisor NOUN cana-4840 195 18 graphs graph NOUN cana-4840 195 19 and and CCONJ cana-4840 195 20 identity identity NOUN cana-4840 195 21 graphs graph NOUN cana-4840 195 22 , , PUNCT cana-4840 195 23 baghdad baghdad PROPN cana-4840 195 24 science science PROPN cana-4840 195 25 journal journal PROPN cana-4840 195 26 , , PUNCT cana-4840 195 27 2023 2023 NUM cana-4840 195 28 , , PUNCT cana-4840 195 29 20(1 20(1 NUM cana-4840 195 30 special special ADJ cana-4840 195 31 issue issue NOUN cana-4840 195 32 ) ) PUNCT cana-4840 195 33 icaam icaam NOUN cana-4840 195 34 : : PUNCT cana-4840 195 35 277 277 NUM cana-4840 195 36 - - SYM cana-4840 195 37 282 282 NUM cana-4840 195 38 . . PUNCT cana-4840 196 1 [ [ X cana-4840 196 2 7 7 NUM cana-4840 196 3 ] ] PUNCT cana-4840 196 4 nataraj nataraj NOUN cana-4840 196 5 , , PUNCT cana-4840 196 6 puttaswamy puttaswamy NOUN cana-4840 196 7 and and CCONJ cana-4840 196 8 purushothama purushothama PROPN cana-4840 196 9 s s PART cana-4840 196 10 laplacian laplacian ADJ cana-4840 196 11 minimum minimum NOUN cana-4840 196 12 dominating dominating NOUN cana-4840 196 13 energy energy NOUN cana-4840 196 14 of of ADP cana-4840 196 15 a a DET cana-4840 196 16 graph graph NOUN cana-4840 196 17 . . PUNCT cana-4840 197 1 tuijin tuijin PROPN cana-4840 197 2 jishu jishu PROPN cana-4840 197 3 / / SYM cana-4840 197 4 journal journal NOUN cana-4840 197 5 of of ADP cana-4840 197 6 propulsion propulsion NOUN cana-4840 197 7 technology technology NOUN cana-4840 197 8 issn issn NOUN cana-4840 197 9 : : PUNCT cana-4840 197 10 1001 1001 NUM cana-4840 197 11 - - SYM cana-4840 197 12 4055 4055 NUM cana-4840 197 13 , , PUNCT cana-4840 197 14 vol vol NOUN cana-4840 197 15 . . PROPN cana-4840 198 1 45 45 NUM cana-4840 198 2 no no NOUN cana-4840 198 3 . . NOUN cana-4840 198 4 3 3 NUM cana-4840 198 5 ( ( PUNCT cana-4840 198 6 2024 2024 NUM cana-4840 198 7 ) ) PUNCT cana-4840 198 8 . . PUNCT cana-4840 199 1 doi doi PROPN cana-4840 199 2 of of ADP cana-4840 199 3 each each DET cana-4840 199 4 article article NOUN cana-4840 199 5 . . PUNCT cana-4840 200 1 communications communication NOUN cana-4840 200 2 on on ADP cana-4840 200 3 applied apply VERB cana-4840 200 4 nonlinear nonlinear ADJ cana-4840 200 5 analysis analysis NOUN cana-4840 200 6 issn issn NOUN cana-4840 200 7 : : PUNCT cana-4840 200 8 1074 1074 NUM cana-4840 200 9 - - PUNCT cana-4840 200 10 133x 133x NUM cana-4840 200 11 vol vol VERB cana-4840 200 12 32 32 NUM cana-4840 200 13 no no NOUN cana-4840 200 14 . . PUNCT cana-4840 201 1 10s 10 NOUN cana-4840 201 2 ( ( PUNCT cana-4840 201 3 2025 2025 NUM cana-4840 201 4 ) ) PUNCT cana-4840 201 5 454 454 NUM cana-4840 201 6 https://internationalpubls.com https://internationalpubls.com X cana-4840 202 1 [ [ X cana-4840 202 2 8 8 X cana-4840 202 3 ] ] X cana-4840 202 4 purushothama purushothama PROPN cana-4840 202 5 s s NOUN cana-4840 202 6 , , PUNCT cana-4840 202 7 puttaswamy puttaswamy NOUN cana-4840 202 8 and and CCONJ cana-4840 202 9 nayaka nayaka NOUN cana-4840 202 10 s s PART cana-4840 202 11 r r NOUN cana-4840 202 12 minimum minimum ADJ cana-4840 202 13 pendant pendant ADJ cana-4840 202 14 dominating dominating NOUN cana-4840 202 15 energy energy NOUN cana-4840 202 16 of of ADP cana-4840 202 17 a a DET cana-4840 202 18 graph graph NOUN cana-4840 202 19 , , PUNCT cana-4840 202 20 aijrstem aijrstem NOUN cana-4840 202 21 , , PUNCT cana-4840 202 22 issn issn PROPN cana-4840 202 23 ( ( PUNCT cana-4840 202 24 print print NOUN cana-4840 202 25 ): ): PUNCT cana-4840 202 26 2328 2328 NUM cana-4840 202 27 - - SYM cana-4840 202 28 3491 3491 NUM cana-4840 202 29 , , PUNCT cana-4840 202 30 issn issn PROPN cana-4840 202 31 ( ( PUNCT cana-4840 202 32 online online ADJ cana-4840 202 33 ): ): PUNCT cana-4840 202 34 2328 2328 NUM cana-4840 202 35 - - SYM cana-4840 202 36 3580 3580 NUM cana-4840 202 37 , , PUNCT cana-4840 202 38 issn issn PROPN cana-4840 202 39 ( ( PUNCT cana-4840 202 40 cd cd PROPN cana-4840 202 41 - - PUNCT cana-4840 202 42 rom rom NOUN cana-4840 202 43 ): ): PUNCT cana-4840 202 44 2328 2328 NUM cana-4840 202 45 - - SYM cana-4840 202 46 3629 3629 NUM cana-4840 202 47 [ [ X cana-4840 202 48 9 9 NUM cana-4840 202 49 ] ] PUNCT cana-4840 202 50 polya polya NOUN cana-4840 202 51 , , PUNCT cana-4840 202 52 g g PROPN cana-4840 202 53 and and CCONJ cana-4840 202 54 szego szego NOUN cana-4840 202 55 , , PUNCT cana-4840 202 56 1972 1972 NUM cana-4840 202 57 , , PUNCT cana-4840 202 58 problems problem NOUN cana-4840 202 59 and and CCONJ cana-4840 202 60 theorems theorem NOUN cana-4840 202 61 in in ADP cana-4840 202 62 analysis analysis NOUN cana-4840 202 63 ' ' PUNCT cana-4840 202 64 series series NOUN cana-4840 202 65 , , PUNCT cana-4840 202 66 integral integral ADJ cana-4840 202 67 calculus calculus NOUN cana-4840 202 68 , , PUNCT cana-4840 202 69 theory theory NOUN cana-4840 202 70 of of ADP cana-4840 202 71 functions function NOUN cana-4840 202 72 , , PUNCT cana-4840 202 73 springer springer NOUN cana-4840 202 74 , , PUNCT cana-4840 202 75 berlin berlin PROPN cana-4840 202 76 [ [ X cana-4840 202 77 10 10 NUM cana-4840 202 78 ] ] X cana-4840 202 79 purushothama purushothama PROPN cana-4840 202 80 s s PART cana-4840 202 81 , , PUNCT cana-4840 202 82 prakasha prakasha VERB cana-4840 202 83 k k PROPN cana-4840 202 84 n n CCONJ cana-4840 202 85 , , PUNCT cana-4840 202 86 minimum minimum ADJ cana-4840 202 87 dominating dominating NOUN cana-4840 202 88 quotient quotient NOUN cana-4840 202 89 energy energy NOUN cana-4840 202 90 of of ADP cana-4840 202 91 graphs graph NOUN cana-4840 202 92 , , PUNCT cana-4840 202 93 in in ADP cana-4840 202 94 communication communication NOUN cana-4840 202 95 . . PUNCT