id sid tid token lemma pos cana-1650 1 1 communications communication NOUN cana-1650 1 2 on on ADP cana-1650 1 3 applied apply VERB cana-1650 1 4 nonlinear nonlinear ADJ cana-1650 1 5 analysis analysis NOUN cana-1650 1 6 issn issn NOUN cana-1650 1 7 : : PUNCT cana-1650 1 8 1074 1074 NUM cana-1650 1 9 - - PUNCT cana-1650 1 10 133x 133x NUM cana-1650 1 11 vol vol NOUN cana-1650 1 12 32 32 NUM cana-1650 1 13 no no NOUN cana-1650 1 14 . . NOUN cana-1650 1 15 1 1 NUM cana-1650 1 16 ( ( PUNCT cana-1650 1 17 2025 2025 NUM cana-1650 1 18 ) ) PUNCT cana-1650 1 19 303 303 NUM cana-1650 1 20 https://internationalpubls.com https://internationalpubls.com NUM cana-1650 1 21 bound bind VERB cana-1650 1 22 inequalities inequality NOUN cana-1650 1 23 on on ADP cana-1650 1 24 degree degree NOUN cana-1650 1 25 sum sum NOUN cana-1650 1 26 energy energy NOUN cana-1650 1 27 of of ADP cana-1650 1 28 graph graph NOUN cana-1650 1 29 ramesha ramesha PROPN cana-1650 1 30 m m PROPN cana-1650 1 31 s s NOUN cana-1650 1 32 1 1 NUM cana-1650 1 33 , , PUNCT cana-1650 1 34 ashwini ashwini NOUN cana-1650 1 35 g g PROPN cana-1650 1 36 2 2 NUM cana-1650 1 37 , , PUNCT cana-1650 1 38 and and CCONJ cana-1650 1 39 shivakumar shivakumar PROPN cana-1650 1 40 swamy swamy PROPN cana-1650 1 41 c c PROPN cana-1650 1 42 s s PART cana-1650 1 43 3 3 NUM cana-1650 1 44 1,2,3 1,2,3 NUM cana-1650 1 45 department department NOUN cana-1650 1 46 of of ADP cana-1650 1 47 mathematics mathematic NOUN cana-1650 1 48 , , PUNCT cana-1650 1 49 government government NOUN cana-1650 1 50 college college NOUN cana-1650 1 51 for for ADP cana-1650 1 52 women woman NOUN cana-1650 1 53 ( ( PUNCT cana-1650 1 54 autonomous autonomous ADJ cana-1650 1 55 ) ) PUNCT cana-1650 1 56 , , PUNCT cana-1650 1 57 mandya-571401 mandya-571401 NOUN cana-1650 1 58 , , PUNCT cana-1650 1 59 india india PROPN cana-1650 1 60 . . PROPN cana-1650 2 1 1 1 NUM cana-1650 2 2 profmsr1978@gmail.com profmsr1978@gmail.com NOUN cana-1650 2 3 , , PUNCT cana-1650 2 4 2 2 NUM cana-1650 2 5 agkn6882@gmail.com*,3 agkn6882@gmail.com*,3 NOUN cana-1650 2 6 cskswamy@gmail.com cskswamy@gmail.com NOUN cana-1650 2 7 * * PUNCT cana-1650 2 8 * * PUNCT cana-1650 2 9 corresponding correspond VERB cana-1650 2 10 author author NOUN cana-1650 2 11 โ€™s โ€™s PART cana-1650 2 12 : : PUNCT cana-1650 2 13 agkn6882@gmail.com agkn6882@gmail.com X cana-1650 2 14 , , PUNCT cana-1650 2 15 cskswamy@gmail.com cskswamy@gmail.com PROPN cana-1650 3 1 article article NOUN cana-1650 3 2 history history NOUN cana-1650 3 3 : : PUNCT cana-1650 3 4 received receive VERB cana-1650 3 5 : : PUNCT cana-1650 3 6 14 14 NUM cana-1650 3 7 - - SYM cana-1650 3 8 07 07 NUM cana-1650 3 9 - - PUNCT cana-1650 3 10 2024 2024 NUM cana-1650 3 11 revised revise VERB cana-1650 3 12 : : PUNCT cana-1650 3 13 29 29 NUM cana-1650 3 14 - - SYM cana-1650 3 15 08 08 NUM cana-1650 3 16 - - PUNCT cana-1650 3 17 2024 2024 NUM cana-1650 3 18 accepted accept VERB cana-1650 3 19 : : PUNCT cana-1650 3 20 11 11 NUM cana-1650 3 21 - - SYM cana-1650 3 22 09 09 NUM cana-1650 3 23 - - PUNCT cana-1650 3 24 2024 2024 NUM cana-1650 3 25 abstract abstract ADJ cana-1650 3 26 ๐ธ๐‘‘๐‘ (๐บ ๐ธ๐‘‘๐‘ (๐บ NUM cana-1650 3 27 ) ) PUNCT cana-1650 3 28 , , PUNCT cana-1650 3 29 the the DET cana-1650 3 30 degree degree NOUN cana-1650 3 31 sum sum NOUN cana-1650 3 32 energy energy NOUN cana-1650 3 33 of of ADP cana-1650 3 34 a a DET cana-1650 3 35 graph graph NOUN cana-1650 3 36 ๐บ ๐บ NOUN cana-1650 3 37 is be AUX cana-1650 3 38 the the DET cana-1650 3 39 total total NOUN cana-1650 3 40 of of ADP cana-1650 3 41 all all DET cana-1650 3 42 the the DET cana-1650 3 43 absolute absolute ADJ cana-1650 3 44 values value NOUN cana-1650 3 45 of of ADP cana-1650 3 46 its its PRON cana-1650 3 47 degree degree NOUN cana-1650 3 48 sum sum NOUN cana-1650 3 49 eigenvalues eigenvalue VERB cana-1650 3 50 . . PUNCT cana-1650 4 1 in in ADP cana-1650 4 2 this this DET cana-1650 4 3 investigation investigation NOUN cana-1650 4 4 one one NUM cana-1650 4 5 upper upper ADJ cana-1650 4 6 and and CCONJ cana-1650 4 7 lower low ADJ cana-1650 4 8 constraints constraint NOUN cana-1650 4 9 on on ADP cana-1650 4 10 the the DET cana-1650 4 11 degree degree NOUN cana-1650 4 12 sum sum NOUN cana-1650 4 13 energy energy NOUN cana-1650 4 14 are be AUX cana-1650 4 15 obtained obtain VERB cana-1650 4 16 in in ADP cana-1650 4 17 this this DET cana-1650 4 18 study study NOUN cana-1650 4 19 . . PUNCT cana-1650 5 1 keywords keyword NOUN cana-1650 5 2 : : PUNCT cana-1650 5 3 degree degree NOUN cana-1650 5 4 sum sum NOUN cana-1650 5 5 matrix matrix NOUN cana-1650 5 6 , , PUNCT cana-1650 5 7 degree degree NOUN cana-1650 5 8 sum sum NOUN cana-1650 5 9 eigenvalues eigenvalue NOUN cana-1650 5 10 , , PUNCT cana-1650 5 11 degree degree NOUN cana-1650 5 12 sum sum NOUN cana-1650 5 13 energy energy NOUN cana-1650 5 14 . . PUNCT cana-1650 6 1 2000 2000 NUM cana-1650 6 2 mathematics mathematic NOUN cana-1650 6 3 subject subject NOUN cana-1650 6 4 classification classification NOUN cana-1650 6 5 . . PUNCT cana-1650 7 1 05c50 05c50 NUM cana-1650 8 1 1 1 X cana-1650 8 2 . . X cana-1650 9 1 introduction introduction NOUN cana-1650 9 2 let let VERB cana-1650 9 3 us we PRON cana-1650 9 4 assume assume VERB cana-1650 9 5 that that SCONJ cana-1650 9 6 ๐บ ๐บ PROPN cana-1650 9 7 is be AUX cana-1650 9 8 a a DET cana-1650 9 9 simple simple ADJ cana-1650 9 10 graph graph NOUN cana-1650 9 11 , , PUNCT cana-1650 9 12 and and CCONJ cana-1650 9 13 that that SCONJ cana-1650 9 14 ๐‘‰ ๐‘‰ PROPN cana-1650 9 15 ( ( PUNCT cana-1650 9 16 ๐บ ๐บ NOUN cana-1650 9 17 ) ) PUNCT cana-1650 9 18 = = PRON cana-1650 9 19 { { PUNCT cana-1650 9 20 ๐‘ฃ1 ๐‘ฃ1 PROPN cana-1650 9 21 , , PUNCT cana-1650 9 22 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1650 9 23 , , PUNCT cana-1650 9 24 โ€ฆ โ€ฆ PUNCT cana-1650 9 25 , , PUNCT cana-1650 9 26 ๐‘ฃ๐‘› ๐‘ฃ๐‘› NOUN cana-1650 9 27 } } PUNCT cana-1650 9 28 is be AUX cana-1650 9 29 its its PRON cana-1650 9 30 vertex vertex NOUN cana-1650 9 31 set set NOUN cana-1650 9 32 . . PUNCT cana-1650 10 1 when when SCONJ cana-1650 10 2 the the DET cana-1650 10 3 vertices vertex NOUN cana-1650 10 4 ๐‘ฃ๐‘– ๐‘ฃ๐‘– VERB cana-1650 10 5 and and CCONJ cana-1650 10 6 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1650 10 7 are be AUX cana-1650 10 8 adjacent adjacent ADJ cana-1650 10 9 , , PUNCT cana-1650 10 10 the the DET cana-1650 10 11 adjacency adjacency NOUN cana-1650 10 12 matrix matrix NOUN cana-1650 10 13 ๐ด(๐บ ๐ด(๐บ PROPN cana-1650 10 14 ) ) PUNCT cana-1650 10 15 of of ADP cana-1650 10 16 the the DET cana-1650 10 17 graph graph NOUN cana-1650 10 18 ๐บ ๐บ NOUN cana-1650 10 19 is be AUX cana-1650 10 20 a a DET cana-1650 10 21 square square ADJ cana-1650 10 22 matrix matrix NOUN cana-1650 10 23 of of ADP cana-1650 10 24 rank rank NOUN cana-1650 10 25 ๐‘› ๐‘› PROPN cana-1650 10 26 with with ADP cana-1650 10 27 the the DET cana-1650 10 28 ( ( PUNCT cana-1650 10 29 ๐‘– ๐‘– NOUN cana-1650 10 30 : : PUNCT cana-1650 10 31 ๐‘— ๐‘— ADJ cana-1650 10 32 ) ) PUNCT cana-1650 10 33 โˆ’ โˆ’ PROPN cana-1650 10 34 entry entry NOUN cana-1650 10 35 equal equal ADJ cana-1650 10 36 to to ADP cana-1650 10 37 unity unity NOUN cana-1650 10 38 , , PUNCT cana-1650 10 39 otherwise otherwise ADV cana-1650 10 40 , , PUNCT cana-1650 10 41 it it PRON cana-1650 10 42 is be AUX cana-1650 10 43 equal equal ADJ cana-1650 10 44 to to ADP cana-1650 10 45 zero zero NUM cana-1650 10 46 . . PUNCT cana-1650 11 1 the the DET cana-1650 11 2 eigenvalues eigenvalue NOUN cana-1650 11 3 of of ADP cana-1650 11 4 the the DET cana-1650 11 5 graph graph NOUN cana-1650 11 6 g g NOUN cana-1650 11 7 are be AUX cana-1650 11 8 are be AUX cana-1650 11 9 ๐œน๐Ÿ ๐œน๐Ÿ ADJ cana-1650 11 10 , , PUNCT cana-1650 11 11 ๐œน๐Ÿ ๐œน๐Ÿ NOUN cana-1650 11 12 , , PUNCT cana-1650 11 13 โ€ฆ โ€ฆ PUNCT cana-1650 11 14 . . PUNCT cana-1650 12 1 , , PUNCT cana-1650 12 2 ๐œน๐’ ๐œน๐’ NOUN cana-1650 12 3 , , PUNCT cana-1650 12 4 of of ADP cana-1650 12 5 ๐ด(๐บ ๐ด(๐บ NOUN cana-1650 12 6 ) ) PUNCT cana-1650 12 7 , , PUNCT cana-1650 12 8 which which PRON cana-1650 12 9 are be AUX cana-1650 12 10 considered consider VERB cana-1650 12 11 to to PART cana-1650 12 12 be be AUX cana-1650 12 13 non non ADJ cana-1650 12 14 - - ADJ cana-1650 12 15 increasing increase VERB cana-1650 12 16 in in ADP cana-1650 12 17 order order NOUN cana-1650 12 18 . . PUNCT cana-1650 13 1 i. i. PROPN cana-1650 13 2 gutman gutman PROPN cana-1650 14 1 [ [ X cana-1650 14 2 6 6 NUM cana-1650 14 3 ] ] PUNCT cana-1650 14 4 originally originally ADV cana-1650 14 5 defined define VERB cana-1650 14 6 the the DET cana-1650 14 7 energy energy NOUN cana-1650 14 8 of of ADP cana-1650 14 9 g g NOUN cana-1650 14 10 in in ADP cana-1650 14 11 1978 1978 NUM cana-1650 14 12 as as ADP cana-1650 14 13 the the DET cana-1650 14 14 total total NOUN cana-1650 14 15 of of ADP cana-1650 14 16 its its PRON cana-1650 14 17 eigenvalues eigenvalue NOUN cana-1650 14 18 absolute absolute ADJ cana-1650 14 19 values value NOUN cana-1650 14 20 : : PUNCT cana-1650 14 21 ๐‘ฌ(๐‘ฎ ๐‘ฌ(๐‘ฎ NUM cana-1650 14 22 ) ) PUNCT cana-1650 14 23 = = PUNCT cana-1650 14 24 โˆ‘ โˆ‘ PUNCT cana-1650 14 25 |๐œน๐’Œ|๐’ |๐œน๐’Œ|๐’ PROPN cana-1650 14 26 ๐’Œ=๐Ÿ ๐’Œ=๐Ÿ PROPN cana-1650 14 27 . . PUNCT cana-1650 15 1 there there PRON cana-1650 15 2 has have AUX cana-1650 15 3 been be AUX cana-1650 15 4 a a DET cana-1650 15 5 steady steady ADJ cana-1650 15 6 flow flow NOUN cana-1650 15 7 of of ADP cana-1650 15 8 articles article NOUN cana-1650 15 9 on on ADP cana-1650 15 10 this this DET cana-1650 15 11 subject subject NOUN cana-1650 15 12 since since SCONJ cana-1650 15 13 i. i. PROPN cana-1650 15 14 gutman gutman PROPN cana-1650 15 15 first first ADV cana-1650 15 16 established establish VERB cana-1650 15 17 the the DET cana-1650 15 18 graph graph NOUN cana-1650 15 19 energy energy NOUN cana-1650 15 20 ๐ธ(๐บ ๐ธ(๐บ PROPN cana-1650 15 21 ) ) PUNCT cana-1650 15 22 of of ADP cana-1650 15 23 a a DET cana-1650 15 24 simple simple ADJ cana-1650 15 25 graph graph NOUN cana-1650 15 26 ๐บ. ๐บ. NOUN cana-1650 15 27 for for ADP cana-1650 15 28 basic basic ADJ cana-1650 15 29 mathematical mathematical ADJ cana-1650 15 30 properties property NOUN cana-1650 15 31 of of ADP cana-1650 15 32 the the DET cana-1650 15 33 theory theory NOUN cana-1650 15 34 of of ADP cana-1650 15 35 graph graph NOUN cana-1650 15 36 energy energy NOUN cana-1650 15 37 including include VERB cana-1650 15 38 its its PRON cana-1650 15 39 upper upper ADJ cana-1650 15 40 and and CCONJ cana-1650 15 41 lower low ADJ cana-1650 15 42 bounds bound NOUN cana-1650 15 43 one one PRON cana-1650 15 44 can can AUX cana-1650 15 45 see see VERB cana-1650 15 46 [ [ X cana-1650 15 47 4 4 NUM cana-1650 15 48 , , PUNCT cana-1650 15 49 11 11 NUM cana-1650 15 50 ] ] PUNCT cana-1650 15 51 . . PUNCT cana-1650 16 1 erich erich PROPN cana-1650 16 2 huckle[8 huckle[8 NOUN cana-1650 16 3 ] ] PUNCT cana-1650 16 4 , , PUNCT cana-1650 16 5 employed employ VERB cana-1650 16 6 the the DET cana-1650 16 7 energy energy NOUN cana-1650 16 8 of of ADP cana-1650 16 9 graphs graph NOUN cana-1650 16 10 technique technique NOUN cana-1650 16 11 in in ADP cana-1650 16 12 the the DET cana-1650 16 13 early early ADJ cana-1650 16 14 1930s 1930 NOUN cana-1650 16 15 to to PART cana-1650 16 16 develop develop VERB cana-1650 16 17 approximations approximation NOUN cana-1650 16 18 solutions solution NOUN cana-1650 16 19 for for ADP cana-1650 16 20 a a DET cana-1650 16 21 family family NOUN cana-1650 16 22 of of ADP cana-1650 16 23 organic organic ADJ cana-1650 16 24 molecules molecule NOUN cana-1650 16 25 known know VERB cana-1650 16 26 as as ADP cana-1650 16 27 conjugated conjugate VERB cana-1650 16 28 hydro hydro NOUN cana-1650 16 29 carbons carbon NOUN cana-1650 16 30 . . PUNCT cana-1650 17 1 numerous numerous ADJ cana-1650 17 2 matrix matrix NOUN cana-1650 17 3 types type NOUN cana-1650 17 4 , , PUNCT cana-1650 17 5 including include VERB cana-1650 17 6 incidence incidence NOUN cana-1650 17 7 [ [ X cana-1650 17 8 10 10 NUM cana-1650 17 9 ] ] PUNCT cana-1650 17 10 , , PUNCT cana-1650 17 11 distance distance NOUN cana-1650 17 12 [ [ X cana-1650 17 13 9 9 NUM cana-1650 17 14 ] ] PUNCT cana-1650 17 15 , , PUNCT cana-1650 17 16 lapalcian lapalcian ADJ cana-1650 17 17 [ [ X cana-1650 17 18 7 7 NUM cana-1650 17 19 ] ] PUNCT cana-1650 17 20 , , PUNCT cana-1650 17 21 maximum maximum ADJ cana-1650 17 22 degree degree NOUN cana-1650 17 23 matrix matrix NOUN cana-1650 18 1 [ [ X cana-1650 18 2 1 1 X cana-1650 18 3 ] ] PUNCT cana-1650 18 4 and and CCONJ cana-1650 18 5 others other NOUN cana-1650 18 6 are be AUX cana-1650 18 7 established establish VERB cana-1650 18 8 and and CCONJ cana-1650 18 9 researched research VERB cana-1650 18 10 for for ADP cana-1650 18 11 graphs graph NOUN cana-1650 18 12 , , PUNCT cana-1650 18 13 with with ADP cana-1650 18 14 inspiration inspiration NOUN cana-1650 18 15 drawn draw VERB cana-1650 18 16 from from ADP cana-1650 18 17 the the DET cana-1650 18 18 adjacency adjacency NOUN cana-1650 18 19 matrix matrix NOUN cana-1650 18 20 of of ADP cana-1650 18 21 a a DET cana-1650 18 22 graph graph NOUN cana-1650 18 23 . . PUNCT cana-1650 19 1 in in ADP cana-1650 19 2 their their PRON cana-1650 19 3 publication publication NOUN cana-1650 19 4 [ [ X cana-1650 19 5 12 12 NUM cana-1650 19 6 ] ] PUNCT cana-1650 19 7 , , PUNCT cana-1650 19 8 ramane ramane PROPN cana-1650 19 9 et et PROPN cana-1650 19 10 al al PROPN cana-1650 19 11 . . PROPN cana-1650 19 12 introduced introduce VERB cana-1650 19 13 and and CCONJ cana-1650 19 14 investigated investigate VERB cana-1650 19 15 sumdegree sumdegree NOUN cana-1650 19 16 energy energy NOUN cana-1650 19 17 of of ADP cana-1650 19 18 ๐บ ๐บ PROPN cana-1650 19 19 , , PUNCT cana-1650 19 20 defined define VERB cana-1650 19 21 as as SCONJ cana-1650 19 22 follows follow VERB cana-1650 19 23 : : PUNCT cana-1650 19 24 let let VERB cana-1650 19 25 g g PRON cana-1650 19 26 be be AUX cana-1650 19 27 a a DET cana-1650 19 28 simple simple ADJ cana-1650 19 29 graph graph NOUN cana-1650 19 30 with with ADP cana-1650 19 31 connections connection NOUN cana-1650 19 32 . . PUNCT cana-1650 20 1 the the DET cana-1650 20 2 matrix matrix NOUN cana-1650 20 3 ๐ท๐‘†๐‘€(๐บ ๐ท๐‘†๐‘€(๐บ VERB cana-1650 20 4 ) ) PUNCT cana-1650 20 5 = = PUNCT cana-1650 21 1 [ [ X cana-1650 21 2 ๐‘‘๐‘˜๐‘— ๐‘‘๐‘˜๐‘— NOUN cana-1650 21 3 ] ] PUNCT cana-1650 21 4 needs need VERB cana-1650 21 5 to to PART cana-1650 21 6 be be AUX cana-1650 21 7 defined define VERB cana-1650 21 8 as as ADP cana-1650 21 9 , , PUNCT cana-1650 21 10 ๐‘‘๐‘˜๐‘— ๐‘‘๐‘˜๐‘— ADJ cana-1650 21 11 = = PUNCT cana-1650 21 12 { { PUNCT cana-1650 21 13 ๐‘‘๐‘˜ ๐‘‘๐‘˜ ADV cana-1650 21 14 + + CCONJ cana-1650 21 15 ๐‘‘๐‘— ๐‘‘๐‘— NOUN cana-1650 21 16 , , PUNCT cana-1650 21 17 ๐‘คโ„Ž๐‘’๐‘› ๐‘คโ„Ž๐‘’๐‘› VERB cana-1650 21 18 ๐‘ฃ๐‘˜ ๐‘ฃ๐‘˜ ADV cana-1650 21 19 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-1650 21 20 ๐‘ฃ๐‘— ๐‘ฃ๐‘— ADP cana-1650 21 21 ๐‘Ž๐‘Ÿ๐‘’ ๐‘Ž๐‘Ÿ๐‘’ PROPN cana-1650 21 22 ๐‘Ž๐‘‘๐‘—๐‘Ž๐‘๐‘’๐‘›๐‘ก ๐‘Ž๐‘‘๐‘—๐‘Ž๐‘๐‘’๐‘›๐‘ก PROPN cana-1650 21 23 0 0 NUM cana-1650 21 24 ๐‘œ๐‘กโ„Ž๐‘’๐‘ค๐‘–๐‘ ๐‘’ ๐‘œ๐‘กโ„Ž๐‘’๐‘ค๐‘–๐‘ ๐‘’ PROPN cana-1650 21 25 , , PUNCT cana-1650 21 26 this this PRON cana-1650 21 27 is be AUX cana-1650 21 28 referred refer VERB cana-1650 21 29 to to ADP cana-1650 21 30 as as ADP cana-1650 21 31 g g NOUN cana-1650 21 32 's 's PART cana-1650 21 33 degree degree NOUN cana-1650 21 34 sum sum NOUN cana-1650 21 35 matrix matrix NOUN cana-1650 21 36 . . PUNCT cana-1650 22 1 the the DET cana-1650 22 2 degree degree NOUN cana-1650 22 3 sum sum NOUN cana-1650 22 4 energy energy NOUN cana-1650 22 5 dse dse PROPN cana-1650 22 6 of of ADP cana-1650 22 7 ๐บ ๐บ PROPN cana-1650 22 8 is be AUX cana-1650 22 9 then then ADV cana-1650 22 10 written write VERB cana-1650 22 11 as as ADP cana-1650 22 12 ๐ธ๐‘‘๐‘ (๐บ ๐ธ๐‘‘๐‘ (๐บ NOUN cana-1650 22 13 ) ) PUNCT cana-1650 22 14 = = SYM cana-1650 22 15 โˆ‘ โˆ‘ PUNCT cana-1650 22 16 |๐œ‰๐’Œ|๐’ |๐œ‰๐’Œ|๐’ PROPN cana-1650 22 17 ๐’Œ=๐Ÿ ๐’Œ=๐Ÿ PROPN cana-1650 22 18 , , PUNCT cana-1650 22 19 where where SCONJ cana-1650 22 20 , , PUNCT cana-1650 22 21 ๐œ‰๐’Œ ๐œ‰๐’Œ PROPN cana-1650 22 22 are be AUX cana-1650 22 23 the the DET cana-1650 22 24 eigenvalues eigenvalue NOUN cana-1650 22 25 of of ADP cana-1650 22 26 dsm(g dsm(g PROPN cana-1650 22 27 ) ) PUNCT cana-1650 22 28 , , PUNCT cana-1650 22 29 furthermore furthermore ADV cana-1650 22 30 , , PUNCT cana-1650 22 31 these these DET cana-1650 22 32 eigenvalues eigenvalue NOUN cana-1650 22 33 are be AUX cana-1650 22 34 real real ADJ cana-1650 22 35 numbers number NOUN cana-1650 22 36 and and CCONJ cana-1650 22 37 are be AUX cana-1650 22 38 sorted sort VERB cana-1650 22 39 in in ADP cana-1650 22 40 ascending ascend VERB cana-1650 22 41 order order NOUN cana-1650 22 42 . . PUNCT cana-1650 23 1 note note VERB cana-1650 23 2 that that SCONJ cana-1650 23 3 dsm(g dsm(g PROPN cana-1650 23 4 ) ) PUNCT cana-1650 23 5 has have VERB cana-1650 23 6 ๐‘ก๐‘Ÿ๐‘Ž๐‘๐‘’ ๐‘ก๐‘Ÿ๐‘Ž๐‘๐‘’ ADJ cana-1650 23 7 = = SYM cana-1650 23 8 0 0 NUM cana-1650 23 9 , , PUNCT cana-1650 23 10 and and CCONJ cana-1650 23 11 โˆ‘ โˆ‘ ADP cana-1650 23 12 ๐œ‰๐‘˜ ๐œ‰๐‘˜ ADJ cana-1650 23 13 2 2 NUM cana-1650 23 14 = = SYM cana-1650 23 15 2๐”ˆ๐‘› 2๐”ˆ๐‘› NUM cana-1650 23 16 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 23 17 , , PUNCT cana-1650 23 18 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ NOUN cana-1650 23 19 ๐”ˆ ๐”ˆ PROPN cana-1650 23 20 = = PRON cana-1650 23 21 โˆ‘ โˆ‘ PROPN cana-1650 23 22 ( ( PUNCT cana-1650 23 23 ๐‘‘๐‘˜ ๐‘‘๐‘˜ ADV cana-1650 23 24 + + CCONJ cana-1650 23 25 ๐‘‘๐‘—)2 ๐‘‘๐‘—)2 PROPN cana-1650 23 26 1โ‰ค๐‘˜<๐‘—โ‰ค๐‘› 1โ‰ค๐‘˜<๐‘—โ‰ค๐‘› NUM cana-1650 23 27 . . PUNCT cana-1650 24 1 communications communication NOUN cana-1650 24 2 on on ADP cana-1650 24 3 applied apply VERB cana-1650 24 4 nonlinear nonlinear ADJ cana-1650 24 5 analysis analysis NOUN cana-1650 24 6 issn issn NOUN cana-1650 24 7 : : PUNCT cana-1650 24 8 1074 1074 NUM cana-1650 24 9 - - PUNCT cana-1650 24 10 133x 133x NUM cana-1650 24 11 vol vol NOUN cana-1650 24 12 32 32 NUM cana-1650 24 13 no no NOUN cana-1650 24 14 . . NOUN cana-1650 24 15 1 1 NUM cana-1650 24 16 ( ( PUNCT cana-1650 24 17 2025 2025 NUM cana-1650 24 18 ) ) PUNCT cana-1650 24 19 304 304 NUM cana-1650 24 20 https://internationalpubls.com https://internationalpubls.com X cana-1650 24 21 2 2 X cana-1650 24 22 . . NUM cana-1650 24 23 bounds bound NOUN cana-1650 24 24 for for ADP cana-1650 24 25 degree degree NOUN cana-1650 24 26 sum sum NOUN cana-1650 24 27 energy energy NOUN cana-1650 24 28 throughout throughout ADP cana-1650 24 29 this this DET cana-1650 24 30 section section NOUN cana-1650 24 31 g g PROPN cana-1650 24 32 denotes denote VERB cana-1650 24 33 a a DET cana-1650 24 34 simple simple ADJ cana-1650 24 35 graph graph NOUN cana-1650 24 36 . . PUNCT cana-1650 25 1 this this DET cana-1650 25 2 section section NOUN cana-1650 25 3 is be AUX cana-1650 25 4 aimed aim VERB cana-1650 25 5 to to PART cana-1650 25 6 discuss discuss VERB cana-1650 25 7 upper upper ADJ cana-1650 25 8 and and CCONJ cana-1650 25 9 lower low ADJ cana-1650 25 10 bounds bound NOUN cana-1650 25 11 for for ADP cana-1650 25 12 degree degree NOUN cana-1650 25 13 sum sum NOUN cana-1650 25 14 energy energy NOUN cana-1650 25 15 ( ( PUNCT cana-1650 25 16 dse dse PROPN cana-1650 25 17 ) ) PUNCT cana-1650 25 18 of of ADP cana-1650 25 19 ๐บ. ๐บ. NOUN cana-1650 25 20 theorem theorem VERB cana-1650 25 21 2.1 2.1 NUM cana-1650 25 22 . . PUNCT cana-1650 26 1 let let VERB cana-1650 26 2 ๐บ ๐บ PRON cana-1650 26 3 be be AUX cana-1650 26 4 a a DET cana-1650 26 5 connected connected ADJ cana-1650 26 6 graph graph NOUN cana-1650 26 7 with with ADP cana-1650 26 8 ๐‘› ๐‘› PROPN cana-1650 26 9 vertices vertex NOUN cana-1650 26 10 and and CCONJ cana-1650 26 11 ๐‘š ๐‘š ADP cana-1650 26 12 edges edge NOUN cana-1650 26 13 and and CCONJ cana-1650 26 14 2๐”ˆ 2๐”ˆ VERB cana-1650 26 15 โ‰ฅ โ‰ฅ NOUN cana-1650 26 16 ๐‘› ๐‘› PRON cana-1650 26 17 then then ADV cana-1650 26 18 ๐ธ๐‘‘๐‘ (๐บ ๐ธ๐‘‘๐‘ (๐บ NUM cana-1650 26 19 ) ) PUNCT cana-1650 26 20 โ‰ค โ‰ค NOUN cana-1650 27 1 2๐”ˆ 2๐”ˆ NOUN cana-1650 27 2 ๐’ ๐’ PROPN cana-1650 27 3 + + CCONJ cana-1650 27 4 ๐Ÿ ๐Ÿ NUM cana-1650 27 5 ๐’ ๐’ PROPN cana-1650 27 6 โˆš2๐”ˆ(๐‘› โˆš2๐”ˆ(๐‘› NOUN cana-1650 27 7 โˆ’ โˆ’ PROPN cana-1650 27 8 1)(๐’๐Ÿ 1)(๐’๐Ÿ NUM cana-1650 27 9 โˆ’ โˆ’ PROPN cana-1650 27 10 2๐”ˆ 2๐”ˆ NOUN cana-1650 27 11 ) ) PUNCT cana-1650 27 12 . . PUNCT cana-1650 28 1 proof proof NOUN cana-1650 28 2 : : PUNCT cana-1650 28 3 cauchy cauchy PROPN cana-1650 28 4 - - PUNCT cana-1650 28 5 schwarz schwarz PROPN cana-1650 28 6 inequality inequality NOUN cana-1650 28 7 states state VERB cana-1650 28 8 that that SCONJ cana-1650 28 9 if if SCONJ cana-1650 28 10 ( ( PUNCT cana-1650 28 11 ๐‘Ž1 ๐‘Ž1 ADV cana-1650 28 12 , , PUNCT cana-1650 28 13 ๐‘Ž2 ๐‘Ž2 PROPN cana-1650 28 14 , , PUNCT cana-1650 28 15 โ€ฆ โ€ฆ PUNCT cana-1650 28 16 , , PUNCT cana-1650 28 17 ๐‘Ž๐‘› ๐‘Ž๐‘› PROPN cana-1650 28 18 ) ) PUNCT cana-1650 28 19 and and CCONJ cana-1650 28 20 ( ( PUNCT cana-1650 28 21 ๐‘1 ๐‘1 NOUN cana-1650 28 22 , , PUNCT cana-1650 28 23 ๐‘2 ๐‘2 PROPN cana-1650 28 24 , , PUNCT cana-1650 28 25 โ€ฆ โ€ฆ PUNCT cana-1650 28 26 , , PUNCT cana-1650 28 27 ๐‘๐‘› ๐‘๐‘› NOUN cana-1650 28 28 ) ) PUNCT cana-1650 28 29 are be AUX cana-1650 28 30 ๐‘› ๐‘› DET cana-1650 28 31 โˆ’ โˆ’ PROPN cana-1650 28 32 ๐‘ฃ๐‘’๐‘๐‘ก๐‘œ๐‘Ÿ๐‘  ๐‘ฃ๐‘’๐‘๐‘ก๐‘œ๐‘Ÿ๐‘  PROPN cana-1650 28 33 then then ADV cana-1650 28 34 : : PUNCT cana-1650 28 35 ( ( PUNCT cana-1650 28 36 โˆ‘ โˆ‘ INTJ cana-1650 28 37 ๐‘Ž๐‘˜๐‘๐‘˜ ๐‘Ž๐‘˜๐‘๐‘˜ PROPN cana-1650 28 38 ๐‘› ๐‘› PROPN cana-1650 28 39 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 28 40 ) ) PUNCT cana-1650 28 41 2 2 NUM cana-1650 28 42 โ‰ค โ‰ค NOUN cana-1650 28 43 ( ( PUNCT cana-1650 28 44 โˆ‘ โˆ‘ NOUN cana-1650 28 45 ๐‘Ž๐‘˜ ๐‘Ž๐‘˜ PROPN cana-1650 28 46 2 2 NUM cana-1650 28 47 ๐‘› ๐‘› PRON cana-1650 28 48 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 28 49 ) ) PUNCT cana-1650 28 50 ( ( PUNCT cana-1650 28 51 โˆ‘ โˆ‘ PROPN cana-1650 28 52 ๐‘๐‘˜ ๐‘๐‘˜ PROPN cana-1650 28 53 2 2 NUM cana-1650 28 54 ๐‘› ๐‘› DET cana-1650 28 55 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 28 56 ) ) PUNCT cana-1650 28 57 . . PUNCT cana-1650 29 1 for for ADP cana-1650 29 2 ๐‘Ž๐‘˜ ๐‘Ž๐‘˜ NOUN cana-1650 29 3 = = SYM cana-1650 29 4 1 1 NUM cana-1650 29 5 , , PUNCT cana-1650 29 6 ๐‘๐‘˜ ๐‘๐‘˜ NOUN cana-1650 29 7 = = SYM cana-1650 29 8 |๐œ‰๐’Œ| |๐œ‰๐’Œ| NOUN cana-1650 29 9 and and CCONJ cana-1650 29 10 2 2 NUM cana-1650 29 11 โ‰ค โ‰ค NOUN cana-1650 29 12 ๐‘˜ ๐‘˜ DET cana-1650 29 13 โ‰ค โ‰ค NUM cana-1650 29 14 ๐‘› ๐‘› NOUN cana-1650 29 15 , , PUNCT cana-1650 29 16 in in ADP cana-1650 29 17 the the DET cana-1650 29 18 above above ADJ cana-1650 29 19 inequality inequality NOUN cana-1650 29 20 , , PUNCT cana-1650 29 21 we we PRON cana-1650 29 22 obtain obtain VERB cana-1650 29 23 ( ( PUNCT cana-1650 29 24 โˆ‘|๐œ‰๐’Œ| โˆ‘|๐œ‰๐’Œ| NOUN cana-1650 29 25 ๐‘› ๐‘› PRON cana-1650 29 26 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 29 27 ) ) PUNCT cana-1650 29 28 2 2 NUM cana-1650 29 29 โ‰ค โ‰ค NOUN cana-1650 29 30 ( ( PUNCT cana-1650 29 31 โˆ‘ โˆ‘ PROPN cana-1650 29 32 12 12 NUM cana-1650 29 33 ๐‘› ๐‘› PRON cana-1650 29 34 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 29 35 ) ) PUNCT cana-1650 29 36 ( ( PUNCT cana-1650 29 37 โˆ‘|๐œ‰๐’Œ|2 โˆ‘|๐œ‰๐’Œ|2 NOUN cana-1650 29 38 ๐‘› ๐‘› PROPN cana-1650 29 39 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 29 40 ) ) PUNCT cana-1650 29 41 . . PUNCT cana-1650 30 1 therefore therefore ADV cana-1650 30 2 , , PUNCT cana-1650 30 3 ( ( PUNCT cana-1650 30 4 ๐ธ๐‘‘๐‘ (๐บ ๐ธ๐‘‘๐‘ (๐บ NOUN cana-1650 30 5 ) ) PUNCT cana-1650 30 6 โˆ’ โˆ’ ADP cana-1650 30 7 ๐œ‰๐Ÿ)๐Ÿ ๐œ‰๐Ÿ)๐Ÿ ADJ cana-1650 30 8 โ‰ค โ‰ค NUM cana-1650 30 9 ( ( PUNCT cana-1650 30 10 ๐’ ๐’ PROPN cana-1650 30 11 โˆ’ โˆ’ PROPN cana-1650 30 12 ๐Ÿ ๐Ÿ NUM cana-1650 30 13 ) ) PUNCT cana-1650 30 14 โˆ‘ โˆ‘ PROPN cana-1650 30 15 ๐œ‰๐’Œ ๐œ‰๐’Œ ADP cana-1650 30 16 2๐‘› 2๐‘› PROPN cana-1650 30 17 ๐‘˜=1 ๐‘˜=1 X cana-1650 31 1 = = PUNCT cana-1650 31 2 ( ( PUNCT cana-1650 31 3 ๐‘› ๐‘› PRON cana-1650 31 4 โˆ’ โˆ’ NUM cana-1650 31 5 1)(2๐”ˆ 1)(2๐”ˆ SYM cana-1650 31 6 โˆ’ โˆ’ PROPN cana-1650 31 7 ๐œ‰๐Ÿ ๐œ‰๐Ÿ PROPN cana-1650 31 8 ๐Ÿ ๐Ÿ NUM cana-1650 31 9 ) ) PUNCT cana-1650 31 10 , , PUNCT cana-1650 31 11 ๐‘ฌ๐’…๐’”(๐‘ฎ ๐‘ฌ๐’…๐’”(๐‘ฎ NUM cana-1650 31 12 ) ) PUNCT cana-1650 31 13 = = SYM cana-1650 31 14 ๐œ‰๐Ÿ ๐œ‰๐Ÿ PROPN cana-1650 31 15 + + CCONJ cana-1650 31 16 โˆš(๐‘› โˆš(๐‘› PROPN cana-1650 32 1 โˆ’ โˆ’ ADP cana-1650 32 2 1)(2๐”ˆ 1)(2๐”ˆ PRON cana-1650 32 3 โˆ’ โˆ’ PROPN cana-1650 32 4 ๐œ‰๐Ÿ ๐œ‰๐Ÿ PROPN cana-1650 32 5 ๐Ÿ ๐Ÿ NUM cana-1650 32 6 ) ) PUNCT cana-1650 32 7 . . PUNCT cana-1650 33 1 now now ADV cana-1650 33 2 consider consider VERB cana-1650 33 3 the the DET cana-1650 33 4 function function NOUN cana-1650 33 5 , , PUNCT cana-1650 33 6 ๐’‡(๐’™ ๐’‡(๐’™ NOUN cana-1650 33 7 ) ) PUNCT cana-1650 33 8 = = SYM cana-1650 34 1 ๐’™ ๐’™ PROPN cana-1650 34 2 + + NUM cana-1650 34 3 โˆš(๐‘› โˆš(๐‘› PROPN cana-1650 35 1 โˆ’ โˆ’ NUM cana-1650 35 2 1)(2๐”ˆ 1)(2๐”ˆ PRON cana-1650 36 1 โˆ’ โˆ’ ADP cana-1650 36 2 ๐’™๐Ÿ ๐’™๐Ÿ NOUN cana-1650 36 3 ) ) PUNCT cana-1650 36 4 note note VERB cana-1650 36 5 that that SCONJ cana-1650 36 6 ๐‘“ ๐‘“ PRON cana-1650 36 7 is be AUX cana-1650 36 8 decreasing decrease VERB cana-1650 36 9 for for ADP cana-1650 36 10 ๐‘ฅ ๐‘ฅ PROPN cana-1650 36 11 โ‰ฅ โ‰ฅ NOUN cana-1650 36 12 โˆš โˆš ADP cana-1650 36 13 2๐”ˆ 2๐”ˆ NOUN cana-1650 36 14 ๐‘› ๐‘› VERB cana-1650 36 15 , , PUNCT cana-1650 36 16 for for ADP cana-1650 36 17 ๐‘“โ€ฒ(๐‘ฅ ๐‘“โ€ฒ(๐‘ฅ PROPN cana-1650 36 18 ) ) PUNCT cana-1650 36 19 = = SYM cana-1650 36 20 1 1 NUM cana-1650 36 21 โˆ’ โˆ’ PROPN cana-1650 36 22 ( ( PUNCT cana-1650 36 23 ๐‘›โˆ’1)๐‘ฅ ๐‘›โˆ’1)๐‘ฅ X cana-1650 36 24 โˆš(๐‘›โˆ’1)(2๐”ˆโˆ’๐‘ฅ2 โˆš(๐‘›โˆ’1)(2๐”ˆโˆ’๐‘ฅ2 ADV cana-1650 36 25 ) ) PUNCT cana-1650 36 26 โ‰ค โ‰ค NOUN cana-1650 36 27 0 0 NUM cana-1650 36 28 , , PUNCT cana-1650 36 29 if if SCONJ cana-1650 36 30 and and CCONJ cana-1650 36 31 only only ADV cana-1650 36 32 if if SCONJ cana-1650 36 33 , , PUNCT cana-1650 36 34 ๐‘ฅ ๐‘ฅ PROPN cana-1650 36 35 โ‰ฅ โ‰ฅ NOUN cana-1650 36 36 โˆš โˆš ADP cana-1650 36 37 2๐”ˆ 2๐”ˆ VERB cana-1650 36 38 ๐‘› ๐‘› PROPN cana-1650 36 39 . . PUNCT cana-1650 37 1 since since SCONJ cana-1650 37 2 , , PUNCT cana-1650 37 3 1 1 NUM cana-1650 37 4 โ‰ค โ‰ค NUM cana-1650 37 5 โˆš โˆš PUNCT cana-1650 37 6 2๐”ˆ 2๐”ˆ PROPN cana-1650 37 7 ๐‘› ๐‘› DET cana-1650 37 8 โ‰ค โ‰ค NUM cana-1650 37 9 2๐”ˆ 2๐”ˆ NOUN cana-1650 37 10 ๐‘› ๐‘› DET cana-1650 37 11 โ‰ค โ‰ค PROPN cana-1650 37 12 ๐œ‰๐Ÿ ๐œ‰๐Ÿ PROPN cana-1650 37 13 , , PUNCT cana-1650 37 14 we we PRON cana-1650 37 15 have have VERB cana-1650 37 16 , , PUNCT cana-1650 37 17 ๐‘“(๐œ‰๐Ÿ ๐‘“(๐œ‰๐Ÿ NOUN cana-1650 37 18 ) ) PUNCT cana-1650 37 19 โ‰ค โ‰ค NOUN cana-1650 37 20 ๐’‡ ๐’‡ X cana-1650 37 21 ( ( PUNCT cana-1650 37 22 2๐”ˆ 2๐”ˆ NOUN cana-1650 37 23 ๐‘› ๐‘› PROPN cana-1650 37 24 ) ) PUNCT cana-1650 37 25 . . PUNCT cana-1650 38 1 therefore therefore ADV cana-1650 38 2 , , PUNCT cana-1650 38 3 ๐‘ฌ๐’…๐’”(๐‘ฎ ๐‘ฌ๐’…๐’”(๐‘ฎ NUM cana-1650 38 4 ) ) PUNCT cana-1650 38 5 โ‰ค โ‰ค NUM cana-1650 38 6 ๐‘“(๐œ‰๐Ÿ ๐‘“(๐œ‰๐Ÿ NOUN cana-1650 38 7 ) ) PUNCT cana-1650 38 8 โ‰ค โ‰ค NOUN cana-1650 38 9 ๐’‡ ๐’‡ X cana-1650 38 10 ( ( PUNCT cana-1650 38 11 2๐”ˆ 2๐”ˆ NOUN cana-1650 38 12 ๐‘› ๐‘› PROPN cana-1650 38 13 ) ) PUNCT cana-1650 38 14 . . PUNCT cana-1650 39 1 hence hence ADV cana-1650 39 2 , , PUNCT cana-1650 39 3 ๐‘ฌ๐’…๐’”(๐‘ฎ ๐‘ฌ๐’…๐’”(๐‘ฎ NUM cana-1650 39 4 ) ) PUNCT cana-1650 39 5 โ‰ค โ‰ค NOUN cana-1650 39 6 2๐”ˆ 2๐”ˆ VERB cana-1650 39 7 ๐’ ๐’ PROPN cana-1650 39 8 + + CCONJ cana-1650 39 9 โˆš(๐’ โˆš(๐’ VERB cana-1650 39 10 โˆ’ โˆ’ NOUN cana-1650 39 11 ๐Ÿ ๐Ÿ NUM cana-1650 39 12 ) ) PUNCT cana-1650 39 13 ( ( PUNCT cana-1650 39 14 2๐”ˆ 2๐”ˆ NOUN cana-1650 39 15 โˆ’ โˆ’ PROPN cana-1650 40 1 ( ( PUNCT cana-1650 40 2 2๐”ˆ 2๐”ˆ NOUN cana-1650 40 3 ๐‘› ๐‘› NOUN cana-1650 40 4 ) ) PUNCT cana-1650 40 5 2 2 X cana-1650 40 6 ) ) PUNCT cana-1650 40 7 communications communication NOUN cana-1650 40 8 on on ADP cana-1650 40 9 applied apply VERB cana-1650 40 10 nonlinear nonlinear ADJ cana-1650 40 11 analysis analysis NOUN cana-1650 40 12 issn issn NOUN cana-1650 40 13 : : PUNCT cana-1650 40 14 1074 1074 NUM cana-1650 40 15 - - PUNCT cana-1650 40 16 133x 133x NUM cana-1650 40 17 vol vol NOUN cana-1650 40 18 32 32 NUM cana-1650 40 19 no no NOUN cana-1650 40 20 . . NOUN cana-1650 40 21 1 1 NUM cana-1650 40 22 ( ( PUNCT cana-1650 40 23 2025 2025 NUM cana-1650 40 24 ) ) PUNCT cana-1650 40 25 305 305 NUM cana-1650 40 26 https://internationalpubls.com https://internationalpubls.com X cana-1650 40 27 or or CCONJ cana-1650 40 28 equivalently equivalently ADV cana-1650 40 29 , , PUNCT cana-1650 40 30 ๐‘ฌ๐’…๐’”(๐‘ฎ ๐‘ฌ๐’…๐’”(๐‘ฎ NUM cana-1650 40 31 ) ) PUNCT cana-1650 40 32 โ‰ค โ‰ค NOUN cana-1650 41 1 2๐”ˆ 2๐”ˆ VERB cana-1650 41 2 ๐’ ๐’ PROPN cana-1650 42 1 + + CCONJ cana-1650 42 2 ๐Ÿ ๐Ÿ NUM cana-1650 42 3 ๐’ ๐’ PROPN cana-1650 42 4 โˆš2๐”ˆ(๐‘› โˆš2๐”ˆ(๐‘› NOUN cana-1650 42 5 โˆ’ โˆ’ PROPN cana-1650 42 6 1)(๐’๐Ÿ 1)(๐’๐Ÿ NUM cana-1650 42 7 โˆ’ โˆ’ PROPN cana-1650 42 8 2๐”ˆ 2๐”ˆ PROPN cana-1650 42 9 ) ) PUNCT cana-1650 42 10 . . PUNCT cana-1650 43 1 theorem theorem VERB cana-1650 43 2 2.2 2.2 NUM cana-1650 43 3 . . PUNCT cana-1650 44 1 let let VERB cana-1650 44 2 g g NOUN cana-1650 44 3 be be AUX cana-1650 44 4 simple simple ADJ cana-1650 44 5 graph graph NOUN cana-1650 44 6 connected connect VERB cana-1650 44 7 having have VERB cana-1650 44 8 order order NOUN cana-1650 44 9 ๐‘› ๐‘› NOUN cana-1650 44 10 and and CCONJ cana-1650 44 11 size size VERB cana-1650 44 12 ๐‘š ๐‘š PROPN cana-1650 44 13 , , PUNCT cana-1650 44 14 then then ADV cana-1650 44 15 ๐ธ๐‘‘๐‘ (๐บ ๐ธ๐‘‘๐‘ (๐บ NOUN cana-1650 44 16 ) ) PUNCT cana-1650 44 17 โ‰ค โ‰ค NOUN cana-1650 44 18 4๐”ˆ 4๐”ˆ NOUN cana-1650 44 19 ( ( PUNCT cana-1650 44 20 ๐œ‰๐Ÿโˆ’๐œ‰๐’ ๐œ‰๐Ÿโˆ’๐œ‰๐’ NUM cana-1650 44 21 ) ) PUNCT cana-1650 44 22 . . PUNCT cana-1650 45 1 2.1 2.1 NUM cana-1650 45 2 proof proof NOUN cana-1650 45 3 : : PUNCT cana-1650 45 4 considering consider VERB cana-1650 45 5 , , PUNCT cana-1650 45 6 ๐‘ฅ ๐‘ฅ PROPN cana-1650 45 7 = = SYM cana-1650 45 8 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ X cana-1650 45 9 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ PROPN cana-1650 45 10 ๐‘ฆ ๐‘ฆ PROPN cana-1650 45 11 = = SYM cana-1650 45 12 ๐‘ฆ๐‘˜ ๐‘ฆ๐‘˜ NOUN cana-1650 45 13 , , PUNCT cana-1650 45 14 1 1 NUM cana-1650 45 15 โ‰ค โ‰ค NOUN cana-1650 45 16 ๐‘˜ ๐‘˜ DET cana-1650 45 17 โ‰ค โ‰ค NUM cana-1650 45 18 ๐‘› ๐‘› PRON cana-1650 45 19 as as ADP cana-1650 45 20 real real ADJ cana-1650 45 21 sequence sequence NOUN cana-1650 45 22 such such ADJ cana-1650 45 23 that that SCONJ cana-1650 45 24 โˆ‘ โˆ‘ PUNCT cana-1650 45 25 |๐‘ฅ๐‘˜| |๐‘ฅ๐‘˜| PROPN cana-1650 45 26 = = NOUN cana-1650 45 27 ๐‘› ๐‘› PRON cana-1650 45 28 ๐‘˜=1 ๐‘˜=1 NUM cana-1650 45 29 1 1 NUM cana-1650 45 30 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-1650 45 31 โˆ‘ โˆ‘ PUNCT cana-1650 45 32 |๐‘ฅ๐‘˜| |๐‘ฅ๐‘˜| PROPN cana-1650 45 33 = = SYM cana-1650 45 34 0 0 NUM cana-1650 45 35 , , PUNCT cana-1650 45 36 ๐‘› ๐‘› PROPN cana-1650 45 37 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 45 38 the the DET cana-1650 45 39 inequality inequality NOUN cana-1650 45 40 stated state VERB cana-1650 45 41 below below ADV cana-1650 45 42 has have AUX cana-1650 45 43 been be AUX cana-1650 45 44 proved prove VERB cana-1650 45 45 in[11 in[11 NOUN cana-1650 45 46 ] ] PUNCT cana-1650 45 47 : : PUNCT cana-1650 45 48 |โˆ‘ |โˆ‘ X cana-1650 45 49 ๐‘ฅ๐‘˜๐‘ฆ๐‘˜ ๐‘ฅ๐‘˜๐‘ฆ๐‘˜ NOUN cana-1650 45 50 ๐‘› ๐‘› PRON cana-1650 45 51 ๐‘˜=1 ๐‘˜=1 PUNCT cana-1650 45 52 | | ADV cana-1650 45 53 โ‰ค โ‰ค NUM cana-1650 45 54 1 1 NUM cana-1650 45 55 2 2 NUM cana-1650 45 56 ( ( PUNCT cana-1650 45 57 max max PROPN cana-1650 45 58 1โ‰ค๐‘˜โ‰ค๐‘› 1โ‰ค๐‘˜โ‰ค๐‘› NUM cana-1650 45 59 ( ( PUNCT cana-1650 45 60 ๐‘ฆ๐‘˜ ๐‘ฆ๐‘˜ NOUN cana-1650 45 61 ) ) PUNCT cana-1650 45 62 โˆ’ โˆ’ PROPN cana-1650 45 63 min min NOUN cana-1650 45 64 1โ‰ค๐‘˜โ‰ค๐‘› 1โ‰ค๐‘˜โ‰ค๐‘› NUM cana-1650 45 65 ( ( PUNCT cana-1650 45 66 ๐‘ฆ๐‘˜ ๐‘ฆ๐‘˜ NOUN cana-1650 45 67 ) ) PUNCT cana-1650 45 68 ) ) PUNCT cana-1650 45 69 2.2 2.2 NUM cana-1650 45 70 since since SCONJ cana-1650 45 71 , , PUNCT cana-1650 45 72 โˆ‘ โˆ‘ PUNCT cana-1650 45 73 |๐œ‰๐’Œ| |๐œ‰๐’Œ| NOUN cana-1650 45 74 = = SYM cana-1650 45 75 0,๐‘› 0,๐‘› ADJ cana-1650 45 76 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 45 77 for for ADP cana-1650 45 78 ๐‘ฆ๐‘˜ ๐‘ฆ๐‘˜ NOUN cana-1650 45 79 = = SYM cana-1650 45 80 ๐œ‰๐’Œ ๐œ‰๐’Œ PROPN cana-1650 45 81 and and CCONJ cana-1650 45 82 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ X cana-1650 45 83 = = SYM cana-1650 45 84 ๐œ‰๐’Œ ๐œ‰๐’Œ PROPN cana-1650 45 85 โˆ‘ โˆ‘ PUNCT cana-1650 45 86 |๐œ‰๐’Œ|,๐‘› |๐œ‰๐’Œ|,๐‘› ADV cana-1650 45 87 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 45 88 for for ADP cana-1650 45 89 each each DET cana-1650 45 90 ๐‘˜ ๐‘˜ PROPN cana-1650 45 91 โˆˆ โˆˆ PROPN cana-1650 45 92 { { PUNCT cana-1650 45 93 1,2 1,2 NUM cana-1650 45 94 , , PUNCT cana-1650 45 95 โ€ฆ โ€ฆ PUNCT cana-1650 45 96 , , PUNCT cana-1650 45 97 3 3 X cana-1650 45 98 } } PUNCT cana-1650 45 99 we we PRON cana-1650 45 100 have have VERB cana-1650 45 101 , , PUNCT cana-1650 45 102 โˆ‘ โˆ‘ PROPN cana-1650 45 103 ๐‘ฅ๐‘˜ ๐‘ฅ๐‘˜ NOUN cana-1650 45 104 ๐‘› ๐‘› NOUN cana-1650 45 105 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 46 1 = = PUNCT cana-1650 46 2 โˆ‘ โˆ‘ PUNCT cana-1650 46 3 ๐œ‰๐’Œ ๐œ‰๐’Œ NOUN cana-1650 46 4 ๐‘› ๐‘› PRON cana-1650 46 5 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 46 6 โˆ‘ โˆ‘ PROPN cana-1650 46 7 |๐œ‰๐’Œ|๐‘› |๐œ‰๐’Œ|๐‘› NOUN cana-1650 46 8 ๐‘˜=1 ๐‘˜=1 X cana-1650 46 9 = = SYM cana-1650 46 10 0 0 NUM cana-1650 46 11 and and CCONJ cana-1650 46 12 โˆ‘|๐‘ฅ๐‘˜| โˆ‘|๐‘ฅ๐‘˜| NOUN cana-1650 46 13 ๐‘› ๐‘› DET cana-1650 46 14 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 46 15 = = PUNCT cana-1650 46 16 โˆ‘ โˆ‘ PUNCT cana-1650 46 17 |๐œ‰๐’Œ|๐‘› |๐œ‰๐’Œ|๐‘› PUNCT cana-1650 46 18 ๐‘˜=1 ๐‘˜=1 SYM cana-1650 46 19 โˆ‘ โˆ‘ PUNCT cana-1650 46 20 |๐œ‰๐’Œ|๐‘› |๐œ‰๐’Œ|๐‘› NOUN cana-1650 46 21 ๐‘˜=1 ๐‘˜=1 X cana-1650 46 22 = = SYM cana-1650 46 23 ๐Ÿ ๐Ÿ NUM cana-1650 46 24 thus thus ADV cana-1650 46 25 , , PUNCT cana-1650 46 26 the the DET cana-1650 46 27 inequality inequality NOUN cana-1650 46 28 ( ( PUNCT cana-1650 46 29 2.2 2.2 NUM cana-1650 46 30 ) ) PUNCT cana-1650 46 31 holds hold VERB cana-1650 46 32 . . PUNCT cana-1650 47 1 since since SCONJ cana-1650 47 2 , , PUNCT cana-1650 47 3 โˆ‘ โˆ‘ PUNCT cana-1650 47 4 ๐œ‰๐‘˜ ๐œ‰๐‘˜ ADJ cana-1650 47 5 2 2 NUM cana-1650 47 6 = = SYM cana-1650 47 7 2๐”ˆ๐‘› 2๐”ˆ๐‘› NUM cana-1650 47 8 ๐‘˜=1 ๐‘˜=1 NOUN cana-1650 47 9 , , PUNCT cana-1650 47 10 we we PRON cana-1650 47 11 have have AUX cana-1650 47 12 |โˆ‘ |โˆ‘ VERB cana-1650 47 13 ๐‘ฅ๐‘˜๐‘ฆ๐‘˜ ๐‘ฅ๐‘˜๐‘ฆ๐‘˜ NOUN cana-1650 47 14 ๐‘› ๐‘› PRON cana-1650 47 15 ๐‘˜=1 ๐‘˜=1 PUNCT cana-1650 47 16 | | NOUN cana-1650 47 17 = = PUNCT cana-1650 47 18 |โˆ‘ |โˆ‘ NUM cana-1650 48 1 |๐œ‰๐’Œ|๐‘› |๐œ‰๐’Œ|๐‘› NOUN cana-1650 48 2 ๐‘˜=1 ๐‘˜=1 X cana-1650 48 3 โˆ™ โˆ™ PROPN cana-1650 48 4 ๐œ‰๐’Œ ๐œ‰๐’Œ ADP cana-1650 48 5 โˆ‘ โˆ‘ PUNCT cana-1650 48 6 |๐œ‰๐’Œ|๐‘› |๐œ‰๐’Œ|๐‘› NOUN cana-1650 48 7 ๐‘˜=1 ๐‘˜=1 PUNCT cana-1650 49 1 | | NOUN cana-1650 49 2 = = SYM cana-1650 50 1 | | ADV cana-1650 50 2 โˆ‘ โˆ‘ INTJ cana-1650 50 3 ( ( PUNCT cana-1650 50 4 ๐œ‰๐’Œ)๐Ÿ๐‘› ๐œ‰๐’Œ)๐Ÿ๐‘› NOUN cana-1650 50 5 ๐‘˜=1 ๐‘˜=1 X cana-1650 50 6 โˆ‘ โˆ‘ PROPN cana-1650 50 7 |๐œ‰๐’Œ|๐‘› |๐œ‰๐’Œ|๐‘› PRON cana-1650 50 8 ๐‘˜=1 ๐‘˜=1 PUNCT cana-1650 51 1 | | ADV cana-1650 51 2 = = NOUN cana-1650 51 3 2๐”ˆ 2๐”ˆ NOUN cana-1650 51 4 ๐ธ๐ท๐‘†(๐บ ๐ธ๐ท๐‘†(๐บ NOUN cana-1650 51 5 ) ) PUNCT cana-1650 51 6 . . PUNCT cana-1650 52 1 applying apply VERB cana-1650 52 2 this this PRON cana-1650 52 3 in in ADP cana-1650 52 4 ( ( PUNCT cana-1650 52 5 2.2 2.2 NUM cana-1650 52 6 ) ) PUNCT cana-1650 52 7 , , PUNCT cana-1650 52 8 we we PRON cana-1650 52 9 get get VERB cana-1650 52 10 , , PUNCT cana-1650 52 11 2๐”ˆ 2๐”ˆ NOUN cana-1650 52 12 ๐ธ๐‘‘๐‘ (๐บ ๐ธ๐‘‘๐‘ (๐บ NOUN cana-1650 52 13 ) ) PUNCT cana-1650 52 14 โ‰ค โ‰ค NUM cana-1650 52 15 ๐Ÿ ๐Ÿ NUM cana-1650 52 16 ๐Ÿ ๐Ÿ NUM cana-1650 52 17 ( ( PUNCT cana-1650 52 18 ๐’Ž๐’‚๐’™(๐œ‰๐’Œ ๐’Ž๐’‚๐’™(๐œ‰๐’Œ NOUN cana-1650 52 19 ) ) PUNCT cana-1650 52 20 โˆ’ โˆ’ PROPN cana-1650 52 21 ๐’Ž๐’Š๐’(๐œ‰๐’Œ ๐’Ž๐’Š๐’(๐œ‰๐’Œ NOUN cana-1650 52 22 ) ) PUNCT cana-1650 52 23 ) ) PUNCT cana-1650 52 24 , , PUNCT cana-1650 52 25 from from ADP cana-1650 52 26 which which PRON cana-1650 52 27 , , PUNCT cana-1650 52 28 we we PRON cana-1650 52 29 have have VERB cana-1650 52 30 2๐”ˆ 2๐”ˆ VERB cana-1650 52 31 ๐ธ๐‘‘๐‘ (๐บ ๐ธ๐‘‘๐‘ (๐บ NOUN cana-1650 52 32 ) ) PUNCT cana-1650 52 33 โ‰ค โ‰ค NUM cana-1650 52 34 ๐Ÿ ๐Ÿ NUM cana-1650 52 35 ๐Ÿ ๐Ÿ NUM cana-1650 52 36 ( ( PUNCT cana-1650 52 37 ๐œ‰๐Ÿ ๐œ‰๐Ÿ PROPN cana-1650 52 38 โˆ’ โˆ’ PROPN cana-1650 52 39 ๐œ‰๐’ ๐œ‰๐’ PROPN cana-1650 52 40 ) ) PUNCT cana-1650 52 41 . . PUNCT cana-1650 53 1 if if SCONJ cana-1650 53 2 ๐บ ๐บ PROPN cana-1650 53 3 โ‰… โ‰… PROPN cana-1650 53 4 ๐พ๐‘› ๐พ๐‘› PROPN cana-1650 53 5 , , PUNCT cana-1650 53 6 then then ADV cana-1650 53 7 we we PRON cana-1650 53 8 see see VERB cana-1650 53 9 that that SCONJ cana-1650 53 10 , , PUNCT cana-1650 53 11 ๐œ‰๐’Œ ๐œ‰๐’Œ ADP cana-1650 53 12 = = SYM cana-1650 53 13 ( ( PUNCT cana-1650 53 14 ๐’ ๐’ PROPN cana-1650 53 15 โˆ’ โˆ’ PROPN cana-1650 53 16 ๐Ÿ)2 ๐Ÿ)2 PROPN cana-1650 53 17 , , PUNCT cana-1650 53 18 ๐œ‰๐Ÿ ๐œ‰๐Ÿ NOUN cana-1650 53 19 = = SYM cana-1650 53 20 โˆ’(๐’ โˆ’(๐’ NOUN cana-1650 53 21 โˆ’ โˆ’ NOUN cana-1650 53 22 ๐Ÿ ๐Ÿ NUM cana-1650 53 23 ) ) PUNCT cana-1650 53 24 , , PUNCT cana-1650 53 25 โ€ฆ โ€ฆ PUNCT cana-1650 53 26 . . PUNCT cana-1650 54 1 , , PUNCT cana-1650 54 2 ๐œ‰๐’ ๐œ‰๐’ VERB cana-1650 54 3 = = ADJ cana-1650 54 4 โˆ’(๐’ โˆ’(๐’ NOUN cana-1650 54 5 โˆ’ โˆ’ NOUN cana-1650 54 6 ๐Ÿ ๐Ÿ NUM cana-1650 54 7 ) ) PUNCT cana-1650 54 8 and and CCONJ cana-1650 54 9 , , PUNCT cana-1650 54 10 ๐œ‰๐Ÿ ๐œ‰๐Ÿ PROPN cana-1650 54 11 โˆ’ โˆ’ PROPN cana-1650 54 12 ๐œ‰๐’ ๐œ‰๐’ PROPN cana-1650 54 13 = = PROPN cana-1650 54 14 ๐’(๐’ ๐’(๐’ PROPN cana-1650 54 15 โˆ’ โˆ’ PROPN cana-1650 54 16 ๐Ÿ ๐Ÿ NUM cana-1650 54 17 ) ) PUNCT cana-1650 54 18 . . PUNCT cana-1650 55 1 so so ADV cana-1650 55 2 the the DET cana-1650 55 3 equality equality NOUN cana-1650 55 4 holds hold VERB cana-1650 55 5 in in ADP cana-1650 55 6 ( ( PUNCT cana-1650 55 7 2.1 2.1 NUM cana-1650 55 8 ) ) PUNCT cana-1650 55 9 . . PUNCT cana-1650 56 1 * * PUNCT cana-1650 56 2 * * PUNCT cana-1650 56 3 * * PUNCT cana-1650 56 4 * * PUNCT cana-1650 56 5 * * PUNCT cana-1650 56 6 * * PUNCT cana-1650 56 7 * * PUNCT cana-1650 56 8 * * PUNCT cana-1650 56 9 * * VERB cana-1650 56 10 communications communication NOUN cana-1650 56 11 on on ADP cana-1650 56 12 applied apply VERB cana-1650 56 13 nonlinear nonlinear ADJ cana-1650 56 14 analysis analysis NOUN cana-1650 56 15 issn issn NOUN cana-1650 56 16 : : PUNCT cana-1650 56 17 1074 1074 NUM cana-1650 56 18 - - PUNCT cana-1650 56 19 133x 133x NUM cana-1650 56 20 vol vol NOUN cana-1650 56 21 32 32 NUM cana-1650 56 22 no no NOUN cana-1650 56 23 . . NOUN cana-1650 56 24 1 1 NUM cana-1650 56 25 ( ( PUNCT cana-1650 56 26 2025 2025 NUM cana-1650 56 27 ) ) PUNCT cana-1650 57 1 306 306 NUM cana-1650 57 2 https://internationalpubls.com https://internationalpubls.com X cana-1650 57 3 acknowledgement acknowledgement NOUN cana-1650 57 4 : : PUNCT cana-1650 57 5 the the DET cana-1650 57 6 authors author NOUN cana-1650 57 7 are be AUX cana-1650 57 8 thankful thankful ADJ cana-1650 57 9 to to PART cana-1650 57 10 prof prof PROPN cana-1650 57 11 . . PUNCT cana-1650 58 1 chandrashekara chandrashekara PROPN cana-1650 58 2 adiga adiga PROPN cana-1650 58 3 for for ADP cana-1650 58 4 his his PRON cana-1650 58 5 encouragement encouragement NOUN cana-1650 58 6 and and CCONJ cana-1650 58 7 suggestions suggestion NOUN cana-1650 58 8 . . PUNCT cana-1650 59 1 references reference NOUN cana-1650 59 2 [ [ X cana-1650 59 3 1 1 NUM cana-1650 59 4 ] ] PUNCT cana-1650 59 5 c. c. PROPN cana-1650 59 6 adiga adiga PROPN cana-1650 59 7 and and CCONJ cana-1650 59 8 smith smith PROPN cana-1650 59 9 m m PROPN cana-1650 59 10 , , PUNCT cana-1650 59 11 on on ADP cana-1650 59 12 maximum maximum ADJ cana-1650 59 13 degree degree NOUN cana-1650 59 14 energy energy NOUN cana-1650 59 15 of of ADP cana-1650 59 16 a a DET cana-1650 59 17 graph graph NOUN cana-1650 59 18 , , PUNCT cana-1650 59 19 int int NOUN cana-1650 59 20 . . PUNCT cana-1650 60 1 j. j. PROPN cana-1650 60 2 contemp contemp PROPN cana-1650 60 3 . . PUNCT cana-1650 61 1 math math NOUN cana-1650 61 2 . . PUNCT cana-1650 62 1 sciences science NOUN cana-1650 62 2 , , PUNCT cana-1650 62 3 4(34 4(34 NUM cana-1650 62 4 ) ) PUNCT cana-1650 62 5 , , PUNCT cana-1650 62 6 ( ( PUNCT cana-1650 62 7 2012 2012 NUM cana-1650 62 8 ) ) PUNCT cana-1650 62 9 . . PUNCT cana-1650 63 1 [ [ X cana-1650 63 2 2 2 X cana-1650 63 3 ] ] X cana-1650 63 4 d. d. NOUN cana-1650 63 5 babi_c babi_c PROPN cana-1650 63 6 and and CCONJ cana-1650 63 7 i. i. PROPN cana-1650 63 8 gutman gutman PROPN cana-1650 63 9 , , PUNCT cana-1650 63 10 more more ADV cana-1650 63 11 lower low ADJ cana-1650 63 12 bounds bound NOUN cana-1650 63 13 for for ADP cana-1650 63 14 the the DET cana-1650 63 15 total total ADJ cana-1650 63 16 ๐œ‹ ๐œ‹ NOUN cana-1650 63 17 โˆ’electron โˆ’electron ADP cana-1650 63 18 energy energy NOUN cana-1650 63 19 of of ADP cana-1650 63 20 alternant alternant ADJ cana-1650 63 21 hydrocarbons hydrocarbon NOUN cana-1650 63 22 , , PUNCT cana-1650 63 23 macth macth PROPN cana-1650 63 24 commun commun PROPN cana-1650 63 25 . . PUNCT cana-1650 64 1 comput comput PROPN cana-1650 64 2 . . PUNCT cana-1650 64 3 , , PUNCT cana-1650 64 4 32 32 NUM cana-1650 64 5 ( ( PUNCT cana-1650 64 6 1995 1995 NUM cana-1650 64 7 ) ) PUNCT cana-1650 64 8 , , PUNCT cana-1650 64 9 7 7 NUM cana-1650 64 10 - - SYM cana-1650 64 11 17 17 NUM cana-1650 64 12 . . PUNCT cana-1650 65 1 [ [ X cana-1650 65 2 3 3 NUM cana-1650 65 3 ] ] PUNCT cana-1650 65 4 e.hโ€ขukel e.hโ€ขukel NOUN cana-1650 65 5 , , PUNCT cana-1650 65 6 quantentheoretische quantentheoretische NOUN cana-1650 65 7 beitrโ€ขage beitrโ€ขage NOUN cana-1650 65 8 zum zum PROPN cana-1650 65 9 benzolproblem benzolproblem NOUN cana-1650 65 10 i. i. PROPN cana-1650 65 11 die die PROPN cana-1650 65 12 elektronenkon elektronenkon PROPN cana-1650 65 13 guration guration PROPN cana-1650 65 14 des des PROPN cana-1650 65 15 benzols benzol NOUN cana-1650 65 16 und und VERB cana-1650 65 17 verwandter verwandter NOUN cana-1650 65 18 vebindungen.z vebindungen.z PROPN cana-1650 65 19 : : PUNCT cana-1650 65 20 phys.70 phys.70 NOUN cana-1650 65 21 ( ( PUNCT cana-1650 65 22 1931 1931 NUM cana-1650 65 23 ) ) PUNCT cana-1650 65 24 204 204 NUM cana-1650 65 25 - - SYM cana-1650 65 26 286 286 NUM cana-1650 65 27 . . PUNCT cana-1650 66 1 [ [ X cana-1650 66 2 4 4 NUM cana-1650 66 3 ] ] X cana-1650 66 4 b.j b.j PROPN cana-1650 66 5 . . PROPN cana-1650 66 6 mcclelland mcclelland PROPN cana-1650 66 7 , , PUNCT cana-1650 66 8 properties property NOUN cana-1650 66 9 of of ADP cana-1650 66 10 the the DET cana-1650 66 11 latent latent NOUN cana-1650 66 12 roots root NOUN cana-1650 66 13 of of ADP cana-1650 66 14 a a DET cana-1650 66 15 matrix matrix NOUN cana-1650 66 16 : : PUNCT cana-1650 66 17 the the DET cana-1650 66 18 estimation estimation NOUN cana-1650 66 19 of of ADP cana-1650 66 20 ๐œ‹ ๐œ‹ PRON cana-1650 66 21 โˆ’electron โˆ’electron ADP cana-1650 66 22 energy energy NOUN cana-1650 66 23 , , PUNCT cana-1650 66 24 j. j. PROPN cana-1650 66 25 chem.phys chem.phys PROPN cana-1650 66 26 . . PROPN cana-1650 66 27 , , PUNCT cana-1650 66 28 41 41 NUM cana-1650 66 29 , , PUNCT cana-1650 66 30 no no INTJ cana-1650 66 31 . . NOUN cana-1650 66 32 1 1 NUM cana-1650 66 33 ( ( PUNCT cana-1650 66 34 2007 2007 NUM cana-1650 66 35 ) ) PUNCT cana-1650 66 36 . . PUNCT cana-1650 67 1 [ [ X cana-1650 67 2 5 5 X cana-1650 67 3 ] ] PUNCT cana-1650 67 4 s. s. PROPN cana-1650 67 5 s. s. PROPN cana-1650 67 6 dragomir dragomir PROPN cana-1650 67 7 , , PUNCT cana-1650 67 8 a a DET cana-1650 67 9 survey survey NOUN cana-1650 67 10 on on ADP cana-1650 67 11 cauchy cauchy NOUN cana-1650 67 12 - - PUNCT cana-1650 67 13 bunyakovsky bunyakovsky NOUN cana-1650 67 14 - - PUNCT cana-1650 67 15 schwarz schwarz NOUN cana-1650 67 16 type type NOUN cana-1650 67 17 discreate discreate PROPN cana-1650 67 18 inequalities inequality NOUN cana-1650 67 19 , , PUNCT cana-1650 67 20 j.inequal j.inequal ADJ cana-1650 67 21 . . PUNCT cana-1650 68 1 pure pure ADJ cana-1650 68 2 appl.math appl.math NOUN cana-1650 68 3 . . PUNCT cana-1650 69 1 4(3 4(3 NUM cana-1650 69 2 ) ) PUNCT cana-1650 69 3 ( ( PUNCT cana-1650 69 4 2003 2003 NUM cana-1650 69 5 ) ) PUNCT cana-1650 69 6 , , PUNCT cana-1650 69 7 1 1 NUM cana-1650 69 8 - - SYM cana-1650 69 9 142 142 NUM cana-1650 69 10 . . PUNCT cana-1650 70 1 [ [ X cana-1650 70 2 6 6 NUM cana-1650 70 3 ] ] PUNCT cana-1650 70 4 i. i. PROPN cana-1650 70 5 gutman gutman PROPN cana-1650 70 6 , , PUNCT cana-1650 70 7 the the DET cana-1650 70 8 energy energy NOUN cana-1650 70 9 of of ADP cana-1650 70 10 a a DET cana-1650 70 11 graph graph NOUN cana-1650 70 12 , , PUNCT cana-1650 70 13 ber ber PROPN cana-1650 70 14 . . PUNCT cana-1650 70 15 math math NOUN cana-1650 70 16 . . PUNCT cana-1650 71 1 stat stat PROPN cana-1650 71 2 . . PUNCT cana-1650 72 1 sekt sekt PROPN cana-1650 72 2 . . PUNCT cana-1650 73 1 forschungsz.graz forschungsz.graz NOUN cana-1650 73 2 , , PUNCT cana-1650 73 3 103 103 NUM cana-1650 73 4 ( ( PUNCT cana-1650 73 5 1978 1978 NUM cana-1650 73 6 ) ) PUNCT cana-1650 73 7 , , PUNCT cana-1650 73 8 1 1 NUM cana-1650 73 9 - - SYM cana-1650 73 10 22 22 NUM cana-1650 73 11 . . PUNCT cana-1650 74 1 [ [ X cana-1650 74 2 7 7 NUM cana-1650 74 3 ] ] X cana-1650 74 4 i. i. PROPN cana-1650 74 5 gutman gutman PROPN cana-1650 74 6 and and CCONJ cana-1650 74 7 b. b. PROPN cana-1650 74 8 jhou jhou PROPN cana-1650 74 9 , , PUNCT cana-1650 74 10 laplacian laplacian ADJ cana-1650 74 11 energy energy NOUN cana-1650 74 12 of of ADP cana-1650 74 13 a a DET cana-1650 74 14 graph graph NOUN cana-1650 74 15 , , PUNCT cana-1650 74 16 lin lin PROPN cana-1650 74 17 . . PUNCT cana-1650 75 1 algebra algebra PROPN cana-1650 75 2 appl appl PROPN cana-1650 75 3 , , PUNCT cana-1650 75 4 414 414 NUM cana-1650 75 5 ( ( PUNCT cana-1650 75 6 2006 2006 NUM cana-1650 75 7 ) ) PUNCT cana-1650 75 8 , , PUNCT cana-1650 75 9 29 29 NUM cana-1650 75 10 - - SYM cana-1650 75 11 37 37 NUM cana-1650 75 12 . . PUNCT cana-1650 76 1 [ [ X cana-1650 76 2 8 8 NUM cana-1650 76 3 ] ] X cana-1650 76 4 g. g. PROPN cana-1650 76 5 indulal indulal PROPN cana-1650 76 6 , , PUNCT cana-1650 76 7 i. i. PROPN cana-1650 76 8 gutman gutman PROPN cana-1650 76 9 , , PUNCT cana-1650 76 10 a. a. NOUN cana-1650 76 11 vijaykumar vijaykumar PROPN cana-1650 76 12 , , PUNCT cana-1650 76 13 on on ADP cana-1650 76 14 distance distance NOUN cana-1650 76 15 energy energy NOUN cana-1650 76 16 of of ADP cana-1650 76 17 graphs graph NOUN cana-1650 76 18 , , PUNCT cana-1650 76 19 match match VERB cana-1650 76 20 commun.math commun.math PROPN cana-1650 76 21 . . PUNCT cana-1650 77 1 comput comput PROPN cana-1650 77 2 . . PUNCT cana-1650 78 1 chem chem NOUN cana-1650 78 2 . . PUNCT cana-1650 79 1 60(2008 60(2008 NOUN cana-1650 79 2 ) ) PUNCT cana-1650 79 3 , , PUNCT cana-1650 79 4 355 355 NUM cana-1650 79 5 - - SYM cana-1650 79 6 372 372 NUM cana-1650 79 7 . . PUNCT cana-1650 80 1 [ [ X cana-1650 80 2 10 10 NUM cana-1650 80 3 ] ] PUNCT cana-1650 80 4 m. m. PROPN cana-1650 80 5 r. r. PROPN cana-1650 80 6 jooyandeh jooyandeh PROPN cana-1650 80 7 , , PUNCT cana-1650 80 8 d. d. PROPN cana-1650 80 9 kiani kiani PROPN cana-1650 80 10 , , PUNCT cana-1650 80 11 m. m. NOUN cana-1650 80 12 mirzakhah mirzakhah PROPN cana-1650 80 13 , , PUNCT cana-1650 80 14 incidence incidence ADJ cana-1650 80 15 energy energy NOUN cana-1650 80 16 of of ADP cana-1650 80 17 graph graph NOUN cana-1650 80 18 , , PUNCT cana-1650 80 19 match match NOUN cana-1650 80 20 commun commun PROPN cana-1650 80 21 . . PUNCT cana-1650 80 22 math math PROPN cana-1650 80 23 . . PUNCT cana-1650 81 1 comput comput NOUN cana-1650 81 2 . . PUNCT cana-1650 82 1 chem chem NOUN cana-1650 82 2 . . PUNCT cana-1650 83 1 bf60 bf60 PROPN cana-1650 83 2 ( ( PUNCT cana-1650 83 3 2008 2008 NUM cana-1650 83 4 ) ) PUNCT cana-1650 83 5 , , PUNCT cana-1650 83 6 561 561 NUM cana-1650 83 7 - - SYM cana-1650 83 8 572 572 NUM cana-1650 83 9 . . PUNCT cana-1650 84 1 [ [ X cana-1650 84 2 11 11 NUM cana-1650 84 3 ] ] X cana-1650 84 4 d. d. PROPN cana-1650 84 5 s. s. PROPN cana-1650 84 6 mitrinovi_c mitrinovi_c PROPN cana-1650 84 7 and and CCONJ cana-1650 84 8 p. p. NOUN cana-1650 84 9 m. m. NOUN cana-1650 84 10 vasi_c vasi_c PROPN cana-1650 84 11 , , PUNCT cana-1650 84 12 analytic analytic ADJ cana-1650 84 13 inequalities inequality NOUN cana-1650 84 14 , , PUNCT cana-1650 84 15 springer springer NOUN cana-1650 84 16 , , PUNCT cana-1650 84 17 berlin berlin PROPN cana-1650 84 18 , , PUNCT cana-1650 84 19 ( ( PUNCT cana-1650 84 20 1970 1970 NUM cana-1650 84 21 ) ) PUNCT cana-1650 84 22 . . PUNCT cana-1650 85 1 [ [ X cana-1650 85 2 12 12 NUM cana-1650 85 3 ] ] PUNCT cana-1650 85 4 h.s.ramane h.s.ramane NOUN cana-1650 85 5 , , PUNCT cana-1650 85 6 d.s.revankar d.s.revankar NOUN cana-1650 85 7 , , PUNCT cana-1650 85 8 and and CCONJ cana-1650 85 9 j.b.patil j.b.patil NOUN cana-1650 85 10 . . PUNCT cana-1650 86 1 bounds bound NOUN cana-1650 86 2 for for ADP cana-1650 86 3 the the DET cana-1650 86 4 degree degree NOUN cana-1650 86 5 sum sum NOUN cana-1650 86 6 eigenvalues eigenvalue NOUN cana-1650 86 7 and and CCONJ cana-1650 86 8 degree degree NOUN cana-1650 86 9 sum sum NOUN cana-1650 86 10 energy energy NOUN cana-1650 86 11 of of ADP cana-1650 86 12 a a DET cana-1650 86 13 graph graph NOUN cana-1650 86 14 . . PUNCT cana-1650 87 1 international international ADJ cana-1650 87 2 journal journal NOUN cana-1650 87 3 of of ADP cana-1650 87 4 pure pure ADJ cana-1650 87 5 and and CCONJ cana-1650 87 6 applied applied ADJ cana-1650 87 7 mathematical mathematical ADJ cana-1650 87 8 sciences science NOUN cana-1650 87 9 , , PUNCT cana-1650 87 10 6(2 6(2 NUM cana-1650 87 11 ) ) PUNCT cana-1650 87 12 , , PUNCT cana-1650 87 13 ( ( PUNCT cana-1650 87 14 2013 2013 NUM cana-1650 87 15 ) ) PUNCT cana-1650 87 16 161 161 NUM cana-1650 87 17 - - SYM cana-1650 87 18 167 167 NUM cana-1650 87 19 . . PUNCT