id sid tid token lemma pos cana-611 1 1 communications communication NOUN cana-611 1 2 on on ADP cana-611 1 3 applied apply VERB cana-611 1 4 nonlinear nonlinear ADJ cana-611 1 5 analysis analysis NOUN cana-611 1 6 issn issn NOUN cana-611 1 7 : : PUNCT cana-611 1 8 1074 1074 NUM cana-611 1 9 - - PUNCT cana-611 1 10 133x 133x NUM cana-611 1 11 vol vol NOUN cana-611 1 12 31 31 NUM cana-611 1 13 no no NOUN cana-611 1 14 . . NOUN cana-611 1 15 2 2 NUM cana-611 1 16 ( ( PUNCT cana-611 1 17 2024 2024 NUM cana-611 1 18 ) ) PUNCT cana-611 1 19 416 416 NUM cana-611 1 20 https://internationalpubls.com https://internationalpubls.com X cana-611 1 21 computing compute VERB cana-611 1 22 topological topological ADJ cana-611 1 23 indices index NOUN cana-611 1 24 of of ADP cana-611 1 25 certain certain ADJ cana-611 1 26 networks network NOUN cana-611 1 27 vidya vidya PROPN cana-611 1 28 s s PART cana-611 1 29 umadi1 umadi1 PROPN cana-611 1 30 , , PUNCT cana-611 1 31 a a DET cana-611 1 32 m m NOUN cana-611 1 33 sangogi2 sangogi2 NOUN cana-611 1 34 1assistant 1assistant NUM cana-611 1 35 professor professor NOUN cana-611 1 36 department department NOUN cana-611 1 37 of of ADP cana-611 1 38 mathematics mathematics PROPN cana-611 1 39 , , PUNCT cana-611 1 40 vidya vidya PROPN cana-611 1 41 samvardhak samvardhak PROPN cana-611 1 42 mandal mandal PROPN cana-611 1 43 ’s ’s PART cana-611 1 44 somashekhar somashekhar NOUN cana-611 1 45 r r NOUN cana-611 1 46 kothiwale kothiwale NOUN cana-611 1 47 institute institute NOUN cana-611 1 48 of of ADP cana-611 1 49 technology technology PROPN cana-611 1 50 , , PUNCT cana-611 1 51 nipani nipani PROPN cana-611 1 52 , , PUNCT cana-611 1 53 karnataka karnataka PROPN cana-611 1 54 state state PROPN cana-611 1 55 , , PUNCT cana-611 1 56 india india PROPN cana-611 1 57 vidyaumadi@gmail.com vidyaumadi@gmail.com PROPN cana-611 2 1 2professor 2professor NUM cana-611 2 2 department department NOUN cana-611 2 3 of of ADP cana-611 2 4 mathematics mathematic NOUN cana-611 2 5 , , PUNCT cana-611 2 6 sgbit sgbit NOUN cana-611 2 7 , , PUNCT cana-611 2 8 belagavi belagavi VERB cana-611 2 9 , , PUNCT cana-611 2 10 karnataka karnataka PROPN cana-611 2 11 state state PROPN cana-611 2 12 , , PUNCT cana-611 2 13 india india PROPN cana-611 2 14 . . PUNCT cana-611 3 1 amsangogi@gmail.com amsangogi@gmail.com PROPN cana-611 3 2 article article NOUN cana-611 3 3 history history NOUN cana-611 3 4 : : PUNCT cana-611 3 5 received receive VERB cana-611 3 6 : : PUNCT cana-611 3 7 01 01 NUM cana-611 3 8 - - PUNCT cana-611 3 9 03 03 NUM cana-611 3 10 - - PUNCT cana-611 3 11 2024 2024 NUM cana-611 3 12 revised revise VERB cana-611 3 13 : : PUNCT cana-611 3 14 29 29 NUM cana-611 3 15 - - PUNCT cana-611 3 16 04 04 NUM cana-611 3 17 - - PUNCT cana-611 3 18 2024 2024 NUM cana-611 3 19 accepted accept VERB cana-611 3 20 : : PUNCT cana-611 3 21 15 15 NUM cana-611 3 22 - - SYM cana-611 3 23 05 05 NUM cana-611 3 24 - - PUNCT cana-611 3 25 2024 2024 NUM cana-611 3 26 abstract abstract NOUN cana-611 3 27 : : PUNCT cana-611 3 28 in in ADP cana-611 3 29 mathematical mathematical ADJ cana-611 3 30 chemistry chemistry NOUN cana-611 3 31 , , PUNCT cana-611 3 32 topological topological ADJ cana-611 3 33 indices index NOUN cana-611 3 34 are be AUX cana-611 3 35 molecular molecular ADJ cana-611 3 36 descriptors descriptor NOUN cana-611 3 37 that that PRON cana-611 3 38 are be AUX cana-611 3 39 calculated calculate VERB cana-611 3 40 on on ADP cana-611 3 41 the the DET cana-611 3 42 molecular molecular ADJ cana-611 3 43 graph graph NOUN cana-611 3 44 of of ADP cana-611 3 45 a a DET cana-611 3 46 chemical chemical NOUN cana-611 3 47 compound compound NOUN cana-611 3 48 . . PUNCT cana-611 4 1 the the DET cana-611 4 2 molecular molecular ADJ cana-611 4 3 graph graph NOUN cana-611 4 4 is be AUX cana-611 4 5 a a DET cana-611 4 6 graph graph NOUN cana-611 4 7 which which PRON cana-611 4 8 is be AUX cana-611 4 9 obtained obtain VERB cana-611 4 10 from from ADP cana-611 4 11 some some DET cana-611 4 12 chemical chemical NOUN cana-611 4 13 structures structure NOUN cana-611 4 14 . . PUNCT cana-611 5 1 the the DET cana-611 5 2 degree degree NOUN cana-611 5 3 of of ADP cana-611 5 4 every every DET cana-611 5 5 molecular molecular ADJ cana-611 5 6 graph graph NOUN cana-611 5 7 can can AUX cana-611 5 8 not not PART cana-611 5 9 exceeds exceeds VERB cana-611 5 10 4 4 NUM cana-611 5 11 . . PUNCT cana-611 6 1 topological topological ADJ cana-611 6 2 indices index NOUN cana-611 6 3 are be AUX cana-611 6 4 numerical numerical ADJ cana-611 6 5 quantities quantity NOUN cana-611 6 6 of of ADP cana-611 6 7 a a DET cana-611 6 8 graph graph NOUN cana-611 6 9 that that PRON cana-611 6 10 describe describe VERB cana-611 6 11 its its PRON cana-611 6 12 topology topology NOUN cana-611 6 13 . . PUNCT cana-611 7 1 an an DET cana-611 7 2 atom atom NOUN cana-611 7 3 represents represent VERB cana-611 7 4 a a DET cana-611 7 5 vertex vertex NOUN cana-611 7 6 and and CCONJ cana-611 7 7 a a DET cana-611 7 8 bond bond NOUN cana-611 7 9 between between ADP cana-611 7 10 two two NUM cana-611 7 11 atoms atom NOUN cana-611 7 12 represents represent VERB cana-611 7 13 an an DET cana-611 7 14 edge edge NOUN cana-611 7 15 in in ADP cana-611 7 16 a a DET cana-611 7 17 molecular molecular ADJ cana-611 7 18 graph graph NOUN cana-611 7 19 . . PUNCT cana-611 8 1 mainly mainly ADV cana-611 8 2 there there PRON cana-611 8 3 are be VERB cana-611 8 4 three three NUM cana-611 8 5 types type NOUN cana-611 8 6 of of ADP cana-611 8 7 topological topological ADJ cana-611 8 8 indices index NOUN cana-611 8 9 viz viz PROPN cana-611 8 10 . . PROPN cana-611 8 11 , , PUNCT cana-611 8 12 degree degree NOUN cana-611 8 13 - - PUNCT cana-611 8 14 based base VERB cana-611 8 15 , , PUNCT cana-611 8 16 distance distance NOUN cana-611 8 17 based base VERB cana-611 8 18 and and CCONJ cana-611 8 19 eigenvalue eigenvalue NOUN cana-611 8 20 - - PUNCT cana-611 8 21 based base VERB cana-611 8 22 topological topological ADJ cana-611 8 23 indices index NOUN cana-611 8 24 . . PUNCT cana-611 9 1 the the DET cana-611 9 2 first first ADJ cana-611 9 3 degree degree NOUN cana-611 9 4 - - PUNCT cana-611 9 5 based base VERB cana-611 9 6 topological topological ADJ cana-611 9 7 indices index NOUN cana-611 9 8 are be AUX cana-611 9 9 the the DET cana-611 9 10 first first ADJ cana-611 9 11 and and CCONJ cana-611 9 12 second second ADJ cana-611 9 13 zagreb zagreb PROPN cana-611 9 14 indices index NOUN cana-611 9 15 . . PUNCT cana-611 10 1 the the DET cana-611 10 2 first first PROPN cana-611 10 3 zagreb zagreb PROPN cana-611 10 4 index index NOUN cana-611 10 5 m_1 m_1 PROPN cana-611 10 6 is be AUX cana-611 10 7 defined define VERB cana-611 10 8 as as ADP cana-611 10 9 the the DET cana-611 10 10 sum sum NOUN cana-611 10 11 of of ADP cana-611 10 12 squares square NOUN cana-611 10 13 of of ADP cana-611 10 14 degrees degree NOUN cana-611 10 15 of of ADP cana-611 10 16 each each DET cana-611 10 17 vertex vertex NOUN cana-611 10 18 in in ADP cana-611 10 19 a a DET cana-611 10 20 graph graph NOUN cana-611 10 21 g g NOUN cana-611 10 22 and and CCONJ cana-611 10 23 the the DET cana-611 10 24 second second ADJ cana-611 10 25 zagreb zagreb PROPN cana-611 10 26 index index PROPN cana-611 10 27 m_2 m_2 PROPN cana-611 10 28 is be AUX cana-611 10 29 the the DET cana-611 10 30 product product NOUN cana-611 10 31 of of ADP cana-611 10 32 degree degree NOUN cana-611 10 33 of of ADP cana-611 10 34 every every DET cana-611 10 35 adjacent adjacent ADJ cana-611 10 36 vertices vertex NOUN cana-611 10 37 . . PUNCT cana-611 11 1 in in ADP cana-611 11 2 this this DET cana-611 11 3 case case NOUN cana-611 11 4 the the DET cana-611 11 5 summation summation NOUN cana-611 11 6 goes go VERB cana-611 11 7 on on ADP cana-611 11 8 the the DET cana-611 11 9 set set NOUN cana-611 11 10 of of ADP cana-611 11 11 edges edge NOUN cana-611 11 12 of of ADP cana-611 11 13 a a DET cana-611 11 14 graph graph NOUN cana-611 11 15 g. g. NOUN cana-611 11 16 the the DET cana-611 11 17 most most ADV cana-611 11 18 studied study VERB cana-611 11 19 topological topological ADJ cana-611 11 20 indices index NOUN cana-611 11 21 are be AUX cana-611 11 22 degree degree NOUN cana-611 11 23 - - PUNCT cana-611 11 24 based base VERB cana-611 11 25 topological topological ADJ cana-611 11 26 indices index NOUN cana-611 11 27 . . PUNCT cana-611 12 1 motivated motivate VERB cana-611 12 2 by by ADP cana-611 12 3 these these DET cana-611 12 4 topological topological ADJ cana-611 12 5 indices index NOUN cana-611 12 6 in in ADP cana-611 12 7 this this DET cana-611 12 8 paper paper NOUN cana-611 12 9 , , PUNCT cana-611 12 10 we we PRON cana-611 12 11 introduce introduce VERB cana-611 12 12 five five NUM cana-611 12 13 new new ADJ cana-611 12 14 degree degree NOUN cana-611 12 15 - - PUNCT cana-611 12 16 based base VERB cana-611 12 17 topological topological ADJ cana-611 12 18 indices index NOUN cana-611 12 19 based base VERB cana-611 12 20 on on ADP cana-611 12 21 the the DET cana-611 12 22 neighborhood neighborhood NOUN cana-611 12 23 degree degree NOUN cana-611 12 24 of of ADP cana-611 12 25 a a DET cana-611 12 26 vertex vertex NOUN cana-611 12 27 . . PUNCT cana-611 13 1 further far ADV cana-611 13 2 , , PUNCT cana-611 13 3 we we PRON cana-611 13 4 compute compute VERB cana-611 13 5 the the DET cana-611 13 6 values value NOUN cana-611 13 7 of of ADP cana-611 13 8 various various ADJ cana-611 13 9 nanostructures nanostructure NOUN cana-611 13 10 like like ADP cana-611 13 11 hexagonal hexagonal ADJ cana-611 13 12 parallelogram parallelogram PROPN cana-611 13 13 p(m p(m PROPN cana-611 13 14 , , PUNCT cana-611 13 15 n n CCONJ cana-611 13 16 ) ) PUNCT cana-611 13 17 nanotube nanotube NOUN cana-611 13 18 , , PUNCT cana-611 13 19 triangular triangular NOUN cana-611 13 20 benzenoid benzenoid NOUN cana-611 13 21 g_n g_n PROPN cana-611 13 22 , , PUNCT cana-611 13 23 zigzag zigzag NOUN cana-611 13 24 - - PUNCT cana-611 13 25 edge edge NOUN cana-611 13 26 coronoid coronoid PROPN cana-611 13 27 fused fuse VERB cana-611 13 28 with with ADP cana-611 13 29 starphene starphene ADJ cana-611 13 30 nanotubes nanotube NOUN cana-611 13 31 zcs(k zcs(k PROPN cana-611 13 32 , , PUNCT cana-611 13 33 l l NOUN cana-611 13 34 , , PUNCT cana-611 13 35 m m NOUN cana-611 13 36 ) ) PUNCT cana-611 13 37 , , PUNCT cana-611 13 38 dominating dominate VERB cana-611 13 39 derived derive VERB cana-611 13 40 networks network NOUN cana-611 13 41 d_1,d_2,d_3 d_1,d_2,d_3 NOUN cana-611 13 42 , , PUNCT cana-611 13 43 porphyrin porphyrin NOUN cana-611 13 44 dendrimer dendrimer NOUN cana-611 13 45 , , PUNCT cana-611 13 46 zinc zinc NOUN cana-611 13 47 - - PUNCT cana-611 13 48 porphyrin porphyrin NOUN cana-611 13 49 dendrimer dendrimer NOUN cana-611 13 50 , , PUNCT cana-611 13 51 propyl propyl ADJ cana-611 13 52 ether ether NOUN cana-611 13 53 imine imine NOUN cana-611 13 54 dendrimer dendrimer NOUN cana-611 13 55 , , PUNCT cana-611 13 56 poly(ethylene poly(ethylene ADJ cana-611 13 57 amido amido NOUN cana-611 13 58 amine amine ADJ cana-611 13 59 dendrimer dendrimer NOUN cana-611 13 60 , , PUNCT cana-611 13 61 pamam pamam NOUN cana-611 13 62 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 PROPN cana-611 13 63 ) ) PUNCT cana-611 13 64 , , PUNCT cana-611 13 65 linear linear PROPN cana-611 13 66 polyomino polyomino PROPN cana-611 13 67 chain chain NOUN cana-611 13 68 l_n l_n PROPN cana-611 13 69 , , PUNCT cana-611 13 70 z_n z_n PRON cana-611 13 71 , , PUNCT cana-611 13 72 b_n^1 b_n^1 PROPN cana-611 13 73 ( ( PUNCT cana-611 13 74 nβ‰₯3),b_n^2 nβ‰₯3),b_n^2 X cana-611 13 75 ( ( PUNCT cana-611 13 76 nβ‰₯3 nβ‰₯3 PROPN cana-611 13 77 ) ) PUNCT cana-611 13 78 and and CCONJ cana-611 13 79 triangular triangular NOUN cana-611 13 80 , , PUNCT cana-611 13 81 hourglass hourglass NOUN cana-611 13 82 , , PUNCT cana-611 13 83 and and CCONJ cana-611 13 84 jagged jagged ADJ cana-611 13 85 - - PUNCT cana-611 13 86 rectangle rectangle NOUN cana-611 13 87 benzenoid benzenoid NOUN cana-611 13 88 systems system NOUN cana-611 13 89 of of ADP cana-611 13 90 these these DET cana-611 13 91 indices index NOUN cana-611 13 92 . . PUNCT cana-611 14 1 the the DET cana-611 14 2 standard standard ADJ cana-611 14 3 computational computational ADJ cana-611 14 4 techniques technique NOUN cana-611 14 5 are be AUX cana-611 14 6 used use VERB cana-611 14 7 for for ADP cana-611 14 8 the the DET cana-611 14 9 computation computation NOUN cana-611 14 10 of of ADP cana-611 14 11 topological topological ADJ cana-611 14 12 indices index NOUN cana-611 14 13 of of ADP cana-611 14 14 nanostructures nanostructure NOUN cana-611 14 15 . . PUNCT cana-611 15 1 for for ADP cana-611 15 2 the the DET cana-611 15 3 edge edge NOUN cana-611 15 4 partition partition NOUN cana-611 15 5 of of ADP cana-611 15 6 the the DET cana-611 15 7 nanostructures nanostructure NOUN cana-611 15 8 the the DET cana-611 15 9 algebraic algebraic ADJ cana-611 15 10 techniques technique NOUN cana-611 15 11 are be AUX cana-611 15 12 used use VERB cana-611 15 13 . . PUNCT cana-611 16 1 using use VERB cana-611 16 2 these these DET cana-611 16 3 techniques technique NOUN cana-611 16 4 computation computation NOUN cana-611 16 5 of of ADP cana-611 16 6 topological topological ADJ cana-611 16 7 indices index NOUN cana-611 16 8 became become VERB cana-611 16 9 easy easy ADJ cana-611 16 10 and and CCONJ cana-611 16 11 also also ADV cana-611 16 12 helped help VERB cana-611 16 13 to to PART cana-611 16 14 get get VERB cana-611 16 15 the the DET cana-611 16 16 more more ADV cana-611 16 17 accurate accurate ADJ cana-611 16 18 results result NOUN cana-611 16 19 . . PUNCT cana-611 17 1 keywords keyword NOUN cana-611 17 2 : : PUNCT cana-611 17 3 molecular molecular ADJ cana-611 17 4 graph graph NOUN cana-611 17 5 , , PUNCT cana-611 17 6 nanostructures nanostructure NOUN cana-611 17 7 , , PUNCT cana-611 17 8 dendrimers dendrimer NOUN cana-611 17 9 , , PUNCT cana-611 17 10 topological topological ADJ cana-611 17 11 indices index NOUN cana-611 17 12 . . PUNCT cana-611 18 1 1 1 X cana-611 18 2 . . X cana-611 18 3 introduction introduction NOUN cana-611 18 4 in in ADP cana-611 18 5 the the DET cana-611 18 6 realm realm NOUN cana-611 18 7 of of ADP cana-611 18 8 modern modern ADJ cana-611 18 9 chemistry chemistry NOUN cana-611 18 10 , , PUNCT cana-611 18 11 the the DET cana-611 18 12 quest quest NOUN cana-611 18 13 to to PART cana-611 18 14 understand understand VERB cana-611 18 15 the the DET cana-611 18 16 intricate intricate ADJ cana-611 18 17 structures structure NOUN cana-611 18 18 of of ADP cana-611 18 19 molecules molecule NOUN cana-611 18 20 and and CCONJ cana-611 18 21 their their PRON cana-611 18 22 impact impact NOUN cana-611 18 23 on on ADP cana-611 18 24 chemical chemical ADJ cana-611 18 25 properties property NOUN cana-611 18 26 has have AUX cana-611 18 27 led lead VERB cana-611 18 28 researchers researcher NOUN cana-611 18 29 to to PART cana-611 18 30 explore explore VERB cana-611 18 31 various various ADJ cana-611 18 32 analytical analytical ADJ cana-611 18 33 tools tool NOUN cana-611 18 34 and and CCONJ cana-611 18 35 methodologies methodology NOUN cana-611 18 36 . . PUNCT cana-611 19 1 among among ADP cana-611 19 2 these these PRON cana-611 19 3 , , PUNCT cana-611 19 4 the the DET cana-611 19 5 field field NOUN cana-611 19 6 of of ADP cana-611 19 7 mathematical mathematical ADJ cana-611 19 8 chemistry chemistry NOUN cana-611 19 9 stands stand VERB cana-611 19 10 out out ADP cana-611 19 11 for for ADP cana-611 19 12 its its PRON cana-611 19 13 emphasis emphasis NOUN cana-611 19 14 on on ADP cana-611 19 15 applying apply VERB cana-611 19 16 mathematical mathematical ADJ cana-611 19 17 concepts concept NOUN cana-611 19 18 and and CCONJ cana-611 19 19 techniques technique NOUN cana-611 19 20 to to PART cana-611 19 21 unravel unravel VERB cana-611 19 22 the the DET cana-611 19 23 communications communication NOUN cana-611 19 24 on on ADP cana-611 19 25 applied apply VERB cana-611 19 26 nonlinear nonlinear ADJ cana-611 19 27 analysis analysis NOUN cana-611 19 28 issn issn NOUN cana-611 19 29 : : PUNCT cana-611 19 30 1074 1074 NUM cana-611 19 31 - - PUNCT cana-611 19 32 133x 133x NUM cana-611 19 33 vol vol NOUN cana-611 19 34 31 31 NUM cana-611 19 35 no no NOUN cana-611 19 36 . . NOUN cana-611 19 37 2 2 NUM cana-611 19 38 ( ( PUNCT cana-611 19 39 2024 2024 NUM cana-611 19 40 ) ) PUNCT cana-611 19 41 417 417 NUM cana-611 19 42 https://internationalpubls.com https://internationalpubls.com X cana-611 19 43 mysteries mystery NOUN cana-611 19 44 of of ADP cana-611 19 45 molecular molecular ADJ cana-611 19 46 structures structure NOUN cana-611 19 47 . . PUNCT cana-611 20 1 central central ADJ cana-611 20 2 to to ADP cana-611 20 3 this this DET cana-611 20 4 endeavor endeavor NOUN cana-611 20 5 is be AUX cana-611 20 6 the the DET cana-611 20 7 study study NOUN cana-611 20 8 of of ADP cana-611 20 9 molecular molecular ADJ cana-611 20 10 graphs graph NOUN cana-611 20 11 and and CCONJ cana-611 20 12 their their PRON cana-611 20 13 characterization characterization NOUN cana-611 20 14 through through ADP cana-611 20 15 topological topological ADJ cana-611 20 16 indices index NOUN cana-611 20 17 , , PUNCT cana-611 20 18 which which PRON cana-611 20 19 serve serve VERB cana-611 20 20 as as ADP cana-611 20 21 powerful powerful ADJ cana-611 20 22 descriptors descriptor NOUN cana-611 20 23 of of ADP cana-611 20 24 molecular molecular ADJ cana-611 20 25 topology topology NOUN cana-611 20 26 . . PUNCT cana-611 21 1 topological topological ADJ cana-611 21 2 indices index NOUN cana-611 21 3 represent represent VERB cana-611 21 4 numerical numerical ADJ cana-611 21 5 quantities quantity NOUN cana-611 21 6 derived derive VERB cana-611 21 7 from from ADP cana-611 21 8 the the DET cana-611 21 9 molecular molecular ADJ cana-611 21 10 graph graph NOUN cana-611 21 11 of of ADP cana-611 21 12 a a DET cana-611 21 13 chemical chemical NOUN cana-611 21 14 compound compound NOUN cana-611 21 15 . . PUNCT cana-611 22 1 the the DET cana-611 22 2 molecular molecular ADJ cana-611 22 3 graph graph NOUN cana-611 22 4 itself itself PRON cana-611 22 5 is be AUX cana-611 22 6 a a DET cana-611 22 7 representation representation NOUN cana-611 22 8 of of ADP cana-611 22 9 the the DET cana-611 22 10 chemical chemical NOUN cana-611 22 11 structure structure NOUN cana-611 22 12 , , PUNCT cana-611 22 13 wherein wherein ADJ cana-611 22 14 atoms atom NOUN cana-611 22 15 are be AUX cana-611 22 16 depicted depict VERB cana-611 22 17 as as ADP cana-611 22 18 vertices vertex NOUN cana-611 22 19 and and CCONJ cana-611 22 20 bonds bond NOUN cana-611 22 21 as as ADP cana-611 22 22 edges edge NOUN cana-611 22 23 . . PUNCT cana-611 23 1 by by ADP cana-611 23 2 analyzing analyze VERB cana-611 23 3 the the DET cana-611 23 4 connectivity connectivity NOUN cana-611 23 5 patterns pattern NOUN cana-611 23 6 and and CCONJ cana-611 23 7 geometrical geometrical ADJ cana-611 23 8 arrangements arrangement NOUN cana-611 23 9 within within ADP cana-611 23 10 these these DET cana-611 23 11 graphs graph NOUN cana-611 23 12 , , PUNCT cana-611 23 13 researchers researcher NOUN cana-611 23 14 can can AUX cana-611 23 15 gain gain VERB cana-611 23 16 valuable valuable ADJ cana-611 23 17 insights insight NOUN cana-611 23 18 into into ADP cana-611 23 19 the the DET cana-611 23 20 structural structural ADJ cana-611 23 21 features feature NOUN cana-611 23 22 that that PRON cana-611 23 23 influence influence VERB cana-611 23 24 the the DET cana-611 23 25 behavior behavior NOUN cana-611 23 26 of of ADP cana-611 23 27 molecules molecule NOUN cana-611 23 28 in in ADP cana-611 23 29 various various ADJ cana-611 23 30 chemical chemical NOUN cana-611 23 31 contexts[1 contexts[1 PROPN cana-611 23 32 ] ] PUNCT cana-611 23 33 . . PUNCT cana-611 24 1 within within ADP cana-611 24 2 the the DET cana-611 24 3 framework framework NOUN cana-611 24 4 of of ADP cana-611 24 5 topological topological ADJ cana-611 24 6 indices index NOUN cana-611 24 7 , , PUNCT cana-611 24 8 three three NUM cana-611 24 9 main main ADJ cana-611 24 10 categories category NOUN cana-611 24 11 emerge emerge VERB cana-611 24 12 : : PUNCT cana-611 24 13 degree degree NOUN cana-611 24 14 - - PUNCT cana-611 24 15 based base VERB cana-611 24 16 , , PUNCT cana-611 24 17 distance distance NOUN cana-611 24 18 - - PUNCT cana-611 24 19 based base VERB cana-611 24 20 , , PUNCT cana-611 24 21 and and CCONJ cana-611 24 22 eigenvalue eigenvalue NOUN cana-611 24 23 - - PUNCT cana-611 24 24 based base VERB cana-611 24 25 indices index NOUN cana-611 24 26 . . PUNCT cana-611 25 1 each each DET cana-611 25 2 category category NOUN cana-611 25 3 offers offer VERB cana-611 25 4 unique unique ADJ cana-611 25 5 perspectives perspective NOUN cana-611 25 6 on on ADP cana-611 25 7 molecular molecular ADJ cana-611 25 8 topology topology NOUN cana-611 25 9 , , PUNCT cana-611 25 10 with with ADP cana-611 25 11 degree degree NOUN cana-611 25 12 - - PUNCT cana-611 25 13 based base VERB cana-611 25 14 indices index NOUN cana-611 25 15 being be AUX cana-611 25 16 particularly particularly ADV cana-611 25 17 prominent prominent ADJ cana-611 25 18 due due ADP cana-611 25 19 to to ADP cana-611 25 20 their their PRON cana-611 25 21 simplicity simplicity NOUN cana-611 25 22 and and CCONJ cana-611 25 23 effectiveness effectiveness NOUN cana-611 25 24 in in ADP cana-611 25 25 capturing capture VERB cana-611 25 26 essential essential ADJ cana-611 25 27 structural structural ADJ cana-611 25 28 information information NOUN cana-611 25 29 . . PUNCT cana-611 26 1 one one NUM cana-611 26 2 of of ADP cana-611 26 3 the the DET cana-611 26 4 cornerstone cornerstone NOUN cana-611 26 5 degree degree NOUN cana-611 26 6 - - PUNCT cana-611 26 7 based base VERB cana-611 26 8 indices index NOUN cana-611 26 9 is be AUX cana-611 26 10 the the DET cana-611 26 11 zagreb zagreb PROPN cana-611 26 12 indices index NOUN cana-611 26 13 , , PUNCT cana-611 26 14 comprising comprise VERB cana-611 26 15 the the DET cana-611 26 16 first first ADJ cana-611 26 17 and and CCONJ cana-611 26 18 second second ADJ cana-611 26 19 zagreb zagreb PROPN cana-611 26 20 indices index NOUN cana-611 26 21 . . PUNCT cana-611 27 1 the the DET cana-611 27 2 first first PROPN cana-611 27 3 zagreb zagreb PROPN cana-611 27 4 index index PROPN cana-611 27 5 , , PUNCT cana-611 27 6 denoted denote VERB cana-611 27 7 as as ADP cana-611 27 8 m1m1[2 m1m1[2 NOUN cana-611 27 9 ] ] PUNCT cana-611 27 10 , , PUNCT cana-611 27 11 quantifies quantify VERB cana-611 27 12 the the DET cana-611 27 13 sum sum NOUN cana-611 27 14 of of ADP cana-611 27 15 the the DET cana-611 27 16 squares square NOUN cana-611 27 17 of of ADP cana-611 27 18 vertex vertex NOUN cana-611 27 19 degrees degree NOUN cana-611 27 20 in in ADP cana-611 27 21 the the DET cana-611 27 22 molecular molecular ADJ cana-611 27 23 graph graph NOUN cana-611 27 24 , , PUNCT cana-611 27 25 providing provide VERB cana-611 27 26 a a DET cana-611 27 27 measure measure NOUN cana-611 27 28 of of ADP cana-611 27 29 overall overall ADJ cana-611 27 30 connectivity connectivity NOUN cana-611 27 31 . . PUNCT cana-611 28 1 on on ADP cana-611 28 2 the the DET cana-611 28 3 other other ADJ cana-611 28 4 hand hand NOUN cana-611 28 5 , , PUNCT cana-611 28 6 the the DET cana-611 28 7 second second ADJ cana-611 28 8 zagreb zagreb PROPN cana-611 28 9 index index PROPN cana-611 28 10 , , PUNCT cana-611 28 11 denoted denote VERB cana-611 28 12 as as ADP cana-611 28 13 m2m2 m2m2 NOUN cana-611 28 14 , , PUNCT cana-611 28 15 captures capture VERB cana-611 28 16 the the DET cana-611 28 17 product product NOUN cana-611 28 18 of of ADP cana-611 28 19 degrees degree NOUN cana-611 28 20 of of ADP cana-611 28 21 adjacent adjacent ADJ cana-611 28 22 vertices vertex NOUN cana-611 28 23 , , PUNCT cana-611 28 24 thus thus ADV cana-611 28 25 highlighting highlight VERB cana-611 28 26 local local ADJ cana-611 28 27 structural structural ADJ cana-611 28 28 motifs motif NOUN cana-611 28 29 within within ADP cana-611 28 30 the the DET cana-611 28 31 molecule[3 molecule[3 NOUN cana-611 28 32 ] ] PUNCT cana-611 28 33 . . PUNCT cana-611 29 1 building build VERB cana-611 29 2 upon upon SCONJ cana-611 29 3 the the DET cana-611 29 4 foundational foundational ADJ cana-611 29 5 concepts concept NOUN cana-611 29 6 of of ADP cana-611 29 7 degree degree NOUN cana-611 29 8 - - PUNCT cana-611 29 9 based base VERB cana-611 29 10 indices index NOUN cana-611 29 11 , , PUNCT cana-611 29 12 this this DET cana-611 29 13 paper paper NOUN cana-611 29 14 introduces introduce VERB cana-611 29 15 five five NUM cana-611 29 16 novel novel ADJ cana-611 29 17 topological topological ADJ cana-611 29 18 indices index NOUN cana-611 29 19 rooted root VERB cana-611 29 20 in in ADP cana-611 29 21 the the DET cana-611 29 22 notion notion NOUN cana-611 29 23 of of ADP cana-611 29 24 neighborhood neighborhood NOUN cana-611 29 25 degree degree NOUN cana-611 29 26 . . PUNCT cana-611 30 1 the the DET cana-611 30 2 neighborhood neighborhood NOUN cana-611 30 3 degree degree NOUN cana-611 30 4 of of ADP cana-611 30 5 a a DET cana-611 30 6 vertex vertex NOUN cana-611 30 7 reflects reflect VERB cana-611 30 8 the the DET cana-611 30 9 cumulative cumulative ADJ cana-611 30 10 degree degree NOUN cana-611 30 11 of of ADP cana-611 30 12 its its PRON cana-611 30 13 neighboring neighboring NOUN cana-611 30 14 vertices vertex NOUN cana-611 30 15 , , PUNCT cana-611 30 16 offering offer VERB cana-611 30 17 insights insight NOUN cana-611 30 18 into into ADP cana-611 30 19 the the DET cana-611 30 20 local local ADJ cana-611 30 21 structural structural ADJ cana-611 30 22 environment environment NOUN cana-611 30 23 of of ADP cana-611 30 24 each each DET cana-611 30 25 vertex vertex NOUN cana-611 30 26 . . PUNCT cana-611 31 1 by by ADP cana-611 31 2 incorporating incorporate VERB cana-611 31 3 this this DET cana-611 31 4 concept concept NOUN cana-611 31 5 into into ADP cana-611 31 6 the the DET cana-611 31 7 design design NOUN cana-611 31 8 of of ADP cana-611 31 9 new new ADJ cana-611 31 10 indices index NOUN cana-611 31 11 , , PUNCT cana-611 31 12 the the DET cana-611 31 13 paper paper NOUN cana-611 31 14 aims aim VERB cana-611 31 15 to to PART cana-611 31 16 enrich enrich VERB cana-611 31 17 the the DET cana-611 31 18 repertoire repertoire NOUN cana-611 31 19 of of ADP cana-611 31 20 tools tool NOUN cana-611 31 21 available available ADJ cana-611 31 22 for for ADP cana-611 31 23 analyzing analyze VERB cana-611 31 24 molecular molecular ADJ cana-611 31 25 graphs graph NOUN cana-611 31 26 and and CCONJ cana-611 31 27 uncovering uncover VERB cana-611 31 28 subtle subtle ADJ cana-611 31 29 structural structural ADJ cana-611 31 30 variations[4 variations[4 PROPN cana-611 31 31 ] ] X cana-611 31 32 . . PUNCT cana-611 32 1 beyond beyond ADP cana-611 32 2 theoretical theoretical ADJ cana-611 32 3 development development NOUN cana-611 32 4 , , PUNCT cana-611 32 5 this this DET cana-611 32 6 paper paper NOUN cana-611 32 7 also also ADV cana-611 32 8 emphasizes emphasize VERB cana-611 32 9 the the DET cana-611 32 10 practical practical ADJ cana-611 32 11 applications application NOUN cana-611 32 12 of of ADP cana-611 32 13 topological topological ADJ cana-611 32 14 indices index NOUN cana-611 32 15 to to ADP cana-611 32 16 a a DET cana-611 32 17 diverse diverse ADJ cana-611 32 18 array array NOUN cana-611 32 19 of of ADP cana-611 32 20 nanostructures nanostructure NOUN cana-611 32 21 . . PUNCT cana-611 33 1 nanostructures nanostructure NOUN cana-611 33 2 , , PUNCT cana-611 33 3 characterized characterize VERB cana-611 33 4 by by ADP cana-611 33 5 their their PRON cana-611 33 6 unique unique ADJ cana-611 33 7 geometries geometry NOUN cana-611 33 8 and and CCONJ cana-611 33 9 properties property NOUN cana-611 33 10 at at ADP cana-611 33 11 the the DET cana-611 33 12 nanoscale nanoscale NOUN cana-611 33 13 , , PUNCT cana-611 33 14 present present ADJ cana-611 33 15 intriguing intriguing ADJ cana-611 33 16 challenges challenge NOUN cana-611 33 17 and and CCONJ cana-611 33 18 opportunities opportunity NOUN cana-611 33 19 for for ADP cana-611 33 20 topological topological ADJ cana-611 33 21 analysis analysis NOUN cana-611 33 22 . . PUNCT cana-611 34 1 examples example NOUN cana-611 34 2 of of ADP cana-611 34 3 such such ADJ cana-611 34 4 nanostructures nanostructure NOUN cana-611 34 5 include include VERB cana-611 34 6 hexagonal hexagonal ADJ cana-611 34 7 parallelogram parallelogram NOUN cana-611 34 8 nanotubes nanotube NOUN cana-611 34 9 , , PUNCT cana-611 34 10 benzenoid benzenoid NOUN cana-611 34 11 systems system NOUN cana-611 34 12 , , PUNCT cana-611 34 13 dendrimers dendrimer NOUN cana-611 34 14 , , PUNCT cana-611 34 15 and and CCONJ cana-611 34 16 polyomino polyomino NOUN cana-611 34 17 chains chain NOUN cana-611 34 18 , , PUNCT cana-611 34 19 among among ADP cana-611 34 20 others other NOUN cana-611 34 21 . . PUNCT cana-611 35 1 by by ADP cana-611 35 2 computing compute VERB cana-611 35 3 the the DET cana-611 35 4 proposed propose VERB cana-611 35 5 indices index NOUN cana-611 35 6 for for ADP cana-611 35 7 these these DET cana-611 35 8 nanostructures nanostructure NOUN cana-611 35 9 , , PUNCT cana-611 35 10 the the DET cana-611 35 11 paper paper NOUN cana-611 35 12 seeks seek VERB cana-611 35 13 to to PART cana-611 35 14 demonstrate demonstrate VERB cana-611 35 15 their their PRON cana-611 35 16 effectiveness effectiveness NOUN cana-611 35 17 in in ADP cana-611 35 18 capturing capture VERB cana-611 35 19 the the DET cana-611 35 20 complex complex ADJ cana-611 35 21 topology topology NOUN cana-611 35 22 inherent inherent ADJ cana-611 35 23 in in ADP cana-611 35 24 these these DET cana-611 35 25 systems.to systems.to NUM cana-611 35 26 facilitate facilitate VERB cana-611 35 27 the the DET cana-611 35 28 computation computation NOUN cana-611 35 29 of of ADP cana-611 35 30 topological topological ADJ cana-611 35 31 indices index NOUN cana-611 35 32 for for ADP cana-611 35 33 nanostructures nanostructure NOUN cana-611 35 34 , , PUNCT cana-611 35 35 the the DET cana-611 35 36 paper paper NOUN cana-611 35 37 employs employ VERB cana-611 35 38 a a DET cana-611 35 39 combination combination NOUN cana-611 35 40 of of ADP cana-611 35 41 standard standard ADJ cana-611 35 42 computational computational ADJ cana-611 35 43 techniques technique NOUN cana-611 35 44 and and CCONJ cana-611 35 45 algebraic algebraic ADJ cana-611 35 46 methods method NOUN cana-611 35 47 for for ADP cana-611 35 48 edge edge NOUN cana-611 35 49 partitioning partitioning NOUN cana-611 35 50 . . PUNCT cana-611 36 1 these these DET cana-611 36 2 techniques technique NOUN cana-611 36 3 not not PART cana-611 36 4 only only ADV cana-611 36 5 streamline streamline VERB cana-611 36 6 the the DET cana-611 36 7 calculation calculation NOUN cana-611 36 8 process process NOUN cana-611 36 9 but but CCONJ cana-611 36 10 also also ADV cana-611 36 11 enhance enhance VERB cana-611 36 12 the the DET cana-611 36 13 accuracy accuracy NOUN cana-611 36 14 and and CCONJ cana-611 36 15 reliability reliability NOUN cana-611 36 16 of of ADP cana-611 36 17 the the DET cana-611 36 18 results result NOUN cana-611 36 19 obtained obtain VERB cana-611 36 20 . . PUNCT cana-611 37 1 by by ADP cana-611 37 2 leveraging leverage VERB cana-611 37 3 computational computational ADJ cana-611 37 4 and and CCONJ cana-611 37 5 algebraic algebraic ADJ cana-611 37 6 tools tool NOUN cana-611 37 7 , , PUNCT cana-611 37 8 researchers researcher NOUN cana-611 37 9 can can AUX cana-611 37 10 explore explore VERB cana-611 37 11 the the DET cana-611 37 12 intricate intricate ADJ cana-611 37 13 details detail NOUN cana-611 37 14 of of ADP cana-611 37 15 molecular molecular ADJ cana-611 37 16 topology topology NOUN cana-611 37 17 with with ADP cana-611 37 18 greater great ADJ cana-611 37 19 efficiency efficiency NOUN cana-611 37 20 and and CCONJ cana-611 37 21 precision[5 precision[5 ADP cana-611 37 22 ] ] PUNCT cana-611 37 23 . . PUNCT cana-611 38 1 introduction introduction NOUN cana-611 38 2 of of ADP cana-611 38 3 novel novel ADJ cana-611 38 4 degree degree NOUN cana-611 38 5 - - PUNCT cana-611 38 6 based base VERB cana-611 38 7 topological topological ADJ cana-611 38 8 indices index NOUN cana-611 38 9 : : PUNCT cana-611 38 10 the the DET cana-611 38 11 primary primary ADJ cana-611 38 12 objective objective NOUN cana-611 38 13 of of ADP cana-611 38 14 this this DET cana-611 38 15 paper paper NOUN cana-611 38 16 is be AUX cana-611 38 17 to to PART cana-611 38 18 introduce introduce VERB cana-611 38 19 five five NUM cana-611 38 20 new new ADJ cana-611 38 21 degree degree NOUN cana-611 38 22 - - PUNCT cana-611 38 23 based base VERB cana-611 38 24 topological topological ADJ cana-611 38 25 indices index NOUN cana-611 38 26 that that PRON cana-611 38 27 are be AUX cana-611 38 28 based base VERB cana-611 38 29 on on ADP cana-611 38 30 the the DET cana-611 38 31 concept concept NOUN cana-611 38 32 of of ADP cana-611 38 33 neighborhood neighborhood NOUN cana-611 38 34 degree degree NOUN cana-611 38 35 . . PUNCT cana-611 39 1 these these DET cana-611 39 2 indices index NOUN cana-611 39 3 are be AUX cana-611 39 4 designed design VERB cana-611 39 5 to to PART cana-611 39 6 provide provide VERB cana-611 39 7 a a DET cana-611 39 8 more more ADV cana-611 39 9 nuanced nuanced ADJ cana-611 39 10 characterization characterization NOUN cana-611 39 11 of of ADP cana-611 39 12 molecular molecular ADJ cana-611 39 13 graphs graph NOUN cana-611 39 14 , , PUNCT cana-611 39 15 with with ADP cana-611 39 16 a a DET cana-611 39 17 focus focus NOUN cana-611 39 18 on on ADP cana-611 39 19 capturing capture VERB cana-611 39 20 local local ADJ cana-611 39 21 structural structural ADJ cana-611 39 22 information information NOUN cana-611 39 23 . . PUNCT cana-611 40 1 communications communication NOUN cana-611 40 2 on on ADP cana-611 40 3 applied apply VERB cana-611 40 4 nonlinear nonlinear ADJ cana-611 40 5 analysis analysis NOUN cana-611 40 6 issn issn NOUN cana-611 40 7 : : PUNCT cana-611 40 8 1074 1074 NUM cana-611 40 9 - - PUNCT cana-611 40 10 133x 133x NUM cana-611 40 11 vol vol NOUN cana-611 40 12 31 31 NUM cana-611 40 13 no no NOUN cana-611 40 14 . . NOUN cana-611 40 15 2 2 NUM cana-611 40 16 ( ( PUNCT cana-611 40 17 2024 2024 NUM cana-611 40 18 ) ) PUNCT cana-611 40 19 418 418 NUM cana-611 41 1 https://internationalpubls.com https://internationalpubls.com X cana-611 41 2 application application NOUN cana-611 41 3 to to ADP cana-611 41 4 various various ADJ cana-611 41 5 nanostructures nanostructure NOUN cana-611 41 6 : : PUNCT cana-611 41 7 another another DET cana-611 41 8 objective objective NOUN cana-611 41 9 is be AUX cana-611 41 10 to to PART cana-611 41 11 demonstrate demonstrate VERB cana-611 41 12 the the DET cana-611 41 13 applicability applicability NOUN cana-611 41 14 of of ADP cana-611 41 15 the the DET cana-611 41 16 proposed propose VERB cana-611 41 17 indices index NOUN cana-611 41 18 to to ADP cana-611 41 19 a a DET cana-611 41 20 diverse diverse ADJ cana-611 41 21 range range NOUN cana-611 41 22 of of ADP cana-611 41 23 nanostructures nanostructure NOUN cana-611 41 24 , , PUNCT cana-611 41 25 including include VERB cana-611 41 26 nanotubes nanotube NOUN cana-611 41 27 , , PUNCT cana-611 41 28 benzenoid benzenoid NOUN cana-611 41 29 systems system NOUN cana-611 41 30 , , PUNCT cana-611 41 31 dendrimers dendrimer NOUN cana-611 41 32 , , PUNCT cana-611 41 33 and and CCONJ cana-611 41 34 polyomino polyomino NOUN cana-611 41 35 chains chain NOUN cana-611 41 36 . . PUNCT cana-611 42 1 by by ADP cana-611 42 2 computing compute VERB cana-611 42 3 these these DET cana-611 42 4 indices index NOUN cana-611 42 5 for for ADP cana-611 42 6 different different ADJ cana-611 42 7 nanostructures nanostructure NOUN cana-611 42 8 , , PUNCT cana-611 42 9 the the DET cana-611 42 10 paper paper NOUN cana-611 42 11 aims aim VERB cana-611 42 12 to to PART cana-611 42 13 showcase showcase VERB cana-611 42 14 their their PRON cana-611 42 15 utility utility NOUN cana-611 42 16 in in ADP cana-611 42 17 analyzing analyze VERB cana-611 42 18 complex complex ADJ cana-611 42 19 molecular molecular ADJ cana-611 42 20 architectures[6,7 architectures[6,7 NOUN cana-611 42 21 ] ] PUNCT cana-611 42 22 . . PUNCT cana-611 43 1 utilization utilization NOUN cana-611 43 2 of of ADP cana-611 43 3 computational computational ADJ cana-611 43 4 and and CCONJ cana-611 43 5 algebraic algebraic ADJ cana-611 43 6 techniques technique NOUN cana-611 43 7 : : PUNCT cana-611 43 8 the the DET cana-611 43 9 paper paper NOUN cana-611 43 10 aims aim VERB cana-611 43 11 to to PART cana-611 43 12 leverage leverage VERB cana-611 43 13 standard standard ADJ cana-611 43 14 computational computational ADJ cana-611 43 15 techniques technique NOUN cana-611 43 16 for for ADP cana-611 43 17 the the DET cana-611 43 18 computation computation NOUN cana-611 43 19 of of ADP cana-611 43 20 topological topological ADJ cana-611 43 21 indices index NOUN cana-611 43 22 , , PUNCT cana-611 43 23 supplemented supplement VERB cana-611 43 24 by by ADP cana-611 43 25 algebraic algebraic ADJ cana-611 43 26 methods method NOUN cana-611 43 27 for for ADP cana-611 43 28 edge edge NOUN cana-611 43 29 partitioning partition VERB cana-611 43 30 in in ADP cana-611 43 31 nanostructures nanostructure NOUN cana-611 43 32 . . PUNCT cana-611 44 1 this this DET cana-611 44 2 approach approach NOUN cana-611 44 3 is be AUX cana-611 44 4 expected expect VERB cana-611 44 5 to to PART cana-611 44 6 enhance enhance VERB cana-611 44 7 the the DET cana-611 44 8 efficiency efficiency NOUN cana-611 44 9 and and CCONJ cana-611 44 10 accuracy accuracy NOUN cana-611 44 11 of of ADP cana-611 44 12 the the DET cana-611 44 13 calculations calculation NOUN cana-611 44 14 , , PUNCT cana-611 44 15 facilitating facilitate VERB cana-611 44 16 more more ADV cana-611 44 17 robust robust ADJ cana-611 44 18 analysis analysis NOUN cana-611 44 19 and and CCONJ cana-611 44 20 interpretation interpretation NOUN cana-611 44 21 of of ADP cana-611 44 22 the the DET cana-611 44 23 results[8 results[8 PROPN cana-611 44 24 ] ] PUNCT cana-611 44 25 . . PUNCT cana-611 45 1 overall overall ADV cana-611 45 2 , , PUNCT cana-611 45 3 the the DET cana-611 45 4 objectives objective NOUN cana-611 45 5 of of ADP cana-611 45 6 the the DET cana-611 45 7 paper paper NOUN cana-611 45 8 encompass encompass VERB cana-611 45 9 both both CCONJ cana-611 45 10 theoretical theoretical ADJ cana-611 45 11 advancements advancement NOUN cana-611 45 12 in in ADP cana-611 45 13 topological topological ADJ cana-611 45 14 index index NOUN cana-611 45 15 theory theory NOUN cana-611 45 16 and and CCONJ cana-611 45 17 practical practical ADJ cana-611 45 18 applications application NOUN cana-611 45 19 to to ADP cana-611 45 20 real real ADJ cana-611 45 21 - - PUNCT cana-611 45 22 world world NOUN cana-611 45 23 nanostructures nanostructure NOUN cana-611 45 24 , , PUNCT cana-611 45 25 with with ADP cana-611 45 26 a a DET cana-611 45 27 focus focus NOUN cana-611 45 28 on on ADP cana-611 45 29 enhancing enhance VERB cana-611 45 30 our our PRON cana-611 45 31 understanding understanding NOUN cana-611 45 32 of of ADP cana-611 45 33 molecular molecular ADJ cana-611 45 34 topology topology NOUN cana-611 45 35 in in ADP cana-611 45 36 mathematical mathematical PROPN cana-611 45 37 chemistry[9,10].in chemistry[9,10].in NUM cana-611 45 38 this this DET cana-611 45 39 section section NOUN cana-611 45 40 , , PUNCT cana-611 45 41 we we PRON cana-611 45 42 consider consider VERB cana-611 45 43 chemical chemical NOUN cana-611 45 44 structures structure NOUN cana-611 45 45 like like ADP cana-611 45 46 hexagonal hexagonal ADJ cana-611 45 47 parallelogram parallelogram NOUN cana-611 45 48 𝑃(π‘š 𝑃(π‘š PROPN cana-611 45 49 , , PUNCT cana-611 45 50 𝑛 𝑛 NOUN cana-611 45 51 ) ) PUNCT cana-611 45 52 nanotube nanotube NOUN cana-611 45 53 , , PUNCT cana-611 45 54 triangular triangular NOUN cana-611 45 55 benzenoid benzenoid ADJ cana-611 45 56 𝐺𝑛,zigzag 𝐺𝑛,zigzag NOUN cana-611 45 57 - - PUNCT cana-611 45 58 edge edge NOUN cana-611 45 59 coronoid coronoid PROPN cana-611 45 60 fused fuse VERB cana-611 45 61 with with ADP cana-611 45 62 starphene starphene NOUN cana-611 45 63 nanotubes nanotube NOUN cana-611 45 64 𝑍𝐢𝑆(π‘˜ 𝑍𝐢𝑆(π‘˜ NUM cana-611 45 65 , , PUNCT cana-611 45 66 𝑙 𝑙 NOUN cana-611 45 67 , , PUNCT cana-611 45 68 π‘š π‘š NOUN cana-611 45 69 ) ) PUNCT cana-611 45 70 , , PUNCT cana-611 45 71 dominating dominate VERB cana-611 45 72 derived derive VERB cana-611 45 73 networks network NOUN cana-611 45 74 𝐷1 𝐷1 NOUN cana-611 45 75 , , PUNCT cana-611 45 76 𝐷2 𝐷2 PROPN cana-611 45 77 , , PUNCT cana-611 45 78 𝐷3 𝐷3 PROPN cana-611 45 79 , , PUNCT cana-611 45 80 porphyrin porphyrin NOUN cana-611 45 81 dendrimer dendrimer NOUN cana-611 45 82 , , PUNCT cana-611 45 83 zinc zinc NOUN cana-611 45 84 - - PUNCT cana-611 45 85 porphyrin porphyrin NOUN cana-611 45 86 dendrimer dendrimer NOUN cana-611 45 87 , , PUNCT cana-611 45 88 propyl propyl ADJ cana-611 45 89 ether ether NOUN cana-611 45 90 imine imine NOUN cana-611 45 91 dendrimer dendrimer NOUN cana-611 45 92 , , PUNCT cana-611 45 93 poly(ethylene poly(ethylene ADJ cana-611 45 94 amido amido NOUN cana-611 45 95 amine amine ADJ cana-611 45 96 dendrimer dendrimer NOUN cana-611 45 97 , , PUNCT cana-611 45 98 pamam pamam NOUN cana-611 45 99 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 PROPN cana-611 45 100 ) ) PUNCT cana-611 45 101 , , PUNCT cana-611 45 102 linear linear PROPN cana-611 45 103 polyomino polyomino PROPN cana-611 45 104 chain chain NOUN cana-611 45 105 𝐿𝑛 𝐿𝑛 PROPN cana-611 45 106 , , PUNCT cana-611 45 107 𝑍𝑛 𝑍𝑛 PROPN cana-611 45 108 , , PUNCT cana-611 45 109 𝐡𝑛 𝐡𝑛 PROPN cana-611 45 110 1(𝑛 1(𝑛 NUM cana-611 45 111 β‰₯ β‰₯ NOUN cana-611 45 112 3 3 NUM cana-611 45 113 ) ) PUNCT cana-611 45 114 , , PUNCT cana-611 45 115 𝐡𝑛 𝐡𝑛 PROPN cana-611 45 116 2(𝑛 2(𝑛 NUM cana-611 45 117 β‰₯ β‰₯ NOUN cana-611 45 118 3 3 NUM cana-611 45 119 ) ) PUNCT cana-611 45 120 and and CCONJ cana-611 45 121 triangular triangular NOUN cana-611 45 122 , , PUNCT cana-611 45 123 hourglass hourglass NOUN cana-611 45 124 , , PUNCT cana-611 45 125 and and CCONJ cana-611 45 126 jagged jagged ADJ cana-611 45 127 - - PUNCT cana-611 45 128 rectangle rectangle NOUN cana-611 45 129 benzenoid benzenoid NOUN cana-611 45 130 systems system NOUN cana-611 45 131 [ [ X cana-611 45 132 2,3,4,5 2,3,4,5 NUM cana-611 45 133 ] ] X cana-611 45 134 . . PUNCT cana-611 46 1 for for ADP cana-611 46 2 notations notation NOUN cana-611 46 3 and and CCONJ cana-611 46 4 terminology terminology NOUN cana-611 46 5 used use VERB cana-611 46 6 in in ADP cana-611 46 7 this this DET cana-611 46 8 paper paper NOUN cana-611 46 9 are be AUX cana-611 46 10 taken take VERB cana-611 46 11 from from ADP cana-611 46 12 . . PROPN cana-611 47 1 2 2 X cana-611 47 2 . . NUM cana-611 47 3 related relate VERB cana-611 47 4 works work NOUN cana-611 47 5 topological topological ADJ cana-611 47 6 indices index NOUN cana-611 47 7 have have AUX cana-611 47 8 got get VERB cana-611 47 9 importance importance NOUN cana-611 47 10 due due ADP cana-611 47 11 to to ADP cana-611 47 12 its its PRON cana-611 47 13 applications application NOUN cana-611 47 14 in in ADP cana-611 47 15 chemistry chemistry NOUN cana-611 47 16 as as ADV cana-611 47 17 well well ADV cana-611 47 18 as as ADP cana-611 47 19 in in ADP cana-611 47 20 life life NOUN cana-611 47 21 science science NOUN cana-611 47 22 . . PUNCT cana-611 48 1 recently recently ADV cana-611 48 2 , , PUNCT cana-611 48 3 hosamani hosamani PROPN cana-611 48 4 et et PROPN cana-611 48 5 al al PROPN cana-611 48 6 . . PUNCT cana-611 49 1 [ [ X cana-611 49 2 11 11 NUM cana-611 49 3 ] ] PUNCT cana-611 49 4 , , PUNCT cana-611 49 5 have have AUX cana-611 49 6 put put VERB cana-611 49 7 forward forward ADV cana-611 49 8 the the DET cana-611 49 9 following follow VERB cana-611 49 10 new new ADJ cana-611 49 11 degreebased degreebase VERB cana-611 49 12 topological topological ADJ cana-611 49 13 indices index NOUN cana-611 49 14 : : PUNCT cana-611 49 15 𝑆1(𝐺 𝑆1(𝐺 NOUN cana-611 49 16 ) ) PUNCT cana-611 49 17 = = PUNCT cana-611 49 18 βˆ‘ βˆ‘ PUNCT cana-611 49 19 𝑑(𝑣𝑖 𝑑(𝑣𝑖 PROPN cana-611 49 20 ) ) PUNCT cana-611 49 21 |𝑑(𝑣𝑖)| |𝑑(𝑣𝑖)| NOUN cana-611 49 22 π‘£π‘–βˆˆπ‘‰ π‘£π‘–βˆˆπ‘‰ PROPN cana-611 49 23 ( ( PUNCT cana-611 49 24 1 1 X cana-611 49 25 ) ) PUNCT cana-611 49 26 𝑆2(𝐺 𝑆2(𝐺 NOUN cana-611 49 27 ) ) PUNCT cana-611 49 28 = = PUNCT cana-611 50 1 βˆ‘ βˆ‘ PUNCT cana-611 50 2 ( ( PUNCT cana-611 50 3 𝑑(𝑣𝑖)|𝑑(𝑣𝑖)| 𝑑(𝑣𝑖)|𝑑(𝑣𝑖)| NOUN cana-611 50 4 π‘£π‘–π‘£π‘—βˆˆπΈ π‘£π‘–π‘£π‘—βˆˆπΈ PROPN cana-611 50 5 + + NOUN cana-611 50 6 𝑑(𝑣𝑗)|𝑑(𝑣𝑗)| 𝑑(𝑣𝑗)|𝑑(𝑣𝑗)| NOUN cana-611 50 7 ) ) PUNCT cana-611 50 8 ( ( PUNCT cana-611 50 9 2 2 X cana-611 50 10 ) ) PUNCT cana-611 50 11 𝑆3(𝐺 𝑆3(𝐺 ADJ cana-611 50 12 ) ) PUNCT cana-611 50 13 = = PUNCT cana-611 50 14 βˆ‘ βˆ‘ PROPN cana-611 50 15 ( ( PUNCT cana-611 50 16 𝑑(𝑣𝑖)|𝑑(𝑣𝑗)| 𝑑(𝑣𝑖)|𝑑(𝑣𝑗)| VERB cana-611 50 17 π‘£π‘–π‘£π‘—βˆˆπΈ π‘£π‘–π‘£π‘—βˆˆπΈ X cana-611 50 18 + + NOUN cana-611 50 19 𝑑(𝑣𝑗)|𝑑(𝑣𝑖)| 𝑑(𝑣𝑗)|𝑑(𝑣𝑖)| NOUN cana-611 50 20 ) ) PUNCT cana-611 50 21 ( ( PUNCT cana-611 50 22 3 3 X cana-611 50 23 ) ) PUNCT cana-611 50 24 𝑆4(𝐺 𝑆4(𝐺 NOUN cana-611 50 25 ) ) PUNCT cana-611 50 26 = = PUNCT cana-611 50 27 βˆ‘ βˆ‘ PUNCT cana-611 50 28 ( ( PUNCT cana-611 50 29 𝑑(𝑣𝑖)|𝑑(𝑣𝑖)| 𝑑(𝑣𝑖)|𝑑(𝑣𝑖)| NOUN cana-611 50 30 π‘£π‘–π‘£π‘—βˆˆπΈ π‘£π‘–π‘£π‘—βˆˆπΈ PROPN cana-611 50 31 + + NOUN cana-611 50 32 𝑑(𝑣𝑗)|𝑑(𝑣𝑗)| 𝑑(𝑣𝑗)|𝑑(𝑣𝑗)| NOUN cana-611 50 33 ) ) PUNCT cana-611 50 34 ( ( PUNCT cana-611 50 35 4 4 X cana-611 50 36 ) ) PUNCT cana-611 50 37 𝑆5(𝐺 𝑆5(𝐺 NOUN cana-611 50 38 ) ) PUNCT cana-611 50 39 = = PUNCT cana-611 50 40 βˆ‘ βˆ‘ PROPN cana-611 50 41 ( ( PUNCT cana-611 50 42 𝑑(𝑣𝑖)|𝑑(𝑣𝑗)| 𝑑(𝑣𝑖)|𝑑(𝑣𝑗)| VERB cana-611 50 43 π‘£π‘–π‘£π‘—βˆˆπΈ π‘£π‘–π‘£π‘—βˆˆπΈ X cana-611 50 44 + + NOUN cana-611 50 45 𝑑(𝑣𝑗)|𝑑(𝑣𝑖)| 𝑑(𝑣𝑗)|𝑑(𝑣𝑖)| NOUN cana-611 50 46 ) ) PUNCT cana-611 50 47 ( ( PUNCT cana-611 50 48 5 5 NUM cana-611 50 49 ) ) SYM cana-611 50 50 3 3 NUM cana-611 50 51 . . X cana-611 50 52 methodology methodology NOUN cana-611 50 53 motivated motivate VERB cana-611 50 54 by by ADP cana-611 50 55 the the DET cana-611 50 56 inverse inverse NOUN cana-611 50 57 degree degree NOUN cana-611 50 58 of of ADP cana-611 50 59 a a DET cana-611 50 60 vertex vertex NOUN cana-611 50 61 and and CCONJ cana-611 50 62 the the DET cana-611 50 63 above above ADV cana-611 50 64 - - PUNCT cana-611 50 65 mentioned mention VERB cana-611 50 66 topological topological ADJ cana-611 50 67 indices index NOUN cana-611 50 68 , , PUNCT cana-611 50 69 here here ADV cana-611 50 70 we we PRON cana-611 50 71 introduced introduce VERB cana-611 50 72 the the DET cana-611 50 73 following follow VERB cana-611 50 74 topological topological ADJ cana-611 50 75 indices index NOUN cana-611 50 76 in in ADP cana-611 50 77 the the DET cana-611 50 78 field field NOUN cana-611 50 79 of of ADP cana-611 50 80 chemical chemical NOUN cana-611 50 81 graph graph NOUN cana-611 50 82 theory theory NOUN cana-611 50 83 . . PUNCT cana-611 51 1 𝑅𝑆1(𝐺 𝑅𝑆1(𝐺 NOUN cana-611 51 2 ) ) PUNCT cana-611 52 1 = = PUNCT cana-611 52 2 βˆ‘ βˆ‘ PUNCT cana-611 52 3 1 1 NUM cana-611 52 4 𝑑(𝑣𝑖)|𝑑(𝑣𝑖)|π‘£π‘–βˆˆπ‘‰ 𝑑(𝑣𝑖)|𝑑(𝑣𝑖)|π‘£π‘–βˆˆπ‘‰ NOUN cana-611 52 5 ( ( PUNCT cana-611 52 6 6 6 NUM cana-611 52 7 ) ) PUNCT cana-611 52 8 𝑅𝑆2(𝐺 𝑅𝑆2(𝐺 NUM cana-611 52 9 ) ) PUNCT cana-611 52 10 = = PUNCT cana-611 53 1 βˆ‘ βˆ‘ PUNCT cana-611 53 2 1 1 NUM cana-611 53 3 𝑑(𝑒)|𝑑(𝑒)|+𝑑(𝑣)|𝑑(𝑣)|π‘’π‘£βˆˆπΈ 𝑑(𝑒)|𝑑(𝑒)|+𝑑(𝑣)|𝑑(𝑣)|π‘’π‘£βˆˆπΈ NUM cana-611 53 4 ( ( PUNCT cana-611 53 5 7 7 NUM cana-611 53 6 ) ) PUNCT cana-611 53 7 𝑅𝑆3(𝐺 𝑅𝑆3(𝐺 NUM cana-611 53 8 ) ) PUNCT cana-611 54 1 = = PUNCT cana-611 54 2 βˆ‘ βˆ‘ PUNCT cana-611 54 3 1 1 NUM cana-611 54 4 𝑑(𝑒)|𝑑(𝑣)|+𝑑(𝑣)|𝑑(𝑒)|π‘’π‘£βˆˆπΈ 𝑑(𝑒)|𝑑(𝑣)|+𝑑(𝑣)|𝑑(𝑒)|π‘’π‘£βˆˆπΈ X cana-611 54 5 ( ( PUNCT cana-611 54 6 8) 8) NUM cana-611 54 7 𝑅𝑆4(𝐺 𝑅𝑆4(𝐺 NUM cana-611 54 8 ) ) PUNCT cana-611 55 1 = = PUNCT cana-611 55 2 βˆ‘ βˆ‘ PUNCT cana-611 55 3 1 1 NUM cana-611 55 4 𝑑(𝑒)|𝑑(𝑒)|𝑑(𝑣)|𝑑(𝑣)|π‘’π‘£βˆˆπΈ 𝑑(𝑒)|𝑑(𝑒)|𝑑(𝑣)|𝑑(𝑣)|π‘’π‘£βˆˆπΈ PROPN cana-611 55 5 ( ( PUNCT cana-611 55 6 9 9 NUM cana-611 55 7 ) ) PUNCT cana-611 55 8 communications communication NOUN cana-611 55 9 on on ADP cana-611 55 10 applied apply VERB cana-611 55 11 nonlinear nonlinear ADJ cana-611 55 12 analysis analysis NOUN cana-611 55 13 issn issn NOUN cana-611 55 14 : : PUNCT cana-611 55 15 1074 1074 NUM cana-611 55 16 - - PUNCT cana-611 55 17 133x 133x NUM cana-611 55 18 vol vol NOUN cana-611 55 19 31 31 NUM cana-611 55 20 no no NOUN cana-611 55 21 . . NOUN cana-611 55 22 2 2 NUM cana-611 55 23 ( ( PUNCT cana-611 55 24 2024 2024 NUM cana-611 55 25 ) ) PUNCT cana-611 55 26 419 419 NUM cana-611 55 27 https://internationalpubls.com https://internationalpubls.com X cana-611 55 28 𝑅𝑆5(𝐺 𝑅𝑆5(𝐺 PROPN cana-611 55 29 ) ) PUNCT cana-611 56 1 = = PUNCT cana-611 56 2 βˆ‘ βˆ‘ PUNCT cana-611 56 3 1 1 NUM cana-611 56 4 𝑑(𝑒)|𝑑(𝑣)|𝑑(𝑣)|𝑑(𝑒)|π‘’π‘£βˆˆπΈ 𝑑(𝑒)|𝑑(𝑣)|𝑑(𝑣)|𝑑(𝑒)|π‘’π‘£βˆˆπΈ PROPN cana-611 56 5 ( ( PUNCT cana-611 56 6 10 10 NUM cana-611 56 7 ) ) PUNCT cana-611 56 8 in in ADP cana-611 56 9 this this DET cana-611 56 10 paper paper NOUN cana-611 56 11 , , PUNCT cana-611 56 12 we we PRON cana-611 56 13 consider consider VERB cana-611 56 14 their their PRON cana-611 56 15 chemical chemical NOUN cana-611 56 16 structures structure NOUN cana-611 56 17 like like ADP cana-611 56 18 hexagonal hexagonal ADJ cana-611 56 19 parallelogram parallelogram NOUN cana-611 56 20 𝑃(π‘š 𝑃(π‘š PROPN cana-611 56 21 , , PUNCT cana-611 56 22 𝑛 𝑛 NOUN cana-611 56 23 ) ) PUNCT cana-611 56 24 nanotube nanotube NOUN cana-611 56 25 , , PUNCT cana-611 56 26 triangular triangular NOUN cana-611 56 27 benzenoid benzenoid NOUN cana-611 56 28 𝐺𝑛 𝐺𝑛 PROPN cana-611 56 29 , , PUNCT cana-611 56 30 zigzag zigzag NOUN cana-611 56 31 - - PUNCT cana-611 56 32 edge edge NOUN cana-611 56 33 coronoid coronoid PROPN cana-611 56 34 fused fuse VERB cana-611 56 35 with with ADP cana-611 56 36 starphene starphene NOUN cana-611 56 37 nanotubes nanotube NOUN cana-611 56 38 𝑍𝐢𝑆(π‘˜ 𝑍𝐢𝑆(π‘˜ NUM cana-611 56 39 , , PUNCT cana-611 56 40 𝑙 𝑙 NOUN cana-611 56 41 , , PUNCT cana-611 56 42 π‘š π‘š NOUN cana-611 56 43 ) ) PUNCT cana-611 56 44 , , PUNCT cana-611 56 45 dominating dominate VERB cana-611 56 46 derived derive VERB cana-611 56 47 networks network NOUN cana-611 56 48 𝐷1 𝐷1 NOUN cana-611 56 49 , , PUNCT cana-611 56 50 𝐷2 𝐷2 PROPN cana-611 56 51 , , PUNCT cana-611 56 52 𝐷3 𝐷3 PROPN cana-611 56 53 , , PUNCT cana-611 56 54 porphyrin porphyrin NOUN cana-611 56 55 dendrimer dendrimer NOUN cana-611 56 56 , , PUNCT cana-611 56 57 zinc zinc NOUN cana-611 56 58 - - PUNCT cana-611 56 59 porphyrin porphyrin NOUN cana-611 56 60 dendrimer dendrimer NOUN cana-611 56 61 , , PUNCT cana-611 56 62 propyl propyl ADJ cana-611 56 63 ether ether NOUN cana-611 56 64 imine imine NOUN cana-611 56 65 dendrimer dendrimer NOUN cana-611 56 66 , , PUNCT cana-611 56 67 poly(ethylene poly(ethylene ADJ cana-611 56 68 amido amido NOUN cana-611 56 69 amine amine ADJ cana-611 56 70 dendrimer dendrimer NOUN cana-611 56 71 , , PUNCT cana-611 56 72 pamam pamam NOUN cana-611 56 73 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 PROPN cana-611 56 74 ) ) PUNCT cana-611 56 75 , , PUNCT cana-611 56 76 linear linear PROPN cana-611 56 77 polyomino polyomino PROPN cana-611 56 78 chain chain NOUN cana-611 56 79 𝐿𝑛 𝐿𝑛 PROPN cana-611 56 80 , , PUNCT cana-611 56 81 𝑍𝑛 𝑍𝑛 PROPN cana-611 56 82 , , PUNCT cana-611 56 83 𝐡𝑛 𝐡𝑛 PROPN cana-611 56 84 1(𝑛 1(𝑛 NUM cana-611 56 85 β‰₯ β‰₯ NOUN cana-611 56 86 3 3 NUM cana-611 56 87 ) ) PUNCT cana-611 56 88 , , PUNCT cana-611 56 89 𝐡𝑛 𝐡𝑛 PROPN cana-611 56 90 2(𝑛 2(𝑛 NUM cana-611 56 91 β‰₯ β‰₯ NOUN cana-611 56 92 3 3 NUM cana-611 56 93 ) ) PUNCT cana-611 56 94 and and CCONJ cana-611 56 95 triangular triangular NOUN cana-611 56 96 , , PUNCT cana-611 56 97 hourglass hourglass NOUN cana-611 56 98 , , PUNCT cana-611 56 99 and and CCONJ cana-611 56 100 jagged jagged ADJ cana-611 56 101 - - PUNCT cana-611 56 102 rectangle rectangle NOUN cana-611 56 103 benzenoid benzenoid NOUN cana-611 56 104 systems system NOUN cana-611 56 105 which which PRON cana-611 56 106 are be AUX cana-611 56 107 depicted depict VERB cana-611 56 108 in in ADP cana-611 56 109 the the DET cana-611 56 110 following follow VERB cana-611 56 111 figures figure NOUN cana-611 56 112 2.1 2.1 NUM cana-611 56 113 , , PUNCT cana-611 56 114 2.2 2.2 NUM cana-611 56 115 , , PUNCT cana-611 56 116 2.3 2.3 NUM cana-611 56 117 , , PUNCT cana-611 56 118 2.4 2.4 NUM cana-611 56 119 , , PUNCT cana-611 56 120 2.5 2.5 NUM cana-611 56 121 and and CCONJ cana-611 56 122 2.6 2.6 NUM cana-611 56 123 respectively respectively ADV cana-611 56 124 : : PUNCT cana-611 56 125 figure figure VERB cana-611 56 126 2.1 2.1 NUM cana-611 56 127 : : PUNCT cana-611 56 128 hexagonal hexagonal ADJ cana-611 56 129 parallelogram parallelogram NOUN cana-611 56 130 . . PUNCT cana-611 57 1 figure figure VERB cana-611 57 2 2.2 2.2 NUM cana-611 57 3 : : PUNCT cana-611 57 4 triangular triangular NOUN cana-611 57 5 benzenoid benzenoid NOUN cana-611 57 6 . . PUNCT cana-611 58 1 communications communication NOUN cana-611 58 2 on on ADP cana-611 58 3 applied apply VERB cana-611 58 4 nonlinear nonlinear ADJ cana-611 58 5 analysis analysis NOUN cana-611 58 6 issn issn NOUN cana-611 58 7 : : PUNCT cana-611 58 8 1074 1074 NUM cana-611 58 9 - - PUNCT cana-611 58 10 133x 133x NUM cana-611 58 11 vol vol NOUN cana-611 58 12 31 31 NUM cana-611 58 13 no no NOUN cana-611 58 14 . . NOUN cana-611 58 15 2 2 NUM cana-611 58 16 ( ( PUNCT cana-611 58 17 2024 2024 NUM cana-611 58 18 ) ) PUNCT cana-611 58 19 420 420 NUM cana-611 58 20 https://internationalpubls.com https://internationalpubls.com X cana-611 58 21 figure figure NOUN cana-611 58 22 2.3 2.3 NUM cana-611 58 23 : : PUNCT cana-611 58 24 the the DET cana-611 58 25 zigzag zigzag NOUN cana-611 58 26 - - PUNCT cana-611 58 27 edge edge NOUN cana-611 58 28 coronoid coronoid PROPN cana-611 58 29 fused fuse VERB cana-611 58 30 with with ADP cana-611 58 31 starphene starphene NOUN cana-611 58 32 . . PUNCT cana-611 59 1 figure figure VERB cana-611 59 2 2.4 2.4 NUM cana-611 59 3 : : PUNCT cana-611 59 4 dominating dominate VERB cana-611 59 5 derived derive VERB cana-611 59 6 network network NOUN cana-611 59 7 . . PUNCT cana-611 60 1 figure figure VERB cana-611 60 2 2.5 2.5 NUM cana-611 60 3 : : PUNCT cana-611 60 4 porphyrin porphyrin NOUN cana-611 60 5 dendrimer dendrimer NOUN cana-611 60 6 . . PUNCT cana-611 61 1 communications communication NOUN cana-611 61 2 on on ADP cana-611 61 3 applied apply VERB cana-611 61 4 nonlinear nonlinear ADJ cana-611 61 5 analysis analysis NOUN cana-611 61 6 issn issn NOUN cana-611 61 7 : : PUNCT cana-611 61 8 1074 1074 NUM cana-611 61 9 - - PUNCT cana-611 61 10 133x 133x NUM cana-611 61 11 vol vol NOUN cana-611 61 12 31 31 NUM cana-611 61 13 no no NOUN cana-611 61 14 . . NOUN cana-611 61 15 2 2 NUM cana-611 61 16 ( ( PUNCT cana-611 61 17 2024 2024 NUM cana-611 61 18 ) ) PUNCT cana-611 61 19 421 421 NUM cana-611 61 20 https://internationalpubls.com https://internationalpubls.com X cana-611 61 21 figure figure NOUN cana-611 61 22 2.6 2.6 NUM cana-611 61 23 : : PUNCT cana-611 61 24 benzoid benzoid PROPN cana-611 61 25 hourglass hourglass NOUN cana-611 61 26 system system NOUN cana-611 61 27 . . PUNCT cana-611 62 1 4 4 X cana-611 62 2 . . NOUN cana-611 62 3 results result NOUN cana-611 62 4 and and CCONJ cana-611 62 5 discussion discussion NOUN cana-611 62 6 theorem theorem VERB cana-611 62 7 1 1 NUM cana-611 62 8 . . PUNCT cana-611 63 1 let let VERB cana-611 63 2 𝐺 𝐺 PROPN cana-611 63 3 denotes denote VERB cana-611 63 4 the the DET cana-611 63 5 line line NOUN cana-611 63 6 graph graph NOUN cana-611 63 7 of of ADP cana-611 63 8 subdivision subdivision NOUN cana-611 63 9 graph graph NOUN cana-611 63 10 of of ADP cana-611 63 11 the the DET cana-611 63 12 hexagonal hexagonal ADJ cana-611 63 13 parallelogram parallelogram NOUN cana-611 63 14 , , PUNCT cana-611 63 15 then then ADV cana-611 63 16 𝑅𝑆2(𝐺 𝑅𝑆2(𝐺 NUM cana-611 63 17 ) ) PUNCT cana-611 64 1 = = SYM cana-611 64 2 1 1 NUM cana-611 64 3 2 2 NUM cana-611 64 4 π‘šπ‘› π‘šπ‘› NOUN cana-611 64 5 + + NOUN cana-611 64 6 1145 1145 NUM cana-611 64 7 3348 3348 NUM cana-611 64 8 ( ( PUNCT cana-611 64 9 π‘š π‘š PROPN cana-611 64 10 + + NOUN cana-611 64 11 𝑛 𝑛 X cana-611 64 12 ) ) PUNCT cana-611 64 13 + + NUM cana-611 64 14 1087 1087 NUM cana-611 64 15 1674 1674 NUM cana-611 64 16 ( ( PUNCT cana-611 64 17 11 11 NUM cana-611 64 18 ) ) PUNCT cana-611 64 19 𝑅𝑆3(𝐺 𝑅𝑆3(𝐺 NUM cana-611 64 20 ) ) PUNCT cana-611 64 21 = = SYM cana-611 64 22 1 1 NUM cana-611 64 23 6 6 NUM cana-611 64 24 π‘šπ‘› π‘šπ‘› NOUN cana-611 64 25 + + CCONJ cana-611 64 26 823 823 NUM cana-611 64 27 1836 1836 NUM cana-611 64 28 ( ( PUNCT cana-611 64 29 π‘š π‘š PROPN cana-611 64 30 + + NOUN cana-611 64 31 𝑛 𝑛 X cana-611 64 32 ) ) PUNCT cana-611 64 33 + + CCONJ cana-611 64 34 401 401 NUM cana-611 64 35 918 918 NUM cana-611 64 36 ( ( PUNCT cana-611 64 37 12 12 NUM cana-611 64 38 ) ) PUNCT cana-611 64 39 𝑅𝑆4(𝐺 𝑅𝑆4(𝐺 PUNCT cana-611 64 40 ) ) PUNCT cana-611 64 41 = = SYM cana-611 65 1 1 1 NUM cana-611 65 2 81 81 NUM cana-611 65 3 π‘šπ‘› π‘šπ‘› NOUN cana-611 65 4 + + CCONJ cana-611 65 5 929 929 NUM cana-611 65 6 5832 5832 NUM cana-611 65 7 ( ( PUNCT cana-611 65 8 π‘š π‘š PROPN cana-611 65 9 + + NOUN cana-611 65 10 𝑛 𝑛 X cana-611 65 11 ) ) PUNCT cana-611 65 12 + + CCONJ cana-611 65 13 611 611 NUM cana-611 65 14 1458 1458 NUM cana-611 65 15 ( ( PUNCT cana-611 65 16 13 13 NUM cana-611 65 17 ) ) PUNCT cana-611 65 18 𝑅𝑆5(𝐺 𝑅𝑆5(𝐺 NUM cana-611 65 19 ) ) PUNCT cana-611 65 20 = = PUNCT cana-611 66 1 1 1 NUM cana-611 66 2 81 81 NUM cana-611 66 3 π‘šπ‘› π‘šπ‘› NOUN cana-611 66 4 + + NUM cana-611 66 5 1037 1037 NUM cana-611 66 6 5832 5832 NUM cana-611 66 7 ( ( PUNCT cana-611 66 8 π‘š π‘š PROPN cana-611 66 9 + + NOUN cana-611 66 10 𝑛 𝑛 X cana-611 66 11 ) ) PUNCT cana-611 66 12 + + CCONJ cana-611 66 13 557 557 NUM cana-611 66 14 1458 1458 NUM cana-611 66 15 ( ( PUNCT cana-611 66 16 14 14 NUM cana-611 66 17 ) ) PUNCT cana-611 66 18 proof proof NOUN cana-611 66 19 . . PUNCT cana-611 67 1 let let VERB cana-611 67 2 ( ( PUNCT cana-611 67 3 π‘š,);,βˆˆβ„€+ π‘š,);,βˆˆβ„€+ X cana-611 67 4 be be AUX cana-611 67 5 a a DET cana-611 67 6 hexagonal hexagonal ADJ cana-611 67 7 parallelogram parallelogram NOUN cana-611 67 8 of of ADP cana-611 67 9 order order NOUN cana-611 67 10 2(π‘š+ 2(π‘š+ NUM cana-611 67 11 𝑛+ 𝑛+ PUNCT cana-611 67 12 π‘šπ‘›i π‘šπ‘›i NOUN cana-611 67 13 and and CCONJ cana-611 67 14 size size NOUN cana-611 67 15 ( ( PUNCT cana-611 67 16 3π‘šπ‘›+ 3π‘šπ‘›+ NUM cana-611 67 17 2π‘š+ 2π‘š+ NUM cana-611 67 18 2𝑛+ 2𝑛+ NUM cana-611 67 19 1 1 NUM cana-611 67 20 ) ) PUNCT cana-611 67 21 respectively respectively ADV cana-611 67 22 . . PUNCT cana-611 68 1 let let VERB cana-611 68 2 𝐺= 𝐺= VERB cana-611 68 3 𝐿(𝑆(𝑃(π‘š,𝑛 𝐿(𝑆(𝑃(π‘š,𝑛 VERB cana-611 68 4 ) ) PUNCT cana-611 68 5 ) ) PUNCT cana-611 68 6 ) ) PUNCT cana-611 69 1 denotes denote VERB cana-611 69 2 the the DET cana-611 69 3 line line NOUN cana-611 69 4 graph graph NOUN cana-611 69 5 of of ADP cana-611 69 6 subdivision subdivision NOUN cana-611 69 7 graph graph NOUN cana-611 69 8 of of ADP cana-611 69 9 𝑃(π‘š,𝑛);π‘š,π‘›βˆˆβ„€+ 𝑃(π‘š,𝑛);π‘š,π‘›βˆˆβ„€+ PROPN cana-611 69 10 then then ADV cana-611 69 11 clearly clearly ADV cana-611 69 12 , , PUNCT cana-611 69 13 the the DET cana-611 69 14 order order NOUN cana-611 69 15 and and CCONJ cana-611 69 16 size size NOUN cana-611 69 17 of of ADP cana-611 69 18 𝐺are 𝐺are PROPN cana-611 69 19 2(3mn+2m+2n+1 2(3mn+2m+2n+1 NUM cana-611 69 20 ) ) PUNCT cana-611 69 21 and and CCONJ cana-611 69 22 9π‘šπ‘›+ 9π‘šπ‘›+ NUM cana-611 70 1 4π‘š+ 4π‘š+ NUM cana-611 70 2 4𝑛+ 4𝑛+ NUM cana-611 70 3 5 5 NUM cana-611 70 4 respectively respectively ADV cana-611 70 5 . . PUNCT cana-611 71 1 the the DET cana-611 71 2 edge edge NOUN cana-611 71 3 set set NOUN cana-611 71 4 of of ADP cana-611 71 5 𝐺 𝐺 PROPN cana-611 71 6 can can AUX cana-611 71 7 be be AUX cana-611 71 8 partitioned partition VERB cana-611 71 9 into into ADP cana-611 71 10 three three NUM cana-611 71 11 disjoint disjoint NOUN cana-611 71 12 sets set NOUN cana-611 71 13 πœ€2,2,πœ€2,3 πœ€2,2,πœ€2,3 PUNCT cana-611 71 14 andi andi PROPN cana-611 71 15 πœ€3,3 πœ€3,3 PROPN cana-611 71 16 , , PUNCT cana-611 71 17 where where SCONJ cana-611 71 18 πœ€(𝐿(𝑆(𝑃(π‘š,𝑛 πœ€(𝐿(𝑆(𝑃(π‘š,𝑛 NOUN cana-611 71 19 ) ) PUNCT cana-611 71 20 ) ) PUNCT cana-611 71 21 ) ) PUNCT cana-611 71 22 ) ) PUNCT cana-611 72 1 = = PUNCT cana-611 72 2 πœ€2,2βˆͺπœ€2,3βˆͺπœ€3,3 πœ€2,2βˆͺπœ€2,3βˆͺπœ€3,3 VERB cana-611 72 3 . . PUNCT cana-611 73 1 further far ADV cana-611 73 2 , , PUNCT cana-611 73 3 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 73 4 = = SYM cana-611 73 5 2(π‘š+𝑛+4 2(π‘š+𝑛+4 NUM cana-611 73 6 ) ) PUNCT cana-611 73 7 , , PUNCT cana-611 73 8 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 73 9 = = SYM cana-611 73 10 4(π‘š+π‘›βˆ’2 4(π‘š+π‘›βˆ’2 NUM cana-611 73 11 ) ) PUNCT cana-611 73 12 , , PUNCT cana-611 73 13 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 73 14 = = SYM cana-611 73 15 9π‘šπ‘›βˆ’2π‘šβˆ’2π‘›βˆ’5 9π‘šπ‘›βˆ’2π‘šβˆ’2π‘›βˆ’5 NUM cana-611 73 16 . . PUNCT cana-611 74 1 such such ADJ cana-611 74 2 that that PRON cana-611 74 3 βˆ£πœ€(𝐿(𝑆(𝑃(π‘š,𝑛))))∣ βˆ£πœ€(𝐿(𝑆(𝑃(π‘š,𝑛))))∣ PROPN cana-611 74 4 = = SYM cana-611 74 5 βˆ£πœ€2,2∣+βˆ£πœ€2,3∣+βˆ£πœ€3,3∣ βˆ£πœ€2,2∣+βˆ£πœ€2,3∣+βˆ£πœ€3,3∣ PROPN cana-611 74 6 = = SYM cana-611 74 7 9π‘šπ‘›+4π‘š+4𝑛+5 9π‘šπ‘›+4π‘š+4𝑛+5 PROPN cana-611 74 8 . . PUNCT cana-611 75 1 thus thus ADV cana-611 75 2 , , PUNCT cana-611 75 3 with with ADP cana-611 75 4 this this DET cana-611 75 5 background background NOUN cana-611 75 6 by by ADP cana-611 75 7 employing employ VERB cana-611 75 8 equations equation NOUN cana-611 75 9 ( ( PUNCT cana-611 75 10 1)-(5 1)-(5 NUM cana-611 75 11 ) ) PUNCT cana-611 75 12 and and CCONJ cana-611 75 13 ( ( PUNCT cana-611 75 14 6)-(10 6)-(10 NUM cana-611 75 15 ) ) PUNCT cana-611 75 16 , , PUNCT cana-611 75 17 we we PRON cana-611 75 18 get get VERB cana-611 75 19 the the DET cana-611 75 20 required require VERB cana-611 75 21 results result NOUN cana-611 75 22 . . PUNCT cana-611 76 1 theorem theorem NOUN cana-611 76 2 2 2 NUM cana-611 76 3 . . X cana-611 77 1 let let AUX cana-611 77 2 ( ( PUNCT cana-611 77 3 ( ( PUNCT cana-611 77 4 𝐺𝑛 𝐺𝑛 NOUN cana-611 77 5 ) ) PUNCT cana-611 77 6 ) ) PUNCT cana-611 77 7 denotes denote VERB cana-611 77 8 their their PRON cana-611 77 9 line line NOUN cana-611 77 10 graphic graphic NOUN cana-611 77 11 of of ADP cana-611 77 12 subdivision subdivision NOUN cana-611 77 13 graph graph NOUN cana-611 77 14 of of ADP cana-611 77 15 the the DET cana-611 77 16 hexagonal hexagonal ADJ cana-611 77 17 parallelogram parallelogram NOUN cana-611 77 18 , , PUNCT cana-611 77 19 then then ADV cana-611 77 20 𝑅𝑆2(𝐿(𝑆(𝐺𝑛 𝑅𝑆2(𝐿(𝑆(𝐺𝑛 NOUN cana-611 77 21 ) ) PUNCT cana-611 77 22 ) ) PUNCT cana-611 77 23 ) ) PUNCT cana-611 78 1 = = SYM cana-611 78 2 83 83 NUM cana-611 78 3 1000 1000 NUM cana-611 78 4 𝑛2 𝑛2 NOUN cana-611 78 5 + + CCONJ cana-611 78 6 3 3 NUM cana-611 78 7 5 5 NUM cana-611 78 8 𝑛 𝑛 NOUN cana-611 78 9 + + NOUN cana-611 78 10 41 41 NUM cana-611 78 11 50 50 NUM cana-611 78 12 ( ( PUNCT cana-611 78 13 15 15 NUM cana-611 78 14 ) ) PUNCT cana-611 78 15 𝑅𝑆3(𝐿(𝑆(𝐺𝑛 𝑅𝑆3(𝐿(𝑆(𝐺𝑛 NOUN cana-611 78 16 ) ) PUNCT cana-611 78 17 ) ) PUNCT cana-611 78 18 ) ) PUNCT cana-611 78 19 = = SYM cana-611 78 20 9 9 NUM cana-611 78 21 108 108 NUM cana-611 78 22 𝑛2 𝑛2 NOUN cana-611 78 23 + + CCONJ cana-611 78 24 19 19 NUM cana-611 78 25 25 25 NUM cana-611 78 26 𝑛 𝑛 PRON cana-611 78 27 βˆ’ βˆ’ NUM cana-611 78 28 9 9 NUM cana-611 78 29 100 100 NUM cana-611 78 30 ( ( PUNCT cana-611 78 31 16 16 NUM cana-611 78 32 ) ) PUNCT cana-611 78 33 𝑅𝑆4(𝐿(𝑆(𝐺𝑛 𝑅𝑆4(𝐿(𝑆(𝐺𝑛 NOUN cana-611 78 34 ) ) PUNCT cana-611 78 35 ) ) PUNCT cana-611 78 36 ) ) PUNCT cana-611 79 1 = = SYM cana-611 79 2 9 9 NUM cana-611 79 3 1458 1458 NUM cana-611 79 4 𝑛2 𝑛2 NOUN cana-611 79 5 + + CCONJ cana-611 79 6 1 1 NUM cana-611 79 7 4 4 NUM cana-611 79 8 𝑛 𝑛 NOUN cana-611 79 9 + + NUM cana-611 79 10 3 3 NUM cana-611 79 11 25 25 NUM cana-611 79 12 ( ( PUNCT cana-611 79 13 17 17 NUM cana-611 79 14 ) ) PUNCT cana-611 79 15 𝑅𝑆5(𝐿(𝑆(𝐺𝑛 𝑅𝑆5(𝐿(𝑆(𝐺𝑛 NOUN cana-611 79 16 ) ) PUNCT cana-611 79 17 ) ) PUNCT cana-611 79 18 ) ) PUNCT cana-611 79 19 = = SYM cana-611 79 20 9 9 NUM cana-611 79 21 1458 1458 NUM cana-611 79 22 𝑛2 𝑛2 NOUN cana-611 79 23 + + CCONJ cana-611 79 24 27 27 NUM cana-611 79 25 100 100 NUM cana-611 79 26 𝑛 𝑛 VERB cana-611 79 27 + + NUM cana-611 79 28 47 47 NUM cana-611 79 29 100 100 NUM cana-611 79 30 ( ( PUNCT cana-611 79 31 18 18 NUM cana-611 79 32 ) ) PUNCT cana-611 79 33 communications communication NOUN cana-611 79 34 on on ADP cana-611 79 35 applied apply VERB cana-611 79 36 nonlinear nonlinear ADJ cana-611 79 37 analysis analysis NOUN cana-611 79 38 issn issn NOUN cana-611 79 39 : : PUNCT cana-611 79 40 1074 1074 NUM cana-611 79 41 - - PUNCT cana-611 79 42 133x 133x NUM cana-611 79 43 vol vol NOUN cana-611 79 44 31 31 NUM cana-611 79 45 no no NOUN cana-611 79 46 . . NOUN cana-611 79 47 2 2 NUM cana-611 79 48 ( ( PUNCT cana-611 79 49 2024 2024 NUM cana-611 79 50 ) ) PUNCT cana-611 79 51 422 422 NUM cana-611 79 52 https://internationalpubls.com https://internationalpubls.com X cana-611 79 53 proof proof NOUN cana-611 79 54 . . PUNCT cana-611 80 1 let let VERB cana-611 80 2 𝐺𝑛;βˆˆβ„€+ 𝐺𝑛;βˆˆβ„€+ NOUN cana-611 80 3 be be AUX cana-611 80 4 a a DET cana-611 80 5 triangular triangular NOUN cana-611 80 6 benzenoid benzenoid NOUN cana-611 80 7 of of ADP cana-611 80 8 order order NOUN cana-611 80 9 𝑛2 𝑛2 NOUN cana-611 80 10 + + CCONJ cana-611 80 11 4𝑛+ 4𝑛+ NUM cana-611 80 12 1 1 NUM cana-611 80 13 and and CCONJ cana-611 80 14 size size NOUN cana-611 80 15 3 3 NUM cana-611 80 16 2 2 NUM cana-611 80 17 𝑛(𝑛 𝑛(𝑛 PROPN cana-611 80 18 + + CCONJ cana-611 80 19 3)respectively 3)respectively ADV cana-611 80 20 . . PUNCT cana-611 81 1 let let VERB cana-611 81 2 ( ( PUNCT cana-611 81 3 𝑆(𝐺𝑛 𝑆(𝐺𝑛 NOUN cana-611 81 4 ) ) PUNCT cana-611 81 5 ) ) PUNCT cana-611 81 6 ) ) PUNCT cana-611 82 1 denotes denote VERB cana-611 82 2 the the DET cana-611 82 3 line line NOUN cana-611 82 4 graph graph NOUN cana-611 82 5 of of ADP cana-611 82 6 subdivision subdivision NOUN cana-611 82 7 graph graph NOUN cana-611 82 8 of of ADP cana-611 82 9 𝐺𝑛 𝐺𝑛 PROPN cana-611 82 10 then then ADV cana-611 82 11 clearly clearly ADV cana-611 82 12 , , PUNCT cana-611 82 13 the the DET cana-611 82 14 order order NOUN cana-611 82 15 and and CCONJ cana-611 82 16 size size NOUN cana-611 82 17 of of ADP cana-611 82 18 𝐿(𝑆(𝐺𝑛 𝐿(𝑆(𝐺𝑛 PROPN cana-611 82 19 ) ) PUNCT cana-611 82 20 ) ) PUNCT cana-611 82 21 ) ) PUNCT cana-611 83 1 are be AUX cana-611 83 2 3𝑛(𝑛+ 3𝑛(𝑛+ PROPN cana-611 83 3 3 3 NUM cana-611 83 4 ) ) PUNCT cana-611 83 5 and and CCONJ cana-611 83 6 3(3𝑛2 3(3𝑛2 NUM cana-611 83 7 + + NOUN cana-611 83 8 7𝑛+2 7𝑛+2 NUM cana-611 83 9 ) ) PUNCT cana-611 83 10 2 2 NUM cana-611 83 11 respectively respectively ADV cana-611 83 12 . . PUNCT cana-611 84 1 the the DET cana-611 84 2 edge edge NOUN cana-611 84 3 set set NOUN cana-611 84 4 of of ADP cana-611 84 5 ( ( PUNCT cana-611 84 6 𝑆(𝐺𝑛 𝑆(𝐺𝑛 NOUN cana-611 84 7 ) ) PUNCT cana-611 84 8 ) ) PUNCT cana-611 84 9 ) ) PUNCT cana-611 84 10 can can AUX cana-611 84 11 be be AUX cana-611 84 12 partitioned partition VERB cana-611 84 13 into into ADP cana-611 84 14 three three NUM cana-611 84 15 disjoint disjoint NOUN cana-611 84 16 sets set NOUN cana-611 84 17 πœ€2,2,πœ€2,3 πœ€2,2,πœ€2,3 PUNCT cana-611 84 18 and and CCONJ cana-611 84 19 πœ€3,3 πœ€3,3 PROPN cana-611 84 20 , , PUNCT cana-611 84 21 where where SCONJ cana-611 84 22 ( ( PUNCT cana-611 84 23 𝐿(𝑆(𝐺𝑛 𝐿(𝑆(𝐺𝑛 PROPN cana-611 84 24 ) ) PUNCT cana-611 84 25 ) ) PUNCT cana-611 84 26 ) ) PUNCT cana-611 84 27 ) ) PUNCT cana-611 85 1 = = PUNCT cana-611 85 2 πœ€2,2 πœ€2,2 PROPN cana-611 85 3 βˆͺπœ€2,3 βˆͺπœ€2,3 PROPN cana-611 85 4 βˆͺπœ€3,3 βˆͺπœ€3,3 NOUN cana-611 85 5 . . PUNCT cana-611 85 6 further far ADV cana-611 85 7 , , PUNCT cana-611 85 8 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 85 9 = = SYM cana-611 85 10 3(𝑛+ 3(𝑛+ NUM cana-611 85 11 3 3 NUM cana-611 85 12 ) ) PUNCT cana-611 85 13 , , PUNCT cana-611 85 14 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 85 15 = = SYM cana-611 85 16 6(π‘›βˆ’ 6(π‘›βˆ’ NUM cana-611 85 17 1 1 NUM cana-611 85 18 ) ) PUNCT cana-611 85 19 , , PUNCT cana-611 85 20 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 85 21 = = SYM cana-611 85 22 3(3𝑛2+π‘›βˆ’4 3(3𝑛2+π‘›βˆ’4 NUM cana-611 85 23 ) ) PUNCT cana-611 85 24 2 2 NUM cana-611 85 25 . . PUNCT cana-611 86 1 such such ADJ cana-611 86 2 that that SCONJ cana-611 86 3 |πœ€ |πœ€ PROPN cana-611 86 4 ( ( PUNCT cana-611 86 5 𝐿(𝑆(𝐺𝑛)))| 𝐿(𝑆(𝐺𝑛)))| PROPN cana-611 86 6 = = SYM cana-611 86 7 |πœ€2,2| |πœ€2,2| PROPN cana-611 86 8 + + CCONJ cana-611 86 9 |πœ€2,3| |πœ€2,3| NOUN cana-611 86 10 + + CCONJ cana-611 86 11 |πœ€3,3| |πœ€3,3| NOUN cana-611 86 12 = = PUNCT cana-611 86 13 3(3𝑛2 3(3𝑛2 NUM cana-611 86 14 + + NUM cana-611 86 15 7𝑛 7𝑛 NUM cana-611 86 16 + + CCONJ cana-611 86 17 2 2 NUM cana-611 86 18 ) ) PUNCT cana-611 86 19 2 2 NUM cana-611 86 20 . . PUNCT cana-611 87 1 thus thus ADV cana-611 87 2 , , PUNCT cana-611 87 3 with with ADP cana-611 87 4 thisbackground thisbackground ADJ cana-611 87 5 by by ADP cana-611 87 6 employing employ VERB cana-611 87 7 equations equation NOUN cana-611 87 8 ( ( PUNCT cana-611 87 9 1)-(5 1)-(5 NUM cana-611 87 10 ) ) PUNCT cana-611 87 11 and and CCONJ cana-611 87 12 ( ( PUNCT cana-611 87 13 6)-(10 6)-(10 NUM cana-611 87 14 ) ) PUNCT cana-611 87 15 , , PUNCT cana-611 87 16 we we PRON cana-611 87 17 get get VERB cana-611 87 18 the the DET cana-611 87 19 required require VERB cana-611 87 20 results result NOUN cana-611 87 21 . . PUNCT cana-611 88 1 theorem theorem NOUN cana-611 88 2 3 3 X cana-611 88 3 . . PUNCT cana-611 89 1 let let VERB cana-611 89 2 ( ( PUNCT cana-611 89 3 ( ( PUNCT cana-611 89 4 𝐼 𝐼 PROPN cana-611 89 5 ) ) PUNCT cana-611 89 6 ) ) PUNCT cana-611 89 7 be be AUX cana-611 89 8 the the DET cana-611 89 9 line line NOUN cana-611 89 10 graph graph NOUN cana-611 89 11 of of ADP cana-611 89 12 the the DET cana-611 89 13 subdivision subdivision NOUN cana-611 89 14 graph graph NOUN cana-611 89 15 of of ADP cana-611 89 16 zigzag zigzag NOUN cana-611 89 17 - - PUNCT cana-611 89 18 edges edge NOUN cana-611 89 19 coronoid coronoid PROPN cana-611 89 20 fused fuse VERB cana-611 89 21 with with ADP cana-611 89 22 starphene starphene NOUN cana-611 89 23 nanotubes nanotube NOUN cana-611 89 24 𝑍𝐢𝑆(π‘˜,𝑙,π‘š 𝑍𝐢𝑆(π‘˜,𝑙,π‘š NOUN cana-611 89 25 ) ) PUNCT cana-611 89 26 for for ADP cana-611 89 27 π‘˜= π‘˜= ADV cana-611 89 28 𝑙= 𝑙= PRON cana-611 89 29 π‘š= π‘š= VERB cana-611 89 30 4 4 NUM cana-611 89 31 . . PUNCT cana-611 89 32 then then ADV cana-611 89 33 𝑅𝑆2(𝐿(𝑆(𝐼 𝑅𝑆2(𝐿(𝑆(𝐼 ADJ cana-611 89 34 ) ) PUNCT cana-611 89 35 ) ) PUNCT cana-611 89 36 ) ) PUNCT cana-611 90 1 = = SYM cana-611 90 2 3 3 NUM cana-611 90 3 2 2 NUM cana-611 90 4 ( ( PUNCT cana-611 90 5 π‘˜ π‘˜ PROPN cana-611 90 6 + + NOUN cana-611 90 7 𝑙 𝑙 NOUN cana-611 90 8 + + NUM cana-611 90 9 π‘š π‘š NOUN cana-611 90 10 ) ) PUNCT cana-611 90 11 βˆ’ βˆ’ PROPN cana-611 90 12 355 355 NUM cana-611 90 13 1116 1116 NUM cana-611 90 14 ( ( PUNCT cana-611 90 15 19 19 NUM cana-611 90 16 ) ) PUNCT cana-611 90 17 𝑅𝑆3(𝐿(𝑆(𝐼 𝑅𝑆3(𝐿(𝑆(𝐼 NOUN cana-611 90 18 ) ) PUNCT cana-611 90 19 ) ) PUNCT cana-611 90 20 ) ) PUNCT cana-611 91 1 = = PUNCT cana-611 91 2 1129 1129 NUM cana-611 91 3 612 612 NUM cana-611 91 4 ( ( PUNCT cana-611 91 5 π‘˜ π‘˜ PROPN cana-611 91 6 + + NOUN cana-611 91 7 𝑙 𝑙 NOUN cana-611 91 8 + + NUM cana-611 91 9 π‘š π‘š NOUN cana-611 91 10 ) ) PUNCT cana-611 91 11 βˆ’ βˆ’ PROPN cana-611 91 12 941 941 NUM cana-611 91 13 100 100 NUM cana-611 91 14 ( ( PUNCT cana-611 91 15 20 20 NUM cana-611 91 16 ) ) PUNCT cana-611 91 17 𝑅𝑆4(𝐿(𝑆(𝐼 𝑅𝑆4(𝐿(𝑆(𝐼 NOUN cana-611 91 18 ) ) PUNCT cana-611 91 19 ) ) PUNCT cana-611 91 20 ) ) PUNCT cana-611 92 1 = = PUNCT cana-611 92 2 103 103 NUM cana-611 92 3 200 200 NUM cana-611 92 4 ( ( PUNCT cana-611 92 5 π‘˜ π‘˜ PROPN cana-611 92 6 + + NOUN cana-611 92 7 𝑙 𝑙 NOUN cana-611 92 8 + + NUM cana-611 92 9 π‘š π‘š NOUN cana-611 92 10 ) ) PUNCT cana-611 92 11 βˆ’ βˆ’ PROPN cana-611 92 12 271 271 NUM cana-611 92 13 100 100 NUM cana-611 92 14 ( ( PUNCT cana-611 92 15 21 21 NUM cana-611 92 16 ) ) PUNCT cana-611 92 17 𝑅𝑆5(𝐿(𝑆(𝐼 𝑅𝑆5(𝐿(𝑆(𝐼 NOUN cana-611 92 18 ) ) PUNCT cana-611 92 19 ) ) PUNCT cana-611 92 20 ) ) PUNCT cana-611 93 1 = = PUNCT cana-611 94 1 57 57 NUM cana-611 94 2 100 100 NUM cana-611 94 3 ( ( PUNCT cana-611 94 4 π‘˜ π‘˜ PROPN cana-611 94 5 + + NOUN cana-611 94 6 𝑙 𝑙 NOUN cana-611 94 7 + + NUM cana-611 94 8 π‘š π‘š NOUN cana-611 94 9 ) ) PUNCT cana-611 94 10 βˆ’ βˆ’ PROPN cana-611 94 11 31 31 NUM cana-611 94 12 10 10 NUM cana-611 94 13 ( ( PUNCT cana-611 94 14 22 22 NUM cana-611 94 15 ) ) PUNCT cana-611 94 16 proof proof NOUN cana-611 94 17 . . PUNCT cana-611 95 1 let let AUX cana-611 95 2 be be AUX cana-611 95 3 zizag zizag NOUN cana-611 95 4 - - PUNCT cana-611 95 5 edge edge NOUN cana-611 95 6 coronoid coronoid PROPN cana-611 95 7 fused fuse VERB cana-611 95 8 with with ADP cana-611 95 9 starphene starphene NOUN cana-611 95 10 nanotubes nanotube NOUN cana-611 95 11 ( ( PUNCT cana-611 95 12 π‘˜ π‘˜ NOUN cana-611 95 13 , , PUNCT cana-611 95 14 , , PUNCT cana-611 95 15 ) ) PUNCT cana-611 95 16 for for ADP cana-611 95 17 π‘˜= π‘˜= ADV cana-611 95 18 𝑙= 𝑙= PRON cana-611 95 19 π‘š= π‘š= VERB cana-611 95 20 4 4 NUM cana-611 95 21 of of ADP cana-611 95 22 order order NOUN cana-611 95 23 36π‘˜+ 36π‘˜+ NUM cana-611 95 24 54 54 NUM cana-611 95 25 and and CCONJ cana-611 95 26 size size NOUN cana-611 95 27 15(π‘˜+ 15(π‘˜+ NUM cana-611 95 28 𝑙+ 𝑙+ PUNCT cana-611 96 1 π‘š π‘š X cana-611 96 2 ) ) PUNCT cana-611 96 3 βˆ’ βˆ’ PROPN cana-611 96 4 63 63 NUM cana-611 96 5 respectively respectively ADV cana-611 96 6 . . PUNCT cana-611 97 1 let let AUX cana-611 97 2 ( ( PUNCT cana-611 97 3 ( ( PUNCT cana-611 97 4 𝐼 𝐼 PROPN cana-611 97 5 ) ) PUNCT cana-611 97 6 ) ) PUNCT cana-611 97 7 be be AUX cana-611 97 8 the the DET cana-611 97 9 line line NOUN cana-611 97 10 graphic graphic NOUN cana-611 97 11 of of ADP cana-611 97 12 the the DET cana-611 97 13 subdivision subdivision NOUN cana-611 97 14 graph graph NOUN cana-611 97 15 of of ADP cana-611 97 16 zigzag zigzag NOUN cana-611 97 17 - - PUNCT cana-611 97 18 edge edge NOUN cana-611 97 19 coronoid coronoid PROPN cana-611 97 20 fused fuse VERB cana-611 97 21 with with ADP cana-611 97 22 starphene starphene NOUN cana-611 97 23 nanotubes nanotube NOUN cana-611 97 24 ( ( PUNCT cana-611 97 25 π‘˜,,π‘š π‘˜,,π‘š NUM cana-611 97 26 ) ) PUNCT cana-611 97 27 for for ADP cana-611 97 28 π‘˜= π‘˜= ADV cana-611 97 29 𝑙= 𝑙= PRON cana-611 97 30 π‘š= π‘š= VERB cana-611 97 31 4 4 NUM cana-611 97 32 . . X cana-611 97 33 theni theni NOUN cana-611 97 34 clearly clearly ADV cana-611 97 35 , , PUNCT cana-611 97 36 the the DET cana-611 97 37 order order NOUN cana-611 97 38 and and CCONJ cana-611 97 39 size size NOUN cana-611 97 40 of of ADP cana-611 97 41 ( ( PUNCT cana-611 97 42 ( ( PUNCT cana-611 97 43 𝐼 𝐼 PROPN cana-611 97 44 ) ) PUNCT cana-611 97 45 ) ) PUNCT cana-611 97 46 are be AUX cana-611 97 47 30(π‘˜+𝑙+π‘š126 30(π‘˜+𝑙+π‘š126 NUM cana-611 97 48 ) ) PUNCT cana-611 97 49 and and CCONJ cana-611 97 50 39(π‘˜+𝑙+π‘š)+153 39(π‘˜+𝑙+π‘š)+153 NUM cana-611 97 51 respectively respectively ADV cana-611 97 52 . . PUNCT cana-611 98 1 the the DET cana-611 98 2 edge edge NOUN cana-611 98 3 set set NOUN cana-611 98 4 of of ADP cana-611 98 5 𝐿(𝑆(𝐼 𝐿(𝑆(𝐼 PROPN cana-611 98 6 ) ) PUNCT cana-611 98 7 ) ) PUNCT cana-611 98 8 can can AUX cana-611 98 9 be be AUX cana-611 98 10 partitioned partition VERB cana-611 98 11 into into ADP cana-611 98 12 three three NUM cana-611 98 13 disjoint disjoint NOUN cana-611 98 14 sets set NOUN cana-611 98 15 πœ€2,2,πœ€2,3 πœ€2,2,πœ€2,3 PUNCT cana-611 98 16 and and CCONJ cana-611 98 17 πœ€3,3,where πœ€3,3,where ADV cana-611 98 18 πœ€(𝐿(𝑆(𝐼 πœ€(𝐿(𝑆(𝐼 NUM cana-611 98 19 ) ) PUNCT cana-611 98 20 ) ) PUNCT cana-611 98 21 ) ) PUNCT cana-611 99 1 = = PUNCT cana-611 100 1 πœ€2,2i πœ€2,2i NOUN cana-611 100 2 βˆͺπœ€2,3 βˆͺπœ€2,3 NOUN cana-611 100 3 βˆͺπœ€3,3 βˆͺπœ€3,3 NOUN cana-611 100 4 . . PUNCT cana-611 100 5 further far ADV cana-611 100 6 , , PUNCT cana-611 100 7 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 100 8 = = SYM cana-611 100 9 6(π‘˜+ 6(π‘˜+ NUM cana-611 100 10 𝑙+π‘šβˆ’ 𝑙+π‘šβˆ’ NOUN cana-611 100 11 5 5 NUM cana-611 100 12 ) ) PUNCT cana-611 100 13 , , PUNCT cana-611 100 14 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 100 15 = = SYM cana-611 100 16 12(π‘˜+ 12(π‘˜+ NUM cana-611 100 17 𝑙+ 𝑙+ PUNCT cana-611 100 18 π‘šβˆ’ π‘šβˆ’ NOUN cana-611 100 19 7 7 NUM cana-611 100 20 ) ) PUNCT cana-611 100 21 , , PUNCT cana-611 100 22 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 100 23 = = SYM cana-611 100 24 21(π‘˜+ 21(π‘˜+ NUM cana-611 100 25 𝑙+ 𝑙+ PUNCT cana-611 100 26 π‘š π‘š X cana-611 100 27 ) ) PUNCT cana-611 100 28 βˆ’ βˆ’ PROPN cana-611 100 29 39 39 NUM cana-611 100 30 . . PUNCT cana-611 101 1 such such ADJ cana-611 101 2 that that DET cana-611 101 3 ∣(𝐿(𝑆(𝐼)))∣ ∣(𝐿(𝑆(𝐼)))∣ NOUN cana-611 101 4 = = SYM cana-611 101 5 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 101 6 + + NUM cana-611 101 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ NOUN cana-611 101 8 + + NUM cana-611 101 9 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 101 10 = = SYM cana-611 101 11 39(π‘˜+ 39(π‘˜+ NUM cana-611 101 12 𝑙+ 𝑙+ PUNCT cana-611 101 13 π‘š π‘š X cana-611 101 14 ) ) PUNCT cana-611 101 15 + + NUM cana-611 101 16 153 153 NUM cana-611 101 17 . . PUNCT cana-611 102 1 thus thus ADV cana-611 102 2 , , PUNCT cana-611 102 3 with with ADP cana-611 102 4 this this DET cana-611 102 5 background background NOUN cana-611 102 6 by by ADP cana-611 102 7 employing employ VERB cana-611 102 8 equations equation NOUN cana-611 102 9 ( ( PUNCT cana-611 102 10 1)-(5 1)-(5 NUM cana-611 102 11 ) ) PUNCT cana-611 102 12 and and CCONJ cana-611 102 13 ( ( PUNCT cana-611 102 14 6)-(10),we 6)-(10),we PRON cana-611 102 15 get get VERB cana-611 102 16 the the DET cana-611 102 17 required require VERB cana-611 102 18 results result NOUN cana-611 102 19 . . PUNCT cana-611 103 1 theorem theorem ADJ cana-611 103 2 4 4 NUM cana-611 103 3 . . PUNCT cana-611 104 1 let let AUX cana-611 104 2 𝐷1(𝑛 𝐷1(𝑛 PRON cana-611 104 3 ) ) PUNCT cana-611 104 4 be be VERB cana-611 104 5 the the DET cana-611 104 6 dominating dominating NOUN cana-611 104 7 derived derive VERB cana-611 104 8 network network NOUN cana-611 104 9 of of ADP cana-611 104 10 1st 1st ADJ cana-611 104 11 type type NOUN cana-611 104 12 . . PUNCT cana-611 105 1 then then ADV cana-611 105 2 𝑅𝑆2(𝐷1(𝑛 𝑅𝑆2(𝐷1(𝑛 ADV cana-611 105 3 ) ) PUNCT cana-611 105 4 ) ) PUNCT cana-611 106 1 = = SYM cana-611 106 2 9 9 NUM cana-611 106 3 25 25 NUM cana-611 106 4 𝑛2 𝑛2 NOUN cana-611 106 5 + + CCONJ cana-611 106 6 1 1 NUM cana-611 106 7 5 5 NUM cana-611 106 8 𝑛 𝑛 NOUN cana-611 106 9 + + NOUN cana-611 106 10 3 3 NUM cana-611 106 11 100 100 NUM cana-611 106 12 ( ( PUNCT cana-611 106 13 23 23 NUM cana-611 106 14 ) ) PUNCT cana-611 106 15 𝑅𝑆3(𝐷1(𝑛 𝑅𝑆3(𝐷1(𝑛 NOUN cana-611 106 16 ) ) PUNCT cana-611 106 17 ) ) PUNCT cana-611 107 1 = = SYM cana-611 107 2 49 49 NUM cana-611 107 3 100 100 NUM cana-611 107 4 𝑛2 𝑛2 NOUN cana-611 107 5 + + CCONJ cana-611 107 6 19 19 NUM cana-611 107 7 25 25 NUM cana-611 107 8 𝑛 𝑛 PRON cana-611 107 9 βˆ’ βˆ’ NUM cana-611 107 10 8 8 NUM cana-611 107 11 25 25 NUM cana-611 107 12 ( ( PUNCT cana-611 107 13 24 24 NUM cana-611 107 14 ) ) PUNCT cana-611 107 15 𝑅𝑆4(𝐷1(𝑛 𝑅𝑆4(𝐷1(𝑛 NUM cana-611 107 16 ) ) PUNCT cana-611 107 17 ) ) PUNCT cana-611 108 1 = = SYM cana-611 108 2 9 9 NUM cana-611 108 3 500 500 NUM cana-611 108 4 𝑛2 𝑛2 NOUN cana-611 108 5 + + CCONJ cana-611 108 6 29 29 NUM cana-611 108 7 100 100 NUM cana-611 108 8 𝑛 𝑛 PRON cana-611 108 9 βˆ’ βˆ’ NUM cana-611 108 10 21 21 NUM cana-611 108 11 500 500 NUM cana-611 108 12 ( ( PUNCT cana-611 108 13 25 25 NUM cana-611 108 14 ) ) PUNCT cana-611 108 15 𝑅𝑆5(𝐷1(𝑛 𝑅𝑆5(𝐷1(𝑛 NOUN cana-611 108 16 ) ) PUNCT cana-611 108 17 ) ) PUNCT cana-611 109 1 = = SYM cana-611 109 2 1 1 NUM cana-611 109 3 50 50 NUM cana-611 109 4 𝑛2 𝑛2 NOUN cana-611 109 5 + + CCONJ cana-611 109 6 39 39 NUM cana-611 109 7 10 10 NUM cana-611 109 8 𝑛 𝑛 PRON cana-611 109 9 βˆ’ βˆ’ NUM cana-611 109 10 11 11 NUM cana-611 109 11 100 100 NUM cana-611 109 12 ( ( PUNCT cana-611 109 13 26 26 NUM cana-611 109 14 ) ) PUNCT cana-611 109 15 proof proof NOUN cana-611 109 16 . . PUNCT cana-611 110 1 let let VERB cana-611 110 2 𝐷1(𝑛 𝐷1(𝑛 PRON cana-611 110 3 ) ) PUNCT cana-611 110 4 be be AUX cana-611 110 5 the the DET cana-611 110 6 dominating dominating NOUN cana-611 110 7 derived derive VERB cana-611 110 8 network network NOUN cana-611 110 9 of of ADP cana-611 110 10 1st 1st ADJ cana-611 110 11 type type NOUN cana-611 110 12 . . PUNCT cana-611 111 1 the the DET cana-611 111 2 edge edge NOUN cana-611 111 3 set set NOUN cana-611 111 4 of of ADP cana-611 111 5 𝐷1(𝑛 𝐷1(𝑛 PROPN cana-611 111 6 ) ) PUNCT cana-611 111 7 can can AUX cana-611 111 8 be be AUX cana-611 111 9 partitioned partition VERB cana-611 111 10 into into ADP cana-611 111 11 six six NUM cana-611 111 12 disjoint disjoint NOUN cana-611 111 13 sets set NOUN cana-611 111 14 πœ€2,2,πœ€2,3,πœ€2,4,πœ€3,3,πœ€3,4 πœ€2,2,πœ€2,3,πœ€2,4,πœ€3,3,πœ€3,4 ADJ cana-611 111 15 and and CCONJ cana-611 111 16 πœ€4,4 πœ€4,4 PROPN cana-611 111 17 , , PUNCT cana-611 111 18 where where SCONJ cana-611 111 19 πœ€(𝐷1(𝑛 πœ€(𝐷1(𝑛 NOUN cana-611 111 20 ) ) PUNCT cana-611 111 21 ) ) PUNCT cana-611 112 1 = = PUNCT cana-611 112 2 communications communication NOUN cana-611 112 3 on on ADP cana-611 112 4 applied apply VERB cana-611 112 5 nonlinear nonlinear ADJ cana-611 112 6 analysis analysis NOUN cana-611 112 7 issn issn NOUN cana-611 112 8 : : PUNCT cana-611 112 9 1074 1074 NUM cana-611 112 10 - - PUNCT cana-611 112 11 133x 133x NUM cana-611 112 12 vol vol NOUN cana-611 112 13 31 31 NUM cana-611 112 14 no no NOUN cana-611 112 15 . . NOUN cana-611 112 16 2 2 NUM cana-611 112 17 ( ( PUNCT cana-611 112 18 2024 2024 NUM cana-611 112 19 ) ) PUNCT cana-611 112 20 423 423 NUM cana-611 112 21 https://internationalpubls.com https://internationalpubls.com X cana-611 112 22 πœ€2,2βˆͺπœ€2,3 πœ€2,2βˆͺπœ€2,3 PROPN cana-611 112 23 βˆͺπœ€2,4 βˆͺπœ€2,4 PROPN cana-611 112 24 βˆͺπœ€3,3 βˆͺπœ€3,3 PROPN cana-611 112 25 βˆͺπœ€3,4 βˆͺπœ€3,4 PROPN cana-611 112 26 βˆͺπœ€4,4 βˆͺπœ€4,4 PROPN cana-611 112 27 . . PUNCT cana-611 113 1 further far ADV cana-611 113 2 , , PUNCT cana-611 113 3 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 113 4 = = SYM cana-611 113 5 4𝑛 4𝑛 NOUN cana-611 113 6 , , PUNCT cana-611 113 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 113 8 = = SYM cana-611 114 1 4π‘›βˆ’ 4π‘›βˆ’ NUM cana-611 114 2 4 4 NUM cana-611 114 3 , , PUNCT cana-611 114 4 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 114 5 = = SYM cana-611 114 6 28𝑛 28𝑛 NUM cana-611 114 7 16 16 NUM cana-611 114 8 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 114 9 = = SYM cana-611 114 10 9𝑛2 9𝑛2 NUM cana-611 114 11 βˆ’ βˆ’ NUM cana-611 114 12 13𝑛+ 13𝑛+ NUM cana-611 114 13 5 5 NUM cana-611 114 14 , , PUNCT cana-611 114 15 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 114 16 = = PUNCT cana-611 115 1 36𝑛2 36𝑛2 NUM cana-611 115 2 βˆ’ βˆ’ NUM cana-611 115 3 56𝑛+ 56𝑛+ NUM cana-611 115 4 24 24 NUM cana-611 115 5 and and CCONJ cana-611 115 6 βˆ£πœ€4,4∣ βˆ£πœ€4,4∣ VERB cana-611 115 7 = = PUNCT cana-611 115 8 36𝑛2βˆ’ 36𝑛2βˆ’ NUM cana-611 115 9 52𝑛+20 52𝑛+20 NUM cana-611 115 10 . . PUNCT cana-611 116 1 such such ADJ cana-611 116 2 that that DET cana-611 116 3 βˆ£πœ€(𝐷1(𝑛))∣ βˆ£πœ€(𝐷1(𝑛))∣ NOUN cana-611 116 4 = = SYM cana-611 116 5 βˆ£πœ€2,2βˆ£βˆ£πœ€2,3∣ βˆ£πœ€2,2βˆ£βˆ£πœ€2,3∣ ADJ cana-611 116 6 + + CCONJ cana-611 116 7 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 116 8 + + CCONJ cana-611 116 9 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 117 1 + + NOUN cana-611 117 2 i i PRON cana-611 117 3 βˆ£πœ€3,4∣+ βˆ£πœ€3,4∣+ VERB cana-611 117 4 βˆ£πœ€4,4∣. βˆ£πœ€4,4∣. VERB cana-611 117 5 thus thus ADV cana-611 117 6 , , PUNCT cana-611 117 7 with with ADP cana-611 117 8 this this DET cana-611 117 9 background background NOUN cana-611 117 10 by by ADP cana-611 117 11 employing employ VERB cana-611 117 12 equations equation NOUN cana-611 117 13 ( ( PUNCT cana-611 117 14 1)-(5 1)-(5 NUM cana-611 117 15 ) ) PUNCT cana-611 117 16 and and CCONJ cana-611 117 17 ( ( PUNCT cana-611 117 18 6)-(10 6)-(10 NUM cana-611 117 19 ) ) PUNCT cana-611 117 20 , , PUNCT cana-611 117 21 we we PRON cana-611 117 22 get get VERB cana-611 117 23 the the DET cana-611 117 24 required require VERB cana-611 117 25 results result NOUN cana-611 117 26 . . PUNCT cana-611 118 1 theorem theorem NOUN cana-611 118 2 5 5 NUM cana-611 118 3 . . PUNCT cana-611 119 1 let let VERB cana-611 119 2 𝐷2(𝑛 𝐷2(𝑛 NUM cana-611 119 3 ) ) PUNCT cana-611 120 1 be be VERB cana-611 120 2 the the DET cana-611 120 3 dominating dominating NOUN cana-611 120 4 derived derive VERB cana-611 120 5 network network NOUN cana-611 120 6 of of ADP cana-611 120 7 2nd 2nd ADJ cana-611 120 8 type type NOUN cana-611 120 9 . . PUNCT cana-611 121 1 then then ADV cana-611 121 2 𝑅𝑆2(𝐷2(𝑛 𝑅𝑆2(𝐷2(𝑛 NOUN cana-611 121 3 ) ) PUNCT cana-611 121 4 ) ) PUNCT cana-611 122 1 = = PUNCT cana-611 122 2 79 79 NUM cana-611 122 3 100 100 NUM cana-611 122 4 𝑛2 𝑛2 NOUN cana-611 122 5 βˆ’ βˆ’ PROPN cana-611 122 6 2 2 NUM cana-611 122 7 5 5 NUM cana-611 122 8 𝑛 𝑛 NOUN cana-611 122 9 + + NUM cana-611 122 10 13 13 NUM cana-611 122 11 50 50 NUM cana-611 122 12 ( ( PUNCT cana-611 122 13 27 27 NUM cana-611 122 14 ) ) PUNCT cana-611 122 15 𝑅𝑆3(𝐷2(𝑛 𝑅𝑆3(𝐷2(𝑛 NOUN cana-611 122 16 ) ) PUNCT cana-611 122 17 ) ) PUNCT cana-611 123 1 = = SYM cana-611 123 2 69 69 NUM cana-611 123 3 50 50 NUM cana-611 123 4 𝑛2 𝑛2 NOUN cana-611 123 5 βˆ’ βˆ’ PROPN cana-611 123 6 41 41 NUM cana-611 123 7 100 100 NUM cana-611 123 8 𝑛 𝑛 PRON cana-611 123 9 + + NUM cana-611 123 10 3 3 NUM cana-611 123 11 50 50 NUM cana-611 123 12 ( ( PUNCT cana-611 123 13 28 28 NUM cana-611 123 14 ) ) PUNCT cana-611 123 15 𝑅𝑆4(𝐷2(𝑛 𝑅𝑆4(𝐷2(𝑛 NOUN cana-611 123 16 ) ) PUNCT cana-611 123 17 ) ) PUNCT cana-611 124 1 = = SYM cana-611 124 2 17 17 NUM cana-611 124 3 100 100 NUM cana-611 124 4 𝑛2 𝑛2 NOUN cana-611 124 5 + + CCONJ cana-611 124 6 13 13 NUM cana-611 124 7 200 200 NUM cana-611 124 8 𝑛 𝑛 DET cana-611 125 1 + + CCONJ cana-611 125 2 11 11 NUM cana-611 125 3 250 250 NUM cana-611 125 4 ( ( PUNCT cana-611 125 5 29 29 NUM cana-611 125 6 ) ) PUNCT cana-611 125 7 𝑅𝑆5(𝐷2(𝑛 𝑅𝑆5(𝐷2(𝑛 NOUN cana-611 125 8 ) ) PUNCT cana-611 125 9 ) ) PUNCT cana-611 126 1 = = SYM cana-611 126 2 13 13 NUM cana-611 126 3 50 50 NUM cana-611 126 4 𝑛2 𝑛2 NOUN cana-611 126 5 + + CCONJ cana-611 126 6 21 21 NUM cana-611 126 7 500 500 NUM cana-611 126 8 𝑛 𝑛 DET cana-611 126 9 + + NUM cana-611 126 10 13 13 NUM cana-611 126 11 500 500 NUM cana-611 126 12 ( ( PUNCT cana-611 126 13 30 30 NUM cana-611 126 14 ) ) PUNCT cana-611 126 15 proof proof NOUN cana-611 126 16 . . PUNCT cana-611 127 1 let let VERB cana-611 127 2 𝐷2(𝑛 𝐷2(𝑛 NUM cana-611 127 3 ) ) PUNCT cana-611 127 4 be be AUX cana-611 127 5 the the DET cana-611 127 6 dominating dominating NOUN cana-611 127 7 derived derive VERB cana-611 127 8 network network NOUN cana-611 127 9 of of ADP cana-611 127 10 2nd 2nd ADJ cana-611 127 11 type type NOUN cana-611 127 12 . . PUNCT cana-611 128 1 the the DET cana-611 128 2 edge edge NOUN cana-611 128 3 set set NOUN cana-611 128 4 of of ADP cana-611 128 5 𝐷1(𝑛 𝐷1(𝑛 PROPN cana-611 128 6 ) ) PUNCT cana-611 128 7 can can AUX cana-611 128 8 be be AUX cana-611 128 9 partitioned partition VERB cana-611 128 10 into into ADP cana-611 128 11 five five NUM cana-611 128 12 disjoint disjoint NOUN cana-611 128 13 sets set NOUN cana-611 128 14 πœ€2,2,πœ€2,3,πœ€2,4,πœ€3,4 πœ€2,2,πœ€2,3,πœ€2,4,πœ€3,4 ADJ cana-611 128 15 and and CCONJ cana-611 128 16 πœ€4,4 πœ€4,4 PROPN cana-611 128 17 , , PUNCT cana-611 128 18 where where SCONJ cana-611 128 19 ( ( PUNCT cana-611 128 20 𝐷2(𝑛 𝐷2(𝑛 NUM cana-611 128 21 ) ) PUNCT cana-611 128 22 ) ) PUNCT cana-611 129 1 = = X cana-611 129 2 πœ€2,2 πœ€2,2 PROPN cana-611 129 3 βˆͺπœ€2,3 βˆͺπœ€2,3 PROPN cana-611 129 4 βˆͺπœ€2,4 βˆͺπœ€2,4 PROPN cana-611 129 5 βˆͺπœ€3,4 βˆͺπœ€3,4 ADJ cana-611 129 6 βˆͺπœ€4,4 βˆͺπœ€4,4 PROPN cana-611 129 7 . . PUNCT cana-611 130 1 further far ADV cana-611 130 2 , , PUNCT cana-611 130 3 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 130 4 = = SYM cana-611 130 5 4𝑛 4𝑛 NOUN cana-611 130 6 , , PUNCT cana-611 130 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 130 8 = = SYM cana-611 130 9 18𝑛2 18𝑛2 NUM cana-611 130 10 βˆ’22𝑛+6,i βˆ’22𝑛+6,i VERB cana-611 130 11 βˆ£πœ€2,4∣= βˆ£πœ€2,4∣= PROPN cana-611 130 12 28π‘›βˆ’16 28π‘›βˆ’16 NUM cana-611 130 13 , , PUNCT cana-611 130 14 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 130 15 = = PUNCT cana-611 131 1 36𝑛2 36𝑛2 NUM cana-611 131 2 βˆ’ βˆ’ NUM cana-611 131 3 56𝑛+ 56𝑛+ NUM cana-611 131 4 24 24 NUM cana-611 131 5 and and CCONJ cana-611 131 6 βˆ£πœ€4,4∣ βˆ£πœ€4,4∣ VERB cana-611 131 7 = = PUNCT cana-611 131 8 36𝑛2 36𝑛2 NUM cana-611 131 9 βˆ’52𝑛+ βˆ’52𝑛+ X cana-611 131 10 20 20 NUM cana-611 131 11 . . PUNCT cana-611 132 1 such such ADJ cana-611 132 2 that that SCONJ cana-611 132 3 ∣(𝐷2(𝑛))∣ ∣(𝐷2(𝑛))∣ ADP cana-611 132 4 = = SYM cana-611 132 5 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ ADJ cana-611 132 6 + + NUM cana-611 132 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 132 8 + + CCONJ cana-611 132 9 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 132 10 + + CCONJ cana-611 132 11 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 132 12 + + CCONJ cana-611 132 13 βˆ£πœ€4,4∣. βˆ£πœ€4,4∣. PUNCT cana-611 132 14 thus thus ADV cana-611 132 15 , , PUNCT cana-611 132 16 with with ADP cana-611 132 17 this this DET cana-611 132 18 background background NOUN cana-611 132 19 by by ADP cana-611 132 20 employing employ VERB cana-611 132 21 equations(1)-(5 equations(1)-(5 NOUN cana-611 132 22 ) ) PUNCT cana-611 132 23 and and CCONJ cana-611 132 24 ( ( PUNCT cana-611 132 25 6)-(10 6)-(10 NUM cana-611 132 26 ) ) PUNCT cana-611 132 27 , , PUNCT cana-611 132 28 we we PRON cana-611 132 29 get get VERB cana-611 132 30 the the DET cana-611 132 31 required require VERB cana-611 132 32 results result NOUN cana-611 132 33 . . PUNCT cana-611 133 1 theorem theorem ADJ cana-611 133 2 6 6 NUM cana-611 133 3 . . PUNCT cana-611 134 1 let let VERB cana-611 134 2 𝐷3(𝑛 𝐷3(𝑛 NOUN cana-611 134 3 ) ) PUNCT cana-611 135 1 be be VERB cana-611 135 2 the the DET cana-611 135 3 dominating dominating NOUN cana-611 135 4 derived derive VERB cana-611 135 5 network network NOUN cana-611 135 6 of of ADP cana-611 135 7 3rd 3rd ADJ cana-611 135 8 type type NOUN cana-611 135 9 . . PUNCT cana-611 136 1 then then ADV cana-611 136 2 𝑅𝑆2(𝐷3(𝑛 𝑅𝑆2(𝐷3(𝑛 NOUN cana-611 136 3 ) ) PUNCT cana-611 136 4 ) ) PUNCT cana-611 137 1 = = SYM cana-611 137 2 7 7 NUM cana-611 137 3 25 25 NUM cana-611 137 4 𝑛2 𝑛2 NOUN cana-611 137 5 + + CCONJ cana-611 137 6 21 21 NUM cana-611 137 7 100 100 NUM cana-611 137 8 𝑛 𝑛 DET cana-611 137 9 + + NUM cana-611 137 10 17 17 NUM cana-611 137 11 200 200 NUM cana-611 137 12 ( ( PUNCT cana-611 137 13 31 31 NUM cana-611 137 14 ) ) PUNCT cana-611 137 15 𝑅𝑆3(𝐷3(𝑛 𝑅𝑆3(𝐷3(𝑛 NUM cana-611 137 16 ) ) PUNCT cana-611 137 17 ) ) PUNCT cana-611 138 1 = = PUNCT cana-611 138 2 127 127 NUM cana-611 138 3 100 100 NUM cana-611 138 4 𝑛2 𝑛2 NOUN cana-611 138 5 βˆ’ βˆ’ PROPN cana-611 138 6 17 17 NUM cana-611 138 7 50 50 NUM cana-611 138 8 𝑛 𝑛 PRON cana-611 138 9 + + NUM cana-611 138 10 43 43 NUM cana-611 138 11 500 500 NUM cana-611 138 12 ( ( PUNCT cana-611 138 13 32 32 NUM cana-611 138 14 ) ) PUNCT cana-611 138 15 𝑅𝑆4(𝐷3(𝑛 𝑅𝑆4(𝐷3(𝑛 NUM cana-611 138 16 ) ) PUNCT cana-611 138 17 ) ) PUNCT cana-611 138 18 = = SYM cana-611 138 19 9 9 NUM cana-611 138 20 250 250 NUM cana-611 138 21 𝑛2 𝑛2 NOUN cana-611 138 22 + + CCONJ cana-611 138 23 23 23 NUM cana-611 138 24 100 100 NUM cana-611 138 25 𝑛 𝑛 DET cana-611 139 1 + + CCONJ cana-611 139 2 7 7 NUM cana-611 139 3 1000 1000 NUM cana-611 139 4 ( ( PUNCT cana-611 139 5 33 33 NUM cana-611 139 6 ) ) PUNCT cana-611 139 7 𝑅𝑆5(𝐷3(𝑛 𝑅𝑆5(𝐷3(𝑛 NOUN cana-611 139 8 ) ) PUNCT cana-611 139 9 ) ) PUNCT cana-611 140 1 = = SYM cana-611 140 2 141 141 NUM cana-611 140 3 1000 1000 NUM cana-611 140 4 𝑛2 𝑛2 NOUN cana-611 140 5 + + CCONJ cana-611 140 6 17 17 NUM cana-611 140 7 100 100 NUM cana-611 140 8 𝑛 𝑛 DET cana-611 140 9 + + CCONJ cana-611 140 10 7 7 NUM cana-611 140 11 1000 1000 NUM cana-611 140 12 ( ( PUNCT cana-611 140 13 34 34 NUM cana-611 140 14 ) ) PUNCT cana-611 140 15 proof proof NOUN cana-611 140 16 . . PUNCT cana-611 141 1 let let VERB cana-611 141 2 𝐷3(𝑛 𝐷3(𝑛 NUM cana-611 141 3 ) ) PUNCT cana-611 142 1 be be AUX cana-611 142 2 the the DET cana-611 142 3 dominating dominating NOUN cana-611 142 4 derived derive VERB cana-611 142 5 network network NOUN cana-611 142 6 of of ADP cana-611 142 7 3rd 3rd ADJ cana-611 142 8 type type NOUN cana-611 142 9 . . PUNCT cana-611 143 1 the the DET cana-611 143 2 edge edge NOUN cana-611 143 3 set set NOUN cana-611 143 4 of of ADP cana-611 143 5 𝐷1(𝑛 𝐷1(𝑛 PROPN cana-611 143 6 ) ) PUNCT cana-611 143 7 can can AUX cana-611 143 8 be be AUX cana-611 143 9 partitioned partition VERB cana-611 143 10 into into ADP cana-611 143 11 three three NUM cana-611 143 12 disjoint disjoint NOUN cana-611 143 13 sets set NOUN cana-611 143 14 πœ€2,2,πœ€2,4 πœ€2,2,πœ€2,4 NOUN cana-611 143 15 and and CCONJ cana-611 143 16 πœ€4,4 πœ€4,4 PROPN cana-611 143 17 , , PUNCT cana-611 143 18 where where SCONJ cana-611 143 19 πœ€(𝐷3(𝑛 πœ€(𝐷3(𝑛 NOUN cana-611 143 20 ) ) PUNCT cana-611 143 21 ) ) PUNCT cana-611 144 1 = = PUNCT cana-611 144 2 πœ€2,2 πœ€2,2 PROPN cana-611 144 3 βˆͺπœ€2,4 βˆͺπœ€2,4 PROPN cana-611 144 4 βˆͺ βˆͺ ADJ cana-611 144 5 πœ€4,4 πœ€4,4 PROPN cana-611 144 6 . . PUNCT cana-611 145 1 further far ADV cana-611 145 2 , , PUNCT cana-611 145 3 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 145 4 = = SYM cana-611 145 5 4𝑛 4𝑛 NOUN cana-611 145 6 , , PUNCT cana-611 145 7 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 145 8 = = PUNCT cana-611 146 1 36𝑛2 36𝑛2 NUM cana-611 146 2 βˆ’ βˆ’ NUM cana-611 146 3 20𝑛 20𝑛 NUM cana-611 146 4 and and CCONJ cana-611 146 5 βˆ£πœ€4,4∣ βˆ£πœ€4,4∣ VERB cana-611 146 6 = = PUNCT cana-611 146 7 72𝑛2 72𝑛2 NUM cana-611 146 8 βˆ’ βˆ’ PROPN cana-611 146 9 108𝑛+ 108𝑛+ NUM cana-611 146 10 44 44 NUM cana-611 146 11 . . PUNCT cana-611 147 1 such such ADJ cana-611 147 2 that that SCONJ cana-611 147 3 ∣(𝐷3(𝑛))∣ ∣(𝐷3(𝑛))∣ NOUN cana-611 147 4 = = SYM cana-611 147 5 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 147 6 + + NUM cana-611 147 7 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 147 8 + + CCONJ cana-611 147 9 βˆ£πœ€4,4∣. βˆ£πœ€4,4∣. PUNCT cana-611 147 10 thus thus ADV cana-611 147 11 , , PUNCT cana-611 147 12 with with ADP cana-611 147 13 this this DET cana-611 147 14 background background NOUN cana-611 147 15 by by ADP cana-611 147 16 employing employ VERB cana-611 147 17 equations equation NOUN cana-611 147 18 ( ( PUNCT cana-611 147 19 1)-(5 1)-(5 NUM cana-611 147 20 ) ) PUNCT cana-611 147 21 and and CCONJ cana-611 147 22 ( ( PUNCT cana-611 147 23 6)-(10 6)-(10 NUM cana-611 147 24 ) ) PUNCT cana-611 147 25 , , PUNCT cana-611 147 26 we we PRON cana-611 147 27 get get VERB cana-611 147 28 the the DET cana-611 147 29 required require VERB cana-611 147 30 results result NOUN cana-611 147 31 . . PUNCT cana-611 148 1 theorem theorem ADJ cana-611 148 2 7 7 NUM cana-611 148 3 . . PUNCT cana-611 149 1 let let VERB cana-611 149 2 𝐷𝑛𝑃𝑛 𝐷𝑛𝑃𝑛 PROPN cana-611 149 3 be be AUX cana-611 149 4 the the DET cana-611 149 5 prophyrin prophyrin NOUN cana-611 149 6 dendrimer dendrimer NOUN cana-611 149 7 . . PUNCT cana-611 150 1 then then ADV cana-611 150 2 𝑅𝑆2(𝐷𝑛𝑃𝑛 𝑅𝑆2(𝐷𝑛𝑃𝑛 X cana-611 150 3 ) ) PUNCT cana-611 150 4 = = SYM cana-611 151 1 323 323 NUM cana-611 151 2 100 100 NUM cana-611 151 3 𝑛 𝑛 DET cana-611 151 4 βˆ’ βˆ’ NUM cana-611 151 5 41 41 NUM cana-611 151 6 50 50 NUM cana-611 151 7 ( ( PUNCT cana-611 151 8 35 35 NUM cana-611 151 9 ) ) PUNCT cana-611 151 10 𝑅𝑆3(𝐷𝑛𝑃𝑛 𝑅𝑆3(𝐷𝑛𝑃𝑛 ADJ cana-611 151 11 ) ) PUNCT cana-611 151 12 = = SYM cana-611 151 13 967 967 NUM cana-611 151 14 100 100 NUM cana-611 151 15 𝑛 𝑛 DET cana-611 151 16 βˆ’ βˆ’ NUM cana-611 151 17 49 49 NUM cana-611 151 18 50 50 NUM cana-611 151 19 ( ( PUNCT cana-611 151 20 36 36 NUM cana-611 151 21 ) ) PUNCT cana-611 151 22 𝑅𝑆4(𝐷𝑛𝑃𝑛 𝑅𝑆4(𝐷𝑛𝑃𝑛 NUM cana-611 151 23 ) ) PUNCT cana-611 151 24 = = PUNCT cana-611 152 1 63 63 NUM cana-611 152 2 50 50 NUM cana-611 152 3 𝑛 𝑛 DET cana-611 152 4 βˆ’ βˆ’ PROPN cana-611 152 5 37 37 NUM cana-611 152 6 100 100 NUM cana-611 152 7 ( ( PUNCT cana-611 152 8 37 37 NUM cana-611 152 9 ) ) PUNCT cana-611 152 10 communications communication NOUN cana-611 152 11 on on ADP cana-611 152 12 applied apply VERB cana-611 152 13 nonlinear nonlinear ADJ cana-611 152 14 analysis analysis NOUN cana-611 152 15 issn issn NOUN cana-611 152 16 : : PUNCT cana-611 152 17 1074 1074 NUM cana-611 152 18 - - PUNCT cana-611 152 19 133x 133x NUM cana-611 152 20 vol vol NOUN cana-611 152 21 31 31 NUM cana-611 152 22 no no NOUN cana-611 152 23 . . NOUN cana-611 152 24 2 2 NUM cana-611 152 25 ( ( PUNCT cana-611 152 26 2024 2024 NUM cana-611 152 27 ) ) PUNCT cana-611 152 28 424 424 NUM cana-611 152 29 https://internationalpubls.com https://internationalpubls.com X cana-611 152 30 𝑅𝑆5(𝐷𝑛𝑃𝑛 𝑅𝑆5(𝐷𝑛𝑃𝑛 X cana-611 152 31 ) ) PUNCT cana-611 152 32 = = PUNCT cana-611 153 1 399 399 NUM cana-611 153 2 50 50 NUM cana-611 153 3 𝑛 𝑛 DET cana-611 153 4 βˆ’ βˆ’ PROPN cana-611 153 5 79 79 NUM cana-611 153 6 200 200 NUM cana-611 153 7 ( ( PUNCT cana-611 153 8 38 38 NUM cana-611 153 9 ) ) PUNCT cana-611 153 10 proof proof NOUN cana-611 153 11 . . PUNCT cana-611 154 1 let let VERB cana-611 154 2 𝐷𝑛𝑃𝑛be 𝐷𝑛𝑃𝑛be NOUN cana-611 154 3 the the DET cana-611 154 4 prophyrin prophyrin NOUN cana-611 154 5 dendrimer dendrimer NOUN cana-611 154 6 of of ADP cana-611 154 7 order order NOUN cana-611 154 8 96π‘›βˆ’10 96π‘›βˆ’10 NUM cana-611 154 9 and and CCONJ cana-611 154 10 size size NOUN cana-611 154 11 105π‘›βˆ’11 105π‘›βˆ’11 NUM cana-611 154 12 respectively respectively ADV cana-611 154 13 . . PUNCT cana-611 155 1 the the DET cana-611 155 2 edge edge NOUN cana-611 155 3 set set NOUN cana-611 155 4 of of ADP cana-611 155 5 𝐷𝑛𝑃𝑛 𝐷𝑛𝑃𝑛 PROPN cana-611 155 6 can can AUX cana-611 155 7 be be AUX cana-611 155 8 partitioned partition VERB cana-611 155 9 into into ADP cana-611 155 10 six six NUM cana-611 155 11 disjoint disjoint NOUN cana-611 155 12 sets set NOUN cana-611 155 13 πœ€1,3,πœ€1,4,πœ€2,2,πœ€2,3,πœ€3,3 πœ€1,3,πœ€1,4,πœ€2,2,πœ€2,3,πœ€3,3 NOUN cana-611 155 14 and and CCONJ cana-611 155 15 πœ€3,4 πœ€3,4 PROPN cana-611 155 16 , , PUNCT cana-611 155 17 where where SCONJ cana-611 155 18 πœ€(𝐷𝑛𝑃𝑛 πœ€(𝐷𝑛𝑃𝑛 NOUN cana-611 155 19 ) ) PUNCT cana-611 155 20 = = PUNCT cana-611 156 1 πœ€1,3 πœ€1,3 PROPN cana-611 156 2 βˆͺπœ€1,4 βˆͺπœ€1,4 ADV cana-611 156 3 βˆͺπœ€2,2 βˆͺπœ€2,2 PRON cana-611 157 1 βˆͺπœ€2,3 βˆͺπœ€2,3 VERB cana-611 157 2 βˆͺπœ€3,3βˆͺπœ€3,4 βˆͺπœ€3,3βˆͺπœ€3,4 ADJ cana-611 157 3 . . PUNCT cana-611 158 1 further far ADV cana-611 158 2 , , PUNCT cana-611 158 3 βˆ£πœ€1,3∣= βˆ£πœ€1,3∣= PROPN cana-611 158 4 2𝑛 2𝑛 NUM cana-611 158 5 , , PUNCT cana-611 158 6 βˆ£πœ€1,4∣ βˆ£πœ€1,4∣ NOUN cana-611 158 7 = = SYM cana-611 158 8 24𝑛 24𝑛 NOUN cana-611 158 9 , , PUNCT cana-611 158 10 βˆ£πœ€2,2∣= βˆ£πœ€2,2∣= VERB cana-611 158 11 10π‘›βˆ’ 10π‘›βˆ’ NUM cana-611 158 12 5 5 NUM cana-611 158 13 , , PUNCT cana-611 158 14 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 158 15 = = SYM cana-611 158 16 48π‘›βˆ’ 48π‘›βˆ’ NUM cana-611 158 17 6 6 NUM cana-611 158 18 , , PUNCT cana-611 158 19 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 158 20 = = SYM cana-611 158 21 13𝑛 13𝑛 NOUN cana-611 158 22 and and CCONJ cana-611 158 23 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 158 24 = = SYM cana-611 158 25 8𝑛. 8𝑛. NUM cana-611 158 26 such such ADJ cana-611 158 27 that that DET cana-611 158 28 ∣(𝐷𝑛𝑃𝑛)∣ ∣(𝐷𝑛𝑃𝑛)∣ NOUN cana-611 159 1 = = NOUN cana-611 159 2 βˆ£πœ€1,3∣ βˆ£πœ€1,3∣ NOUN cana-611 159 3 + + CCONJ cana-611 159 4 βˆ£πœ€1,4∣ βˆ£πœ€1,4∣ NOUN cana-611 159 5 + + NOUN cana-611 159 6 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 159 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ NOUN cana-611 159 8 + + CCONJ cana-611 159 9 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 159 10 + + CCONJ cana-611 159 11 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 159 12 = = SYM cana-611 159 13 105π‘›βˆ’ 105π‘›βˆ’ NUM cana-611 159 14 11 11 NUM cana-611 159 15 . . PUNCT cana-611 160 1 thus thus ADV cana-611 160 2 , , PUNCT cana-611 160 3 with with ADP cana-611 160 4 this this DET cana-611 160 5 background background NOUN cana-611 160 6 by by ADP cana-611 160 7 employing employ VERB cana-611 160 8 equations equation NOUN cana-611 160 9 ( ( PUNCT cana-611 160 10 1)-(5 1)-(5 NUM cana-611 160 11 ) ) PUNCT cana-611 160 12 and and CCONJ cana-611 160 13 ( ( PUNCT cana-611 160 14 6)-(10 6)-(10 NUM cana-611 160 15 ) ) PUNCT cana-611 160 16 , , PUNCT cana-611 160 17 we we PRON cana-611 160 18 get get VERB cana-611 160 19 the the DET cana-611 160 20 required require VERB cana-611 160 21 results result NOUN cana-611 160 22 . . PUNCT cana-611 161 1 theorem theorem ADJ cana-611 161 2 8 8 NUM cana-611 161 3 . . PUNCT cana-611 162 1 let let VERB cana-611 162 2 𝐷𝑃𝑍𝑛 𝐷𝑃𝑍𝑛 NOUN cana-611 162 3 be be AUX cana-611 162 4 the the DET cana-611 162 5 zinc zinc NOUN cana-611 162 6 - - PUNCT cana-611 162 7 porphyrin porphyrin NOUN cana-611 162 8 dendrimer dendrimer NOUN cana-611 162 9 . . PUNCT cana-611 163 1 then then ADV cana-611 163 2 𝑅𝑆2(𝐷𝑃𝑍𝑛 𝑅𝑆2(𝐷𝑃𝑍𝑛 X cana-611 163 3 ) ) PUNCT cana-611 163 4 = = SYM cana-611 163 5 343 343 NUM cana-611 163 6 100 100 NUM cana-611 163 7 2𝑛 2𝑛 NUM cana-611 163 8 βˆ’ βˆ’ PROPN cana-611 163 9 13 13 NUM cana-611 163 10 10 10 NUM cana-611 163 11 ( ( PUNCT cana-611 163 12 39 39 NUM cana-611 163 13 ) ) PUNCT cana-611 163 14 𝑅𝑆3(𝐷𝑃𝑍𝑛 𝑅𝑆3(𝐷𝑃𝑍𝑛 NUM cana-611 163 15 ) ) PUNCT cana-611 164 1 = = SYM cana-611 164 2 9 9 NUM cana-611 164 3 2 2 NUM cana-611 164 4 2𝑛 2𝑛 NOUN cana-611 164 5 βˆ’ βˆ’ PROPN cana-611 164 6 171 171 NUM cana-611 164 7 100 100 NUM cana-611 164 8 ( ( PUNCT cana-611 164 9 40 40 NUM cana-611 164 10 ) ) PUNCT cana-611 164 11 𝑅𝑆4(𝐷𝑃𝑍𝑛 𝑅𝑆4(𝐷𝑃𝑍𝑛 NUM cana-611 164 12 ) ) PUNCT cana-611 165 1 = = SYM cana-611 165 2 69 69 NUM cana-611 165 3 50 50 NUM cana-611 165 4 2𝑛 2𝑛 NOUN cana-611 165 5 βˆ’ βˆ’ PROPN cana-611 165 6 21 21 NUM cana-611 165 7 50 50 NUM cana-611 165 8 ( ( PUNCT cana-611 165 9 41 41 NUM cana-611 165 10 ) ) PUNCT cana-611 165 11 𝑅𝑆5(𝐷𝑃𝑍𝑛 𝑅𝑆5(𝐷𝑃𝑍𝑛 NUM cana-611 165 12 ) ) PUNCT cana-611 165 13 = = SYM cana-611 165 14 8 8 NUM cana-611 165 15 5 5 NUM cana-611 165 16 2𝑛 2𝑛 NOUN cana-611 165 17 βˆ’ βˆ’ PROPN cana-611 165 18 1 1 NUM cana-611 165 19 2 2 NUM cana-611 165 20 ( ( PUNCT cana-611 165 21 42 42 NUM cana-611 165 22 ) ) PUNCT cana-611 165 23 proof proof NOUN cana-611 165 24 . . PUNCT cana-611 166 1 let let VERB cana-611 166 2 𝐷𝑃𝑍𝑛 𝐷𝑃𝑍𝑛 NOUN cana-611 166 3 be be AUX cana-611 166 4 the the DET cana-611 166 5 zinc zinc NOUN cana-611 166 6 - - PUNCT cana-611 166 7 porphyrin porphyrin NOUN cana-611 166 8 dendrimer dendrimer NOUN cana-611 166 9 . . PUNCT cana-611 167 1 of of ADP cana-611 167 2 order order NOUN cana-611 167 3 96π‘›βˆ’10 96π‘›βˆ’10 NUM cana-611 167 4 and and CCONJ cana-611 167 5 size size NOUN cana-611 167 6 105π‘›βˆ’11 105π‘›βˆ’11 NUM cana-611 167 7 respectively respectively ADV cana-611 167 8 . . PUNCT cana-611 168 1 the the DET cana-611 168 2 edge edge NOUN cana-611 168 3 set set NOUN cana-611 168 4 of of ADP cana-611 168 5 𝐷𝑃𝑍𝑛can 𝐷𝑃𝑍𝑛can PROPN cana-611 168 6 be be AUX cana-611 168 7 partitioned partition VERB cana-611 168 8 into into ADP cana-611 168 9 four four NUM cana-611 168 10 disjoint disjoint NOUN cana-611 168 11 sets set NOUN cana-611 168 12 πœ€2,2,πœ€2,3,πœ€3,3 πœ€2,2,πœ€2,3,πœ€3,3 ADJ cana-611 168 13 and and CCONJ cana-611 168 14 πœ€3,4 πœ€3,4 ADJ cana-611 168 15 , , PUNCT cana-611 168 16 where where SCONJ cana-611 168 17 πœ€(𝐷𝑃𝑍𝑛 πœ€(𝐷𝑃𝑍𝑛 VERB cana-611 169 1 ) ) PUNCT cana-611 169 2 = = SYM cana-611 169 3 πœ€2,2 πœ€2,2 PROPN cana-611 169 4 βˆͺπœ€2,3 βˆͺπœ€2,3 NOUN cana-611 169 5 βˆͺπœ€3,3 βˆͺπœ€3,3 ADP cana-611 169 6 βˆͺπœ€3,4 βˆͺπœ€3,4 NOUN cana-611 169 7 . . PUNCT cana-611 169 8 further far ADV cana-611 169 9 , , PUNCT cana-611 169 10 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 169 11 = = SYM cana-611 169 12 16 16 NUM cana-611 169 13 β‹… β‹… PROPN cana-611 169 14 2π‘›βˆ’ 2π‘›βˆ’ NUM cana-611 169 15 4 4 NUM cana-611 169 16 , , PUNCT cana-611 169 17 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 169 18 = = SYM cana-611 169 19 40 40 NUM cana-611 169 20 β‹… β‹… PROPN cana-611 169 21 2π‘›βˆ’ 2π‘›βˆ’ NUM cana-611 169 22 16 16 NUM cana-611 169 23 , , PUNCT cana-611 169 24 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 169 25 = = SYM cana-611 169 26 8 8 NUM cana-611 169 27 β‹… β‹… PROPN cana-611 169 28 2π‘›βˆ’ 2π‘›βˆ’ PROPN cana-611 169 29 16i 16i NOUN cana-611 169 30 and and CCONJ cana-611 169 31 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 169 32 = = SYM cana-611 169 33 4 4 X cana-611 169 34 . . X cana-611 169 35 such such ADJ cana-611 169 36 that that SCONJ cana-611 169 37 ∣(𝐷𝑃𝑍𝑛)∣ ∣(𝐷𝑃𝑍𝑛)∣ PROPN cana-611 169 38 = = SYM cana-611 169 39 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 169 40 + + NUM cana-611 169 41 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ NOUN cana-611 169 42 + + NUM cana-611 169 43 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 169 44 + + CCONJ cana-611 169 45 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 169 46 = = SYM cana-611 169 47 105π‘›βˆ’ 105π‘›βˆ’ NUM cana-611 169 48 11.thus 11.thus NUM cana-611 169 49 , , PUNCT cana-611 169 50 with with ADP cana-611 169 51 this this DET cana-611 169 52 background background NOUN cana-611 169 53 by by ADP cana-611 169 54 employing employ VERB cana-611 169 55 equations equation NOUN cana-611 169 56 ( ( PUNCT cana-611 169 57 1)-(5 1)-(5 NUM cana-611 169 58 ) ) PUNCT cana-611 169 59 and and CCONJ cana-611 169 60 ( ( PUNCT cana-611 169 61 6)-(10),we 6)-(10),we PRON cana-611 169 62 get get VERB cana-611 169 63 the the DET cana-611 169 64 required require VERB cana-611 169 65 results result NOUN cana-611 169 66 . . PUNCT cana-611 170 1 theorem theorem VERB cana-611 170 2 9 9 NUM cana-611 170 3 . . PUNCT cana-611 171 1 for for ADP cana-611 171 2 the the DET cana-611 171 3 pamam pamam NOUN cana-611 171 4 dendrimers dendrimers PROPN cana-611 171 5 𝑃𝐷1 𝑃𝐷1 PROPN cana-611 171 6 , , PUNCT cana-611 171 7 we we PRON cana-611 171 8 have have VERB cana-611 171 9 𝑅𝑆2(𝑃𝐷1 𝑅𝑆2(𝑃𝐷1 VERB cana-611 171 10 ) ) PUNCT cana-611 171 11 = = SYM cana-611 172 1 37 37 NUM cana-611 172 2 10 10 NUM cana-611 172 3 2𝑛 2𝑛 NOUN cana-611 172 4 βˆ’ βˆ’ PROPN cana-611 172 5 81 81 NUM cana-611 172 6 50 50 NUM cana-611 172 7 ( ( PUNCT cana-611 172 8 43 43 NUM cana-611 172 9 ) ) PUNCT cana-611 172 10 𝑅𝑆3(𝑃𝐷1 𝑅𝑆3(𝑃𝐷1 NUM cana-611 172 11 ) ) PUNCT cana-611 172 12 = = SYM cana-611 172 13 599 599 NUM cana-611 172 14 100 100 NUM cana-611 172 15 2𝑛 2𝑛 NUM cana-611 172 16 βˆ’ βˆ’ PROPN cana-611 172 17 13 13 NUM cana-611 172 18 5 5 NUM cana-611 172 19 ( ( PUNCT cana-611 172 20 44 44 NUM cana-611 172 21 ) ) PUNCT cana-611 172 22 𝑅𝑆4(𝑃𝐷1 𝑅𝑆4(𝑃𝐷1 PROPN cana-611 172 23 ) ) PUNCT cana-611 172 24 = = PUNCT cana-611 173 1 23 23 NUM cana-611 173 2 10 10 NUM cana-611 173 3 2𝑛 2𝑛 NOUN cana-611 173 4 βˆ’ βˆ’ PROPN cana-611 173 5 39 39 NUM cana-611 173 6 50 50 NUM cana-611 173 7 ( ( PUNCT cana-611 173 8 45 45 NUM cana-611 173 9 ) ) PUNCT cana-611 173 10 𝑅𝑆5(𝑃𝐷1 𝑅𝑆5(𝑃𝐷1 NUM cana-611 173 11 ) ) PUNCT cana-611 173 12 = = SYM cana-611 174 1 59 59 NUM cana-611 174 2 12 12 NUM cana-611 174 3 2𝑛 2𝑛 NOUN cana-611 174 4 βˆ’ βˆ’ PROPN cana-611 174 5 173 173 NUM cana-611 174 6 100 100 NUM cana-611 174 7 ( ( PUNCT cana-611 174 8 46 46 NUM cana-611 174 9 ) ) PUNCT cana-611 174 10 proof proof NOUN cana-611 174 11 . . PUNCT cana-611 175 1 let let VERB cana-611 175 2 𝑃𝐷1 𝑃𝐷1 PROPN cana-611 175 3 denote denote NOUN cana-611 175 4 pamam pamam NOUN cana-611 175 5 dendrimers dendrimer NOUN cana-611 175 6 with with ADP cana-611 175 7 trifunctional trifunctional ADJ cana-611 175 8 core core NOUN cana-611 175 9 unit unit NOUN cana-611 175 10 generated generate VERB cana-611 175 11 by by ADP cana-611 175 12 𝐺𝑛 𝐺𝑛 PROPN cana-611 175 13 with with ADP cana-611 175 14 𝑛 𝑛 DET cana-611 175 15 growth growth NOUN cana-611 175 16 stages stage NOUN cana-611 175 17 . . PUNCT cana-611 176 1 the the DET cana-611 176 2 edge edge NOUN cana-611 176 3 set set NOUN cana-611 176 4 of of ADP cana-611 176 5 𝑃𝐷1 𝑃𝐷1 PROPN cana-611 176 6 can can AUX cana-611 176 7 be be AUX cana-611 176 8 partitioned partition VERB cana-611 176 9 into into ADP cana-611 176 10 four four NUM cana-611 176 11 disjoint disjoint NOUN cana-611 176 12 sets set NOUN cana-611 176 13 πœ€1,2,πœ€1,3,πœ€2,2 πœ€1,2,πœ€1,3,πœ€2,2 NOUN cana-611 176 14 and and CCONJ cana-611 176 15 πœ€2,3 πœ€2,3 NOUN cana-611 176 16 , , PUNCT cana-611 176 17 where where SCONJ cana-611 176 18 πœ€(𝑃𝐷1 πœ€(𝑃𝐷1 ADJ cana-611 176 19 ) ) PUNCT cana-611 177 1 = = SYM cana-611 177 2 πœ€1,2βˆͺπœ€1,3 πœ€1,2βˆͺπœ€1,3 X cana-611 177 3 βˆͺπœ€2,2 βˆͺπœ€2,2 PUNCT cana-611 178 1 βˆͺπœ€2,3 βˆͺπœ€2,3 NOUN cana-611 178 2 . . PUNCT cana-611 179 1 further far ADV cana-611 179 2 , , PUNCT cana-611 179 3 βˆ£πœ€1,2∣ βˆ£πœ€1,2∣ NOUN cana-611 179 4 = = SYM cana-611 179 5 3 3 NUM cana-611 179 6 β‹… β‹… PROPN cana-611 179 7 2𝑛,βˆ£πœ€,3∣ 2𝑛,βˆ£πœ€,3∣ NUM cana-611 179 8 = = SYM cana-611 179 9 6 6 NUM cana-611 179 10 β‹… β‹… PROPN cana-611 179 11 2π‘›βˆ’ 2π‘›βˆ’ NUM cana-611 179 12 3 3 NUM cana-611 179 13 , , PUNCT cana-611 179 14 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 179 15 = = SYM cana-611 179 16 18 18 NUM cana-611 179 17 β‹… β‹… PROPN cana-611 179 18 2π‘›βˆ’ 2π‘›βˆ’ NUM cana-611 179 19 9 9 NUM cana-611 179 20 and and CCONJ cana-611 179 21 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 179 22 = = SYM cana-611 179 23 21 21 NUM cana-611 179 24 β‹… β‹… PROPN cana-611 179 25 2π‘›βˆ’ 2π‘›βˆ’ PROPN cana-611 179 26 12 12 NUM cana-611 179 27 . . PUNCT cana-611 180 1 such such ADJ cana-611 180 2 that that DET cana-611 180 3 ∣(𝑃𝐷1)∣ ∣(𝑃𝐷1)∣ ADJ cana-611 180 4 = = SYM cana-611 180 5 βˆ£πœ€1,2∣ βˆ£πœ€1,2∣ NOUN cana-611 180 6 + + NUM cana-611 180 7 βˆ£πœ€1,3∣ βˆ£πœ€1,3∣ NOUN cana-611 180 8 + + CCONJ cana-611 180 9 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 180 10 + + CCONJ cana-611 180 11 βˆ£πœ€2,3 βˆ£πœ€2,3 ADJ cana-611 180 12 . . PUNCT cana-611 181 1 thus thus ADV cana-611 181 2 , , PUNCT cana-611 181 3 with with ADP cana-611 181 4 this this DET cana-611 181 5 background background NOUN cana-611 181 6 by by ADP cana-611 181 7 employing employ VERB cana-611 181 8 equations equation NOUN cana-611 181 9 ( ( PUNCT cana-611 181 10 1)-(5 1)-(5 NUM cana-611 181 11 ) ) PUNCT cana-611 181 12 and and CCONJ cana-611 181 13 ( ( PUNCT cana-611 181 14 6)-(10 6)-(10 NUM cana-611 181 15 ) ) PUNCT cana-611 181 16 , , PUNCT cana-611 181 17 we we PRON cana-611 181 18 get get VERB cana-611 181 19 the the DET cana-611 181 20 required require VERB cana-611 181 21 results result NOUN cana-611 181 22 . . PUNCT cana-611 182 1 theorem theorem ADJ cana-611 182 2 10 10 NUM cana-611 182 3 . . PUNCT cana-611 183 1 for for ADP cana-611 183 2 their their PRON cana-611 183 3 pamam pamam NOUN cana-611 183 4 dendrimers dendrimers PROPN cana-611 183 5 𝑃𝐷2 𝑃𝐷2 PROPN cana-611 183 6 , , PUNCT cana-611 183 7 we we PRON cana-611 183 8 have have VERB cana-611 183 9 𝑅𝑆2(𝑃𝐷2 𝑅𝑆2(𝑃𝐷2 NUM cana-611 183 10 ) ) PUNCT cana-611 183 11 = = SYM cana-611 184 1 499 499 NUM cana-611 184 2 100 100 NUM cana-611 184 3 2𝑛 2𝑛 NUM cana-611 184 4 βˆ’ βˆ’ PROPN cana-611 184 5 49 49 NUM cana-611 184 6 25 25 NUM cana-611 184 7 ( ( PUNCT cana-611 184 8 47 47 NUM cana-611 184 9 ) ) PUNCT cana-611 184 10 𝑅𝑆3(𝑃𝐷2 𝑅𝑆3(𝑃𝐷2 NUM cana-611 184 11 ) ) PUNCT cana-611 185 1 = = PUNCT cana-611 185 2 399 399 NUM cana-611 185 3 50 50 NUM cana-611 185 4 2𝑛 2𝑛 NOUN cana-611 185 5 βˆ’ βˆ’ PROPN cana-611 185 6 16 16 NUM cana-611 185 7 5 5 NUM cana-611 185 8 ( ( PUNCT cana-611 185 9 48 48 NUM cana-611 185 10 ) ) PUNCT cana-611 185 11 𝑅𝑆4(𝑃𝐷2 𝑅𝑆4(𝑃𝐷2 PROPN cana-611 185 12 ) ) PUNCT cana-611 185 13 = = SYM cana-611 186 1 61 61 NUM cana-611 186 2 20 20 NUM cana-611 186 3 2𝑛 2𝑛 NOUN cana-611 186 4 βˆ’ βˆ’ PROPN cana-611 186 5 24 24 NUM cana-611 186 6 25 25 NUM cana-611 186 7 ( ( PUNCT cana-611 186 8 49 49 NUM cana-611 186 9 ) ) PUNCT cana-611 186 10 𝑅𝑆5(𝑃𝐷2 𝑅𝑆5(𝑃𝐷2 ADJ cana-611 186 11 ) ) PUNCT cana-611 187 1 = = SYM cana-611 187 2 59 59 NUM cana-611 187 3 9 9 NUM cana-611 187 4 2𝑛 2𝑛 NOUN cana-611 187 5 βˆ’ βˆ’ PROPN cana-611 187 6 11 11 NUM cana-611 187 7 5 5 NUM cana-611 187 8 ( ( PUNCT cana-611 187 9 50 50 NUM cana-611 187 10 ) ) PUNCT cana-611 187 11 communications communication NOUN cana-611 187 12 on on ADP cana-611 187 13 applied apply VERB cana-611 187 14 nonlinear nonlinear ADJ cana-611 187 15 analysis analysis NOUN cana-611 187 16 issn issn NOUN cana-611 187 17 : : PUNCT cana-611 187 18 1074 1074 NUM cana-611 187 19 - - PUNCT cana-611 187 20 133x 133x NUM cana-611 187 21 vol vol NOUN cana-611 187 22 31 31 NUM cana-611 187 23 no no NOUN cana-611 187 24 . . NOUN cana-611 187 25 2 2 NUM cana-611 187 26 ( ( PUNCT cana-611 187 27 2024 2024 NUM cana-611 187 28 ) ) PUNCT cana-611 187 29 425 425 NUM cana-611 187 30 https://internationalpubls.com https://internationalpubls.com X cana-611 187 31 proof proof NOUN cana-611 187 32 . . PUNCT cana-611 188 1 let let VERB cana-611 188 2 𝑃𝐷2i 𝑃𝐷2i PRON cana-611 188 3 denote denote VERB cana-611 188 4 pamam pamam NOUN cana-611 188 5 dendrimers dendrimer NOUN cana-611 188 6 with with ADP cana-611 188 7 different different ADJ cana-611 188 8 core core NOUN cana-611 188 9 unit unit NOUN cana-611 188 10 generated generate VERB cana-611 188 11 by by ADP cana-611 188 12 dendrimer dendrimer NOUN cana-611 188 13 𝐺𝑛 𝐺𝑛 PROPN cana-611 188 14 with with ADP cana-611 188 15 𝑛 𝑛 DET cana-611 188 16 growth growth NOUN cana-611 188 17 stages stage NOUN cana-611 188 18 . . PUNCT cana-611 189 1 the the DET cana-611 189 2 edge edge NOUN cana-611 189 3 set set NOUN cana-611 189 4 of of ADP cana-611 189 5 𝑃𝐷2i 𝑃𝐷2i PUNCT cana-611 189 6 can can AUX cana-611 189 7 be be AUX cana-611 189 8 partitioned partition VERB cana-611 189 9 into into ADP cana-611 189 10 four four NUM cana-611 189 11 disjoint disjoint NOUN cana-611 189 12 sets set NOUN cana-611 189 13 πœ€1,2,πœ€1,3,πœ€2,2 πœ€1,2,πœ€1,3,πœ€2,2 NOUN cana-611 189 14 and and CCONJ cana-611 189 15 πœ€2,3 πœ€2,3 NOUN cana-611 189 16 , , PUNCT cana-611 189 17 where where SCONJ cana-611 189 18 πœ€(𝑃𝐷2 πœ€(𝑃𝐷2 NOUN cana-611 189 19 ) ) PUNCT cana-611 190 1 = = PUNCT cana-611 190 2 πœ€1,2i πœ€1,2i PROPN cana-611 190 3 βˆͺπœ€1,3 βˆͺπœ€1,3 PROPN cana-611 190 4 βˆͺπœ€2,2 βˆͺπœ€2,2 PRON cana-611 190 5 βˆͺπœ€2,3 βˆͺπœ€2,3 NOUN cana-611 190 6 . . PUNCT cana-611 191 1 further far ADV cana-611 191 2 , , PUNCT cana-611 191 3 βˆ£πœ€1,2∣ βˆ£πœ€1,2∣ NOUN cana-611 191 4 = = SYM cana-611 191 5 4 4 NUM cana-611 191 6 β‹… β‹… PROPN cana-611 191 7 2𝑛 2𝑛 NUM cana-611 191 8 , , PUNCT cana-611 191 9 βˆ£πœ€1,3∣ βˆ£πœ€1,3∣ NOUN cana-611 191 10 = = SYM cana-611 191 11 i8 i8 NOUN cana-611 191 12 β‹… β‹… NOUN cana-611 192 1 2π‘›βˆ’ 2π‘›βˆ’ NUM cana-611 192 2 4 4 NUM cana-611 192 3 , , PUNCT cana-611 192 4 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 192 5 = = SYM cana-611 192 6 24 24 NUM cana-611 192 7 β‹… β‹… PROPN cana-611 192 8 2π‘›βˆ’ 2π‘›βˆ’ NUM cana-611 192 9 11 11 NUM cana-611 192 10 and and CCONJ cana-611 192 11 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 192 12 = = SYM cana-611 192 13 28 28 NUM cana-611 192 14 β‹… β‹… PROPN cana-611 192 15 2π‘›βˆ’ 2π‘›βˆ’ PROPN cana-611 192 16 14 14 NUM cana-611 192 17 . . PUNCT cana-611 193 1 such such ADJ cana-611 193 2 that that DET cana-611 193 3 ∣(𝑃𝐷2)∣ ∣(𝑃𝐷2)∣ NOUN cana-611 193 4 = = SYM cana-611 193 5 βˆ£πœ€1,2∣ βˆ£πœ€1,2∣ NOUN cana-611 193 6 + + NUM cana-611 193 7 βˆ£πœ€1,3∣ βˆ£πœ€1,3∣ NOUN cana-611 193 8 + + CCONJ cana-611 193 9 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 193 10 + + CCONJ cana-611 193 11 βˆ£πœ€2,3 βˆ£πœ€2,3 ADJ cana-611 193 12 . . PUNCT cana-611 194 1 thus thus ADV cana-611 194 2 , , PUNCT cana-611 194 3 with with ADP cana-611 194 4 this this DET cana-611 194 5 background background NOUN cana-611 194 6 by by ADP cana-611 194 7 employing employ VERB cana-611 194 8 equations equation NOUN cana-611 194 9 ( ( PUNCT cana-611 194 10 1)-(5 1)-(5 NUM cana-611 194 11 ) ) PUNCT cana-611 194 12 and and CCONJ cana-611 194 13 ( ( PUNCT cana-611 194 14 6)-(10 6)-(10 NUM cana-611 194 15 ) ) PUNCT cana-611 194 16 , , PUNCT cana-611 194 17 we we PRON cana-611 194 18 get get VERB cana-611 194 19 the the DET cana-611 194 20 required require VERB cana-611 194 21 results result NOUN cana-611 194 22 . . PUNCT cana-611 195 1 theorem theorem ADJ cana-611 195 2 11 11 NUM cana-611 195 3 . . PUNCT cana-611 196 1 for for ADP cana-611 196 2 their their PRON cana-611 196 3 pamam pamam NOUN cana-611 196 4 dendrimers dendrimer NOUN cana-611 196 5 𝐷𝑆1 𝐷𝑆1 PROPN cana-611 196 6 , , PUNCT cana-611 196 7 we we PRON cana-611 196 8 have have VERB cana-611 196 9 𝑅𝑆2(𝐷𝑆1 𝑅𝑆2(𝐷𝑆1 NUM cana-611 196 10 ) ) PUNCT cana-611 196 11 = = SYM cana-611 196 12 32 32 NUM cana-611 196 13 25 25 NUM cana-611 196 14 3𝑛 3𝑛 NUM cana-611 196 15 βˆ’ βˆ’ NOUN cana-611 196 16 253 253 NUM cana-611 196 17 200 200 NUM cana-611 196 18 ( ( PUNCT cana-611 196 19 51 51 NUM cana-611 196 20 ) ) PUNCT cana-611 196 21 𝑅𝑆3(𝐷𝑆1 𝑅𝑆3(𝐷𝑆1 NUM cana-611 196 22 ) ) PUNCT cana-611 196 23 = = SYM cana-611 197 1 87 87 NUM cana-611 197 2 40 40 NUM cana-611 197 3 3𝑛 3𝑛 NUM cana-611 197 4 βˆ’ βˆ’ PROPN cana-611 197 5 11 11 NUM cana-611 197 6 8 8 NUM cana-611 197 7 ( ( PUNCT cana-611 197 8 52 52 NUM cana-611 197 9 ) ) PUNCT cana-611 197 10 𝑅𝑆3(𝐷𝑆1 𝑅𝑆3(𝐷𝑆1 NUM cana-611 197 11 ) ) PUNCT cana-611 197 12 = = PUNCT cana-611 198 1 165 165 NUM cana-611 198 2 256 256 NUM cana-611 198 3 3𝑛 3𝑛 NUM cana-611 198 4 βˆ’ βˆ’ NOUN cana-611 198 5 161 161 NUM cana-611 198 6 256 256 NUM cana-611 198 7 ( ( PUNCT cana-611 198 8 53 53 NUM cana-611 198 9 ) ) PUNCT cana-611 198 10 𝑅𝑆2(𝐷𝑆1 𝑅𝑆2(𝐷𝑆1 NUM cana-611 198 11 ) ) PUNCT cana-611 198 12 = = SYM cana-611 198 13 105 105 NUM cana-611 198 14 64 64 NUM cana-611 198 15 3𝑛 3𝑛 NUM cana-611 198 16 βˆ’ βˆ’ PROPN cana-611 198 17 16 16 NUM cana-611 198 18 25 25 NUM cana-611 198 19 ( ( PUNCT cana-611 198 20 54 54 NUM cana-611 198 21 ) ) PUNCT cana-611 198 22 proof proof NOUN cana-611 198 23 . . PUNCT cana-611 199 1 let let VERB cana-611 199 2 𝐷𝑆1i 𝐷𝑆1i PRON cana-611 199 3 denote denote VERB cana-611 199 4 pamam pamam NOUN cana-611 199 5 dendrimers dendrimer NOUN cana-611 199 6 with with ADP cana-611 199 7 different different ADJ cana-611 199 8 core core NOUN cana-611 199 9 unit unit NOUN cana-611 199 10 generated generate VERB cana-611 199 11 by by ADP cana-611 199 12 dendrimer dendrimer NOUN cana-611 199 13 𝐺𝑛 𝐺𝑛 PROPN cana-611 199 14 with with ADP cana-611 199 15 𝑛 𝑛 DET cana-611 199 16 growth growth NOUN cana-611 199 17 stages stage NOUN cana-611 199 18 . . PUNCT cana-611 200 1 the the DET cana-611 200 2 edge edge NOUN cana-611 200 3 set set NOUN cana-611 200 4 of of ADP cana-611 200 5 𝐷𝑆1 𝐷𝑆1 PROPN cana-611 200 6 can can AUX cana-611 200 7 be be AUX cana-611 200 8 partitioned partition VERB cana-611 200 9 into into ADP cana-611 200 10 three three NUM cana-611 200 11 disjoint disjoint NOUN cana-611 200 12 sets set NOUN cana-611 200 13 πœ€1,4,πœ€2,2 πœ€1,4,πœ€2,2 NUM cana-611 200 14 and and CCONJ cana-611 200 15 πœ€2,4 πœ€2,4 PROPN cana-611 200 16 , , PUNCT cana-611 200 17 where where SCONJ cana-611 200 18 πœ€(𝐷𝑆1 πœ€(𝐷𝑆1 VERB cana-611 200 19 ) ) PUNCT cana-611 200 20 = = PUNCT cana-611 201 1 πœ€1,4i πœ€1,4i PROPN cana-611 201 2 βˆͺπœ€2,2 βˆͺπœ€2,2 PROPN cana-611 201 3 βˆͺπœ€2,4 βˆͺπœ€2,4 PROPN cana-611 201 4 . . PUNCT cana-611 201 5 further far ADV cana-611 201 6 , , PUNCT cana-611 201 7 βˆ£πœ€1,4∣ βˆ£πœ€1,4∣ NOUN cana-611 201 8 = = SYM cana-611 201 9 4 4 NUM cana-611 201 10 β‹… β‹… NUM cana-611 201 11 3𝑛 3𝑛 NUM cana-611 201 12 , , PUNCT cana-611 201 13 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 201 14 = = SYM cana-611 201 15 10β‹…3π‘›βˆ’10 10β‹…3π‘›βˆ’10 NUM cana-611 201 16 , , PUNCT cana-611 201 17 and and CCONJ cana-611 201 18 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 201 19 = = SYM cana-611 201 20 4β‹…3π‘›βˆ’4 4β‹…3π‘›βˆ’4 PROPN cana-611 201 21 . . PUNCT cana-611 202 1 such such ADJ cana-611 202 2 that that PRON cana-611 202 3 ∣(𝐷𝑆1)∣ ∣(𝐷𝑆1)∣ PUNCT cana-611 202 4 = = NOUN cana-611 202 5 βˆ£πœ€1,4∣+βˆ£πœ€2,2∣+βˆ£πœ€2,4∣. βˆ£πœ€1,4∣+βˆ£πœ€2,2∣+βˆ£πœ€2,4∣. PROPN cana-611 202 6 thus thus ADV cana-611 202 7 , , PUNCT cana-611 202 8 with with ADP cana-611 202 9 this this DET cana-611 202 10 background background NOUN cana-611 202 11 by by ADP cana-611 202 12 employing employ VERB cana-611 202 13 equations equation NOUN cana-611 202 14 ( ( PUNCT cana-611 202 15 1)-(5 1)-(5 NUM cana-611 202 16 ) ) PUNCT cana-611 202 17 and and CCONJ cana-611 202 18 ( ( PUNCT cana-611 202 19 6)-(10 6)-(10 NUM cana-611 202 20 ) ) PUNCT cana-611 202 21 , , PUNCT cana-611 202 22 we we PRON cana-611 202 23 get get VERB cana-611 202 24 the the DET cana-611 202 25 required require VERB cana-611 202 26 results result NOUN cana-611 202 27 . . PUNCT cana-611 203 1 theorem theorem NOUN cana-611 203 2 12 12 NUM cana-611 203 3 . . PUNCT cana-611 204 1 for for ADP cana-611 204 2 a a DET cana-611 204 3 linear linear ADJ cana-611 204 4 polyomino polyomino NOUN cana-611 204 5 chain chain NOUN cana-611 204 6 𝐿𝑛we 𝐿𝑛we PROPN cana-611 204 7 have have VERB cana-611 204 8 𝑅𝑆2(𝐿𝑛 𝑅𝑆2(𝐿𝑛 NUM cana-611 204 9 ) ) PUNCT cana-611 205 1 = = SYM cana-611 205 2 1 1 NUM cana-611 205 3 18 18 NUM cana-611 205 4 𝑛 𝑛 NOUN cana-611 205 5 + + NOUN cana-611 205 6 143 143 NUM cana-611 205 7 500 500 NUM cana-611 205 8 ( ( PUNCT cana-611 205 9 55 55 NUM cana-611 205 10 ) ) PUNCT cana-611 205 11 𝑅𝑆3(𝐿𝑛 𝑅𝑆3(𝐿𝑛 PROPN cana-611 205 12 ) ) PUNCT cana-611 205 13 = = SYM cana-611 205 14 1 1 NUM cana-611 205 15 18 18 NUM cana-611 205 16 𝑛 𝑛 DET cana-611 205 17 + + CCONJ cana-611 205 18 393 393 NUM cana-611 205 19 1000 1000 NUM cana-611 205 20 ( ( PUNCT cana-611 205 21 56 56 NUM cana-611 205 22 ) ) PUNCT cana-611 205 23 𝑅𝑆4(𝐿𝑛 𝑅𝑆4(𝐿𝑛 VERB cana-611 205 24 ) ) PUNCT cana-611 205 25 = = SYM cana-611 205 26 1 1 NUM cana-611 205 27 243 243 NUM cana-611 205 28 𝑛 𝑛 DET cana-611 205 29 + + NUM cana-611 205 30 31 31 NUM cana-611 205 31 200 200 NUM cana-611 205 32 ( ( PUNCT cana-611 205 33 57 57 NUM cana-611 205 34 ) ) PUNCT cana-611 205 35 𝑅𝑆5(𝐿𝑛 𝑅𝑆5(𝐿𝑛 PROPN cana-611 205 36 ) ) PUNCT cana-611 205 37 = = PUNCT cana-611 205 38 1 1 NUM cana-611 205 39 243 243 NUM cana-611 205 40 𝑛 𝑛 DET cana-611 205 41 + + NUM cana-611 205 42 873 873 NUM cana-611 205 43 500 500 NUM cana-611 205 44 ( ( PUNCT cana-611 205 45 58 58 NUM cana-611 205 46 ) ) PUNCT cana-611 205 47 proof proof NOUN cana-611 205 48 . . PUNCT cana-611 206 1 let let VERB cana-611 206 2 𝐿𝑛be 𝐿𝑛be PROPN cana-611 206 3 their their PRON cana-611 206 4 polyomino polyomino NOUN cana-611 206 5 chain chain NOUN cana-611 206 6 with with ADP cana-611 206 7 𝑛squares 𝑛square NOUN cana-611 206 8 where where SCONJ cana-611 206 9 𝑙1 𝑙1 PROPN cana-611 206 10 = = PROPN cana-611 207 1 i i PRON cana-611 207 2 𝑛and 𝑛and VERB cana-611 207 3 π‘š= π‘š= NOUN cana-611 207 4 1 1 NUM cana-611 207 5 . . PUNCT cana-611 208 1 the the DET cana-611 208 2 edge edge NOUN cana-611 208 3 set set NOUN cana-611 208 4 of of ADP cana-611 208 5 𝐿𝑛can 𝐿𝑛can PROPN cana-611 208 6 be be AUX cana-611 208 7 partitioned partition VERB cana-611 208 8 into into ADP cana-611 208 9 three three NUM cana-611 208 10 disjoint disjoint NOUN cana-611 208 11 sets set NOUN cana-611 208 12 πœ€2,2,πœ€2,3 πœ€2,2,πœ€2,3 PUNCT cana-611 208 13 and and CCONJ cana-611 208 14 πœ€3,3 πœ€3,3 PROPN cana-611 208 15 , , PUNCT cana-611 208 16 i i PRON cana-611 208 17 where where SCONJ cana-611 208 18 πœ€(𝐿𝑛 πœ€(𝐿𝑛 VERB cana-611 208 19 ) ) PUNCT cana-611 208 20 = = PUNCT cana-611 209 1 πœ€2,2i πœ€2,2i NOUN cana-611 209 2 βˆͺπœ€2,3 βˆͺπœ€2,3 NOUN cana-611 209 3 βˆͺπœ€3,3 βˆͺπœ€3,3 NOUN cana-611 209 4 . . PUNCT cana-611 209 5 further far ADV cana-611 209 6 , , PUNCT cana-611 209 7 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 209 8 = = SYM cana-611 209 9 2,i 2,i NUM cana-611 209 10 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ NOUN cana-611 209 11 = = SYM cana-611 209 12 4 4 NUM cana-611 209 13 , , PUNCT cana-611 209 14 and and CCONJ cana-611 209 15 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 209 16 = = SYM cana-611 209 17 3π‘›βˆ’ 3π‘›βˆ’ NUM cana-611 209 18 5 5 NUM cana-611 209 19 . . PUNCT cana-611 209 20 such such ADJ cana-611 209 21 that that SCONJ cana-611 209 22 ∣(𝐿𝑛)∣ ∣(𝐿𝑛)∣ PROPN cana-611 209 23 = = SYM cana-611 209 24 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 209 25 + + NUM cana-611 209 26 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 209 27 + + CCONJ cana-611 209 28 βˆ£πœ€3,3∣. βˆ£πœ€3,3∣. PROPN cana-611 209 29 thus thus ADV cana-611 209 30 , , PUNCT cana-611 209 31 with with ADP cana-611 209 32 this this DET cana-611 209 33 background background NOUN cana-611 209 34 by by ADP cana-611 209 35 employing employ VERB cana-611 209 36 equations equation NOUN cana-611 209 37 ( ( PUNCT cana-611 209 38 1)-(5 1)-(5 NUM cana-611 209 39 ) ) PUNCT cana-611 209 40 and and CCONJ cana-611 209 41 ( ( PUNCT cana-611 209 42 6)-(10 6)-(10 NUM cana-611 209 43 ) ) PUNCT cana-611 209 44 , , PUNCT cana-611 209 45 we we PRON cana-611 209 46 get get VERB cana-611 209 47 the the DET cana-611 209 48 required require VERB cana-611 209 49 results result NOUN cana-611 209 50 . . PUNCT cana-611 210 1 theorem theorem NOUN cana-611 210 2 13 13 NUM cana-611 210 3 . . PUNCT cana-611 211 1 let let VERB cana-611 211 2 𝑍𝑛be 𝑍𝑛be PROPN cana-611 211 3 zigzag zigzag VERB cana-611 211 4 polyomino polyomino NOUN cana-611 211 5 chain chain NOUN cana-611 211 6 with with ADP cana-611 211 7 𝑛 𝑛 PROPN cana-611 211 8 squares square NOUN cana-611 211 9 such such ADJ cana-611 211 10 that that DET cana-611 211 11 𝑙= 𝑙= NUM cana-611 211 12 2 2 NUM cana-611 211 13 and and CCONJ cana-611 211 14 π‘š= π‘š= VERB cana-611 211 15 π‘›βˆ’ π‘›βˆ’ NUM cana-611 211 16 1 1 NUM cana-611 211 17 . . PUNCT cana-611 212 1 then then ADV cana-611 212 2 𝑅𝑆2(𝑍𝑛 𝑅𝑆2(𝑍𝑛 NUM cana-611 212 3 ) ) PUNCT cana-611 213 1 = = PUNCT cana-611 213 2 63 63 NUM cana-611 213 3 16640 16640 NUM cana-611 213 4 π‘š π‘š X cana-611 213 5 + + PROPN cana-611 213 6 3 3 NUM cana-611 213 7 512 512 NUM cana-611 213 8 𝑛 𝑛 NOUN cana-611 214 1 + + NUM cana-611 214 2 37 37 NUM cana-611 214 3 100 100 NUM cana-611 214 4 ( ( PUNCT cana-611 214 5 59 59 NUM cana-611 214 6 ) ) PUNCT cana-611 214 7 𝑅𝑆3(𝑍𝑛 𝑅𝑆3(𝑍𝑛 ADV cana-611 214 8 ) ) PUNCT cana-611 215 1 = = PUNCT cana-611 215 2 15 15 NUM cana-611 215 3 256 256 NUM cana-611 215 4 π‘š π‘š NOUN cana-611 215 5 + + PROPN cana-611 215 6 3 3 NUM cana-611 215 7 512 512 NUM cana-611 215 8 𝑛 𝑛 NOUN cana-611 216 1 + + NUM cana-611 216 2 43 43 NUM cana-611 216 3 100 100 NUM cana-611 216 4 ( ( PUNCT cana-611 216 5 60 60 NUM cana-611 216 6 ) ) PUNCT cana-611 216 7 𝑅𝑆4(𝑍𝑛 𝑅𝑆4(𝑍𝑛 ADV cana-611 216 8 ) ) PUNCT cana-611 217 1 = = PUNCT cana-611 217 2 63 63 NUM cana-611 217 3 32768 32768 NUM cana-611 217 4 π‘š π‘š ADP cana-611 217 5 + + NOUN cana-611 217 6 3 3 NUM cana-611 217 7 65536 65536 NUM cana-611 217 8 𝑛 𝑛 DET cana-611 217 9 + + NUM cana-611 217 10 4 4 NUM cana-611 217 11 25 25 NUM cana-611 217 12 ( ( PUNCT cana-611 217 13 61 61 NUM cana-611 217 14 ) ) PUNCT cana-611 217 15 communications communication NOUN cana-611 217 16 on on ADP cana-611 217 17 applied apply VERB cana-611 217 18 nonlinear nonlinear ADJ cana-611 217 19 analysis analysis NOUN cana-611 217 20 issn issn NOUN cana-611 217 21 : : PUNCT cana-611 217 22 1074 1074 NUM cana-611 217 23 - - PUNCT cana-611 217 24 133x 133x NUM cana-611 217 25 vol vol NOUN cana-611 217 26 31 31 NUM cana-611 217 27 no no NOUN cana-611 217 28 . . NOUN cana-611 217 29 2 2 NUM cana-611 217 30 ( ( PUNCT cana-611 217 31 2024 2024 NUM cana-611 217 32 ) ) PUNCT cana-611 218 1 426 426 NUM cana-611 218 2 https://internationalpubls.com https://internationalpubls.com X cana-611 218 3 𝑅𝑆5(𝑍𝑛 𝑅𝑆5(𝑍𝑛 ADV cana-611 218 4 ) ) PUNCT cana-611 218 5 = = SYM cana-611 219 1 255 255 NUM cana-611 219 2 32768 32768 NUM cana-611 219 3 π‘š π‘š ADP cana-611 219 4 + + NOUN cana-611 219 5 3 3 NUM cana-611 219 6 65536 65536 NUM cana-611 219 7 𝑛 𝑛 PRON cana-611 219 8 + + NUM cana-611 219 9 17 17 NUM cana-611 219 10 100 100 NUM cana-611 219 11 ( ( PUNCT cana-611 219 12 62 62 NUM cana-611 219 13 ) ) PUNCT cana-611 219 14 proof proof NOUN cana-611 219 15 let let VERB cana-611 219 16 𝑍𝑛be 𝑍𝑛be PROPN cana-611 219 17 zigzag zigzag VERB cana-611 219 18 polyomino polyomino NOUN cana-611 219 19 chain chain NOUN cana-611 219 20 with with ADP cana-611 219 21 𝑛squares 𝑛square NOUN cana-611 219 22 such such DET cana-611 219 23 that that DET cana-611 219 24 𝑙= 𝑙= NUM cana-611 219 25 2 2 NUM cana-611 219 26 and and CCONJ cana-611 219 27 π‘š= π‘š= VERB cana-611 219 28 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-611 219 29 . . PUNCT cana-611 220 1 polyomino polyomino NOUN cana-611 220 2 chain chain NOUN cana-611 220 3 consists consist VERB cana-611 220 4 of of ADP cana-611 220 5 a a DET cana-611 220 6 sequence sequence NOUN cana-611 220 7 of of ADP cana-611 220 8 segments segment NOUN cana-611 220 9 𝑆1,2,β‹…β‹…β‹…,π‘†π‘šand 𝑆1,2,β‹…β‹…β‹…,π‘†π‘šand ADV cana-611 220 10 𝑙(𝑆 𝑙(𝑆 PROPN cana-611 220 11 ) ) PUNCT cana-611 221 1 = = VERB cana-611 221 2 𝑙where 𝑙where ADJ cana-611 221 3 π‘šβ‰₯ π‘šβ‰₯ ADV cana-611 221 4 1 1 NUM cana-611 221 5 and and CCONJ cana-611 221 6 ∈ ∈ PROPN cana-611 221 7 { { PUNCT cana-611 221 8 1,2,β‹…β‹…β‹…,π‘š 1,2,β‹…β‹…β‹…,π‘š NUM cana-611 221 9 } } PUNCT cana-611 221 10 . . PUNCT cana-611 222 1 the the DET cana-611 222 2 edge edge NOUN cana-611 222 3 set set NOUN cana-611 222 4 of of ADP cana-611 222 5 𝑍𝑛can 𝑍𝑛can PROPN cana-611 222 6 be be AUX cana-611 222 7 partitioned partition VERB cana-611 222 8 into into ADP cana-611 222 9 five five NUM cana-611 222 10 disjoint disjoint NOUN cana-611 222 11 sets set NOUN cana-611 222 12 πœ€2,2,πœ€2,3 πœ€2,2,πœ€2,3 NOUN cana-611 222 13 , , PUNCT cana-611 222 14 πœ€2,4 πœ€2,4 PROPN cana-611 222 15 , , PUNCT cana-611 222 16 πœ€3,4 πœ€3,4 ADJ cana-611 222 17 and and CCONJ cana-611 222 18 πœ€4,4 πœ€4,4 PROPN cana-611 222 19 , , PUNCT cana-611 222 20 where where SCONJ cana-611 222 21 πœ€(𝑍𝑛 πœ€(𝑍𝑛 ADV cana-611 222 22 ) ) PUNCT cana-611 223 1 = = PUNCT cana-611 224 1 πœ€2,2i πœ€2,2i PUNCT cana-611 224 2 βˆͺπœ€2,3 βˆͺπœ€2,3 NOUN cana-611 224 3 βˆͺπœ€2,4 βˆͺπœ€2,4 PROPN cana-611 224 4 βˆͺπœ€3,4 βˆͺπœ€3,4 ADJ cana-611 224 5 βˆͺπœ€4,4 βˆͺπœ€4,4 PROPN cana-611 224 6 . . PUNCT cana-611 224 7 further far ADV cana-611 224 8 , , PUNCT cana-611 224 9 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 224 10 = = SYM cana-611 224 11 2 2 NUM cana-611 224 12 , , PUNCT cana-611 224 13 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 224 14 = = SYM cana-611 224 15 4 4 NUM cana-611 224 16 , , PUNCT cana-611 224 17 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 224 18 = = SYM cana-611 224 19 2(π‘šβˆ’ 2(π‘šβˆ’ PROPN cana-611 224 20 1 1 NUM cana-611 224 21 ) ) PUNCT cana-611 224 22 , , PUNCT cana-611 224 23 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 224 24 = = SYM cana-611 224 25 2 2 NUM cana-611 224 26 and and CCONJ cana-611 224 27 βˆ£πœ€4,4∣ βˆ£πœ€4,4∣ VERB cana-611 224 28 = = SYM cana-611 224 29 3π‘›βˆ’ 3π‘›βˆ’ NUM cana-611 224 30 2π‘šβˆ’ 2π‘šβˆ’ NUM cana-611 224 31 5 5 NUM cana-611 224 32 . . PUNCT cana-611 224 33 such such ADJ cana-611 224 34 that that SCONJ cana-611 224 35 ∣(𝑍𝑛)∣ ∣(𝑍𝑛)∣ NOUN cana-611 224 36 = = NOUN cana-611 224 37 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ ADJ cana-611 224 38 + + NUM cana-611 224 39 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 224 40 + + CCONJ cana-611 224 41 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 224 42 + + CCONJ cana-611 224 43 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 224 44 + + ADV cana-611 224 45 i i PRON cana-611 224 46 βˆ£πœ€4,4∣. βˆ£πœ€4,4∣. VERB cana-611 224 47 thus thus ADV cana-611 224 48 , , PUNCT cana-611 224 49 with with ADP cana-611 224 50 this this DET cana-611 224 51 background background NOUN cana-611 224 52 by by ADP cana-611 224 53 employing employ VERB cana-611 224 54 equations equation NOUN cana-611 224 55 ( ( PUNCT cana-611 224 56 1)-(5 1)-(5 NUM cana-611 224 57 ) ) PUNCT cana-611 224 58 and and CCONJ cana-611 224 59 ( ( PUNCT cana-611 224 60 6)-(10 6)-(10 NUM cana-611 224 61 ) ) PUNCT cana-611 224 62 , , PUNCT cana-611 224 63 we we PRON cana-611 224 64 get get VERB cana-611 224 65 the the DET cana-611 224 66 required require VERB cana-611 224 67 results result NOUN cana-611 224 68 . . PUNCT cana-611 225 1 theorem theorem VERB cana-611 225 2 14 14 NUM cana-611 225 3 . . PUNCT cana-611 226 1 for for ADP cana-611 226 2 the the DET cana-611 226 3 polyomino polyomino NOUN cana-611 226 4 chain chain NOUN cana-611 226 5 with with ADP cana-611 226 6 𝑛 𝑛 PROPN cana-611 226 7 squares square NOUN cana-611 226 8 and and CCONJ cana-611 226 9 of of ADP cana-611 226 10 π‘š π‘š PROPN cana-611 226 11 segments segment NOUN cana-611 226 12 𝑆1 𝑆1 VERB cana-611 226 13 and and CCONJ cana-611 226 14 𝑆2i 𝑆2i NOUN cana-611 226 15 satisfying satisfy VERB cana-611 226 16 𝑙1 𝑙1 NOUN cana-611 226 17 = = SYM cana-611 226 18 2 2 NUM cana-611 226 19 and and CCONJ cana-611 226 20 𝑙2 𝑙2 NOUN cana-611 226 21 = = SYM cana-611 226 22 π‘›βˆ’ π‘›βˆ’ PROPN cana-611 226 23 1 1 NUM cana-611 226 24 , , PUNCT cana-611 226 25 𝐡𝑛 𝐡𝑛 PROPN cana-611 226 26 1(𝑛β‰₯ 1(𝑛β‰₯ NUM cana-611 226 27 3 3 X cana-611 226 28 ) ) PUNCT cana-611 226 29 we we PRON cana-611 226 30 have have VERB cana-611 226 31 the the DET cana-611 226 32 following following NOUN cana-611 226 33 : : PUNCT cana-611 227 1 𝑅𝑆2(𝐡𝑛 𝑅𝑆2(𝐡𝑛 PROPN cana-611 227 2 1 1 NUM cana-611 227 3 ) ) PUNCT cana-611 227 4 = = SYM cana-611 227 5 1 1 NUM cana-611 227 6 18 18 NUM cana-611 227 7 𝑛 𝑛 NOUN cana-611 227 8 + + NUM cana-611 227 9 6 6 NUM cana-611 227 10 25 25 NUM cana-611 227 11 ( ( PUNCT cana-611 227 12 63 63 NUM cana-611 227 13 ) ) PUNCT cana-611 227 14 𝑅𝑆3(𝐡𝑛 𝑅𝑆3(𝐡𝑛 PROPN cana-611 227 15 1 1 NUM cana-611 227 16 ) ) PUNCT cana-611 227 17 = = SYM cana-611 227 18 1 1 NUM cana-611 227 19 18 18 NUM cana-611 227 20 𝑛 𝑛 NOUN cana-611 227 21 + + NUM cana-611 227 22 41 41 NUM cana-611 227 23 100 100 NUM cana-611 227 24 ( ( PUNCT cana-611 227 25 64 64 NUM cana-611 227 26 ) ) PUNCT cana-611 227 27 𝑅𝑆4(𝐡𝑛 𝑅𝑆4(𝐡𝑛 PROPN cana-611 227 28 1 1 NUM cana-611 227 29 ) ) PUNCT cana-611 227 30 = = SYM cana-611 228 1 1 1 NUM cana-611 228 2 243 243 NUM cana-611 228 3 𝑛 𝑛 DET cana-611 228 4 + + NUM cana-611 228 5 4 4 NUM cana-611 228 6 25 25 NUM cana-611 228 7 ( ( PUNCT cana-611 228 8 65 65 NUM cana-611 228 9 ) ) PUNCT cana-611 228 10 𝑅𝑆5(𝐡𝑛 𝑅𝑆5(𝐡𝑛 ADP cana-611 228 11 1 1 X cana-611 228 12 ) ) PUNCT cana-611 228 13 = = SYM cana-611 228 14 1 1 NUM cana-611 228 15 243 243 NUM cana-611 228 16 𝑛 𝑛 DET cana-611 229 1 + + CCONJ cana-611 229 2 9 9 NUM cana-611 229 3 50 50 NUM cana-611 229 4 ( ( PUNCT cana-611 229 5 66 66 NUM cana-611 229 6 ) ) PUNCT cana-611 229 7 proof proof NOUN cana-611 229 8 . . PUNCT cana-611 230 1 let let VERB cana-611 230 2 𝐡𝑛 𝐡𝑛 PROPN cana-611 230 3 1(𝑛β‰₯ 1(𝑛β‰₯ NUM cana-611 230 4 3 3 X cana-611 230 5 ) ) PUNCT cana-611 230 6 be be AUX cana-611 230 7 the the DET cana-611 230 8 polyomino polyomino NOUN cana-611 230 9 chain chain NOUN cana-611 230 10 with with ADP cana-611 230 11 𝑛squares 𝑛square NOUN cana-611 230 12 and and CCONJ cana-611 230 13 π‘šsegments π‘šsegment VERB cana-611 230 14 𝑆1i 𝑆1i PROPN cana-611 230 15 and and CCONJ cana-611 230 16 𝑆2i 𝑆2i PROPN cana-611 230 17 satisfying satisfy VERB cana-611 230 18 𝑙1 𝑙1 NOUN cana-611 230 19 = = SYM cana-611 230 20 2 2 NUM cana-611 230 21 and and CCONJ cana-611 230 22 𝑙2 𝑙2 NOUN cana-611 231 1 = = SYM cana-611 231 2 π‘›βˆ’ π‘›βˆ’ PROPN cana-611 231 3 1 1 NUM cana-611 231 4 . . PUNCT cana-611 232 1 the the DET cana-611 232 2 edge edge NOUN cana-611 232 3 set set NOUN cana-611 232 4 of of ADP cana-611 232 5 𝐡𝑛 𝐡𝑛 PROPN cana-611 232 6 1(𝑛β‰₯ 1(𝑛β‰₯ PROPN cana-611 232 7 3 3 NUM cana-611 232 8 ) ) PUNCT cana-611 232 9 can can AUX cana-611 232 10 be be AUX cana-611 232 11 partitioned partition VERB cana-611 232 12 into into ADP cana-611 232 13 five five NUM cana-611 232 14 disjoint disjoint NOUN cana-611 232 15 sets set NOUN cana-611 232 16 πœ€ πœ€ ADP cana-611 232 17 2,2,πœ€2,3 2,2,πœ€2,3 NUM cana-611 232 18 , , PUNCT cana-611 232 19 πœ€2,4 πœ€2,4 PROPN cana-611 232 20 , , PUNCT cana-611 232 21 πœ€3,3i πœ€3,3i PROPN cana-611 232 22 and and CCONJ cana-611 232 23 πœ€3,4 πœ€3,4 ADJ cana-611 232 24 , , PUNCT cana-611 232 25 whereπœ€(𝐡𝑛 whereπœ€(𝐡𝑛 PROPN cana-611 232 26 1(𝑛 1(𝑛 NUM cana-611 232 27 β‰₯ β‰₯ NOUN cana-611 232 28 3 3 NUM cana-611 232 29 ) ) PUNCT cana-611 232 30 ) ) PUNCT cana-611 233 1 = = PUNCT cana-611 233 2 πœ€22 πœ€22 NOUN cana-611 233 3 βˆͺ βˆͺ VERB cana-611 233 4 πœ€23 πœ€23 NOUN cana-611 233 5 βˆͺ βˆͺ ADP cana-611 233 6 πœ€24 πœ€24 PROPN cana-611 233 7 βˆͺ βˆͺ PROPN cana-611 233 8 πœ€33 πœ€33 NOUN cana-611 233 9 βˆͺ βˆͺ ADP cana-611 233 10 πœ€34 πœ€34 NOUN cana-611 233 11 . . PUNCT cana-611 234 1 further far ADV cana-611 234 2 , , PUNCT cana-611 234 3 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 234 4 = = SYM cana-611 234 5 2 2 NUM cana-611 234 6 , , PUNCT cana-611 234 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 234 8 = = SYM cana-611 234 9 5 5 NUM cana-611 234 10 , , PUNCT cana-611 234 11 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 234 12 = = SYM cana-611 234 13 1 1 NUM cana-611 234 14 , , PUNCT cana-611 234 15 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 234 16 = = SYM cana-611 234 17 3π‘›βˆ’ 3π‘›βˆ’ NUM cana-611 234 18 10 10 NUM cana-611 234 19 and and CCONJ cana-611 234 20 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 234 21 = = SYM cana-611 234 22 3 3 X cana-611 234 23 . . X cana-611 234 24 such such ADJ cana-611 234 25 that that SCONJ cana-611 234 26 ∣(𝐡𝑛 ∣(𝐡𝑛 PROPN cana-611 234 27 1(𝑛β‰₯ 1(𝑛β‰₯ PROPN cana-611 234 28 3 3 NUM cana-611 234 29 ) ) PUNCT cana-611 234 30 ) ) PUNCT cana-611 234 31 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 234 32 + + NUM cana-611 234 33 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 234 34 + + CCONJ cana-611 234 35 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 234 36 + + CCONJ cana-611 234 37 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 234 38 + + CCONJ cana-611 234 39 βˆ£πœ€3,4∣. βˆ£πœ€3,4∣. NOUN cana-611 234 40 thus thus ADV cana-611 234 41 , , PUNCT cana-611 234 42 with with ADP cana-611 234 43 this this DET cana-611 234 44 background background NOUN cana-611 234 45 by by ADP cana-611 234 46 employing employ VERB cana-611 234 47 equations equation NOUN cana-611 234 48 ( ( PUNCT cana-611 234 49 1)-(5 1)-(5 NUM cana-611 234 50 ) ) PUNCT cana-611 234 51 and and CCONJ cana-611 234 52 ( ( PUNCT cana-611 234 53 6)(10 6)(10 NUM cana-611 234 54 ) ) PUNCT cana-611 234 55 , , PUNCT cana-611 234 56 we we PRON cana-611 234 57 get get VERB cana-611 234 58 the the DET cana-611 234 59 required require VERB cana-611 234 60 results result NOUN cana-611 234 61 . . PUNCT cana-611 235 1 theorem theorem ADJ cana-611 235 2 15 15 NUM cana-611 235 3 . . PUNCT cana-611 236 1 for for ADP cana-611 236 2 their their PRON cana-611 236 3 polyomino polyomino NOUN cana-611 236 4 chain chain NOUN cana-611 236 5 with with ADP cana-611 236 6 𝑛 𝑛 PROPN cana-611 236 7 squares square NOUN cana-611 236 8 and and CCONJ cana-611 236 9 of of ADP cana-611 236 10 π‘š π‘š ADP cana-611 236 11 segments segment NOUN cana-611 236 12 𝑆1,2,β‹…β‹…β‹…π‘ π‘šsatisfying 𝑆1,2,β‹…β‹…β‹…π‘ π‘šsatisfye VERB cana-611 236 13 𝑙1 𝑙1 NOUN cana-611 236 14 = = SYM cana-611 236 15 π‘™π‘š= π‘™π‘š= PROPN cana-611 236 16 2 2 NUM cana-611 236 17 and and CCONJ cana-611 236 18 𝑙2,𝑙3,β‹…β‹…β‹…,β‰₯ 𝑙2,𝑙3,β‹…β‹…β‹…,β‰₯ PROPN cana-611 236 19 3 3 NUM cana-611 236 20 𝐡𝑛 𝐡𝑛 PROPN cana-611 236 21 2(𝑛β‰₯ 2(𝑛β‰₯ PROPN cana-611 236 22 4 4 X cana-611 236 23 ) ) PUNCT cana-611 236 24 we we PRON cana-611 236 25 have have VERB cana-611 236 26 the the DET cana-611 236 27 following following NOUN cana-611 236 28 : : PUNCT cana-611 237 1 𝑅𝑆2(𝐡𝑛 𝑅𝑆2(𝐡𝑛 PROPN cana-611 237 2 2 2 NUM cana-611 237 3 ) ) PUNCT cana-611 237 4 = = SYM cana-611 238 1 βˆ’ βˆ’ PROPN cana-611 238 2 2543 2543 NUM cana-611 238 3 78957 78957 NUM cana-611 239 1 π‘š π‘š ADP cana-611 239 2 + + PROPN cana-611 239 3 1 1 NUM cana-611 239 4 18 18 NUM cana-611 239 5 𝑛 𝑛 NOUN cana-611 239 6 + + NUM cana-611 239 7 29 29 NUM cana-611 239 8 100 100 NUM cana-611 239 9 ( ( PUNCT cana-611 239 10 67 67 NUM cana-611 239 11 ) ) PUNCT cana-611 239 12 𝑅𝑆3(𝐡𝑛 𝑅𝑆3(𝐡𝑛 PROPN cana-611 239 13 2 2 NUM cana-611 239 14 ) ) PUNCT cana-611 239 15 = = SYM cana-611 240 1 17 17 NUM cana-611 240 2 500 500 NUM cana-611 240 3 π‘š π‘š PRON cana-611 240 4 + + NOUN cana-611 240 5 1 1 NUM cana-611 240 6 18 18 NUM cana-611 240 7 𝑛 𝑛 NOUN cana-611 240 8 + + NUM cana-611 240 9 33 33 NUM cana-611 240 10 100 100 NUM cana-611 240 11 ( ( PUNCT cana-611 240 12 68 68 NUM cana-611 240 13 ) ) PUNCT cana-611 240 14 𝑅𝑆4(𝐡𝑛 𝑅𝑆4(𝐡𝑛 PROPN cana-611 240 15 2 2 NUM cana-611 240 16 ) ) PUNCT cana-611 240 17 = = SYM cana-611 240 18 1 1 NUM cana-611 240 19 100 100 NUM cana-611 240 20 π‘š π‘š ADP cana-611 240 21 + + NOUN cana-611 240 22 1 1 NUM cana-611 240 23 243 243 NUM cana-611 240 24 𝑛 𝑛 DET cana-611 241 1 + + NUM cana-611 241 2 13 13 NUM cana-611 241 3 100 100 NUM cana-611 241 4 ( ( PUNCT cana-611 241 5 69 69 NUM cana-611 241 6 ) ) PUNCT cana-611 241 7 𝑅𝑆5(𝐡𝑛 𝑅𝑆5(𝐡𝑛 ADP cana-611 241 8 2 2 X cana-611 241 9 ) ) PUNCT cana-611 241 10 = = SYM cana-611 241 11 1 1 NUM cana-611 241 12 50 50 NUM cana-611 241 13 π‘š π‘š NOUN cana-611 241 14 + + PROPN cana-611 241 15 1 1 NUM cana-611 241 16 243 243 NUM cana-611 241 17 𝑛 𝑛 DET cana-611 242 1 + + NUM cana-611 242 2 13 13 NUM cana-611 242 3 100 100 NUM cana-611 242 4 ( ( PUNCT cana-611 242 5 70 70 NUM cana-611 242 6 ) ) PUNCT cana-611 242 7 proof proof NOUN cana-611 242 8 . . PUNCT cana-611 243 1 let let VERB cana-611 243 2 𝐡𝑛 𝐡𝑛 PROPN cana-611 243 3 2(𝑛β‰₯4 2(𝑛β‰₯4 PROPN cana-611 243 4 ) ) PUNCT cana-611 243 5 be be AUX cana-611 243 6 the the DET cana-611 243 7 polyomino polyomino NOUN cana-611 243 8 chain chain NOUN cana-611 243 9 with with ADP cana-611 243 10 𝑛squares 𝑛square NOUN cana-611 243 11 and and CCONJ cana-611 243 12 π‘šsegments π‘šsegment VERB cana-611 243 13 π‘šsegments π‘šsegment NOUN cana-611 243 14 𝑆1,𝑆2,β‹…β‹…β‹…π‘ π‘šsatisfying 𝑆1,𝑆2,β‹…β‹…β‹…π‘ π‘šsatisfye VERB cana-611 244 1 𝑙1 𝑙1 NOUN cana-611 244 2 = = SYM cana-611 244 3 π‘™π‘š= π‘™π‘š= PROPN cana-611 244 4 2 2 NUM cana-611 244 5 and and CCONJ cana-611 244 6 𝑙2,𝑙3,β‹…β‹…β‹…,β‰₯ 𝑙2,𝑙3,β‹…β‹…β‹…,β‰₯ PROPN cana-611 244 7 3 3 NUM cana-611 244 8 . . PUNCT cana-611 245 1 the the DET cana-611 245 2 edge edge NOUN cana-611 245 3 set set NOUN cana-611 245 4 of of ADP cana-611 245 5 𝐡𝑛 𝐡𝑛 PROPN cana-611 245 6 2(𝑛β‰₯4 2(𝑛β‰₯4 PROPN cana-611 245 7 ) ) PUNCT cana-611 245 8 can can AUX cana-611 245 9 be be AUX cana-611 245 10 partitioned partition VERB cana-611 245 11 into into ADP cana-611 245 12 five five NUM cana-611 245 13 disjoint disjoint NOUN cana-611 245 14 sets set NOUN cana-611 245 15 πœ€2,2,πœ€2,3 πœ€2,2,πœ€2,3 NOUN cana-611 245 16 , , PUNCT cana-611 245 17 πœ€2,4 πœ€2,4 PROPN cana-611 245 18 , , PUNCT cana-611 245 19 πœ€3,3 πœ€3,3 PROPN cana-611 245 20 and and CCONJ cana-611 245 21 πœ€3,4 πœ€3,4 PROPN cana-611 245 22 where where SCONJ cana-611 245 23 πœ€(𝐡𝑛 πœ€(𝐡𝑛 PROPN cana-611 245 24 2(𝑛 2(𝑛 NUM cana-611 245 25 β‰₯ β‰₯ NOUN cana-611 245 26 3 3 NUM cana-611 245 27 ) ) PUNCT cana-611 245 28 ) ) PUNCT cana-611 246 1 = = PUNCT cana-611 246 2 πœ€22 πœ€22 NOUN cana-611 246 3 βˆͺ βˆͺ VERB cana-611 246 4 πœ€23 πœ€23 NOUN cana-611 246 5 βˆͺ βˆͺ ADP cana-611 246 6 πœ€24 πœ€24 PROPN cana-611 246 7 βˆͺ βˆͺ PROPN cana-611 246 8 πœ€33 πœ€33 NOUN cana-611 246 9 βˆͺ βˆͺ ADP cana-611 246 10 πœ€34.further πœ€34.further PROPN cana-611 246 11 , , PUNCT cana-611 246 12 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 246 13 = = SYM cana-611 246 14 2 2 NUM cana-611 246 15 , , PUNCT cana-611 246 16 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 246 17 = = SYM cana-611 246 18 2π‘š 2π‘š NOUN cana-611 246 19 , , PUNCT cana-611 246 20 βˆ£πœ€2,4∣ βˆ£πœ€2,4∣ NOUN cana-611 246 21 = = SYM cana-611 246 22 2 2 NUM cana-611 246 23 , , PUNCT cana-611 246 24 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 246 25 = = SYM cana-611 246 26 3π‘›βˆ’6π‘š+3 3π‘›βˆ’6π‘š+3 NUM cana-611 246 27 and and CCONJ cana-611 246 28 βˆ£πœ€3,4∣ βˆ£πœ€3,4∣ NOUN cana-611 246 29 = = SYM cana-611 246 30 4π‘šβˆ’6 4π‘šβˆ’6 NUM cana-611 246 31 . . PUNCT cana-611 247 1 such such ADJ cana-611 247 2 that that SCONJ cana-611 247 3 ∣(𝐡𝑛 ∣(𝐡𝑛 PROPN cana-611 247 4 2(𝑛β‰₯ 2(𝑛β‰₯ PROPN cana-611 247 5 3))∣ 3))∣ NUM cana-611 247 6 = = PUNCT cana-611 247 7 βˆ£πœ€2,2∣+βˆ£πœ€2,3∣+βˆ£πœ€2,4∣+βˆ£πœ€3,3∣+βˆ£πœ€3,4∣. βˆ£πœ€2,2∣+βˆ£πœ€2,3∣+βˆ£πœ€2,4∣+βˆ£πœ€3,3∣+βˆ£πœ€3,4∣. NOUN cana-611 247 8 thus thus ADV cana-611 247 9 , , PUNCT cana-611 247 10 with with ADP cana-611 247 11 this this DET cana-611 247 12 background background NOUN cana-611 247 13 by by ADP cana-611 247 14 employing employ VERB cana-611 247 15 equations equation NOUN cana-611 247 16 ( ( PUNCT cana-611 247 17 1)-(5 1)-(5 NUM cana-611 247 18 ) ) PUNCT cana-611 247 19 and and CCONJ cana-611 247 20 ( ( PUNCT cana-611 247 21 6)-(10 6)-(10 NUM cana-611 247 22 ) ) PUNCT cana-611 247 23 , , PUNCT cana-611 247 24 we we PRON cana-611 247 25 get get VERB cana-611 247 26 the the DET cana-611 247 27 required require VERB cana-611 247 28 results result NOUN cana-611 247 29 . . PUNCT cana-611 248 1 theorem theorem VERB cana-611 248 2 16 16 NUM cana-611 248 3 . . PUNCT cana-611 249 1 let let VERB cana-611 249 2 𝑇𝑝be 𝑇𝑝be PROPN cana-611 249 3 a a DET cana-611 249 4 triangular triangular NOUN cana-611 249 5 benzenoid benzenoid NOUN cana-611 249 6 where where SCONJ cana-611 249 7 𝑝i 𝑝i NOUN cana-611 249 8 shows show VERB cana-611 249 9 the the DET cana-611 249 10 number number NOUN cana-611 249 11 of of ADP cana-611 249 12 hexagons hexagon NOUN cana-611 249 13 in in ADP cana-611 249 14 the the DET cana-611 249 15 base base NOUN cana-611 249 16 graphic graphic NOUN cana-611 249 17 and and CCONJ cana-611 249 18 total total ADJ cana-611 249 19 number number NOUN cana-611 249 20 of of ADP cana-611 249 21 hexagons hexagon NOUN cana-611 249 22 in in ADP cana-611 249 23 𝑇𝑝 𝑇𝑝 PROPN cana-611 249 24 is be AUX cana-611 249 25 𝑝(𝑝+1 𝑝(𝑝+1 NOUN cana-611 249 26 ) ) PUNCT cana-611 249 27 2 2 NUM cana-611 249 28 . . PUNCT cana-611 250 1 then then ADV cana-611 250 2 communications communication NOUN cana-611 250 3 on on ADP cana-611 250 4 applied apply VERB cana-611 250 5 nonlinear nonlinear ADJ cana-611 250 6 analysis analysis NOUN cana-611 250 7 issn issn NOUN cana-611 250 8 : : PUNCT cana-611 250 9 1074 1074 NUM cana-611 250 10 - - PUNCT cana-611 250 11 133x 133x NUM cana-611 250 12 vol vol NOUN cana-611 250 13 31 31 NUM cana-611 250 14 no no NOUN cana-611 250 15 . . NOUN cana-611 250 16 2 2 NUM cana-611 250 17 ( ( PUNCT cana-611 250 18 2024 2024 NUM cana-611 250 19 ) ) PUNCT cana-611 250 20 427 427 NUM cana-611 250 21 https://internationalpubls.com https://internationalpubls.com X cana-611 250 22 𝑅𝑆2(𝑇𝑝 𝑅𝑆2(𝑇𝑝 NOUN cana-611 250 23 ) ) PUNCT cana-611 250 24 = = SYM cana-611 250 25 1 1 NUM cana-611 250 26 36 36 NUM cana-611 250 27 𝑝2 𝑝2 NOUN cana-611 250 28 + + CCONJ cana-611 250 29 185 185 NUM cana-611 250 30 1116 1116 NUM cana-611 250 31 𝑝 𝑝 NOUN cana-611 251 1 + + CCONJ cana-611 251 2 69 69 NUM cana-611 251 3 124 124 NUM cana-611 251 4 ( ( PUNCT cana-611 251 5 71 71 NUM cana-611 251 6 ) ) PUNCT cana-611 251 7 𝑅𝑆3(𝑇𝑝 𝑅𝑆3(𝑇𝑝 PROPN cana-611 251 8 ) ) PUNCT cana-611 251 9 = = SYM cana-611 251 10 1 1 NUM cana-611 251 11 36 36 NUM cana-611 251 12 𝑝2 𝑝2 NOUN cana-611 251 13 + + CCONJ cana-611 251 14 199 199 NUM cana-611 251 15 612 612 NUM cana-611 251 16 𝑝 𝑝 NOUN cana-611 251 17 + + CCONJ cana-611 251 18 27 27 NUM cana-611 251 19 68 68 NUM cana-611 251 20 ( ( PUNCT cana-611 251 21 72 72 NUM cana-611 251 22 ) ) PUNCT cana-611 251 23 𝑅𝑆4(𝑇𝑝 𝑅𝑆4(𝑇𝑝 PUNCT cana-611 251 24 ) ) PUNCT cana-611 251 25 = = SYM cana-611 251 26 1 1 NUM cana-611 251 27 486 486 NUM cana-611 251 28 𝑝2 𝑝2 NOUN cana-611 251 29 + + CCONJ cana-611 251 30 13 13 NUM cana-611 251 31 243 243 NUM cana-611 251 32 𝑝 𝑝 NOUN cana-611 251 33 + + CCONJ cana-611 251 34 23 23 NUM cana-611 251 35 72 72 NUM cana-611 251 36 ( ( PUNCT cana-611 251 37 73 73 NUM cana-611 251 38 ) ) PUNCT cana-611 251 39 𝑅𝑆5(𝑇𝑝 𝑅𝑆5(𝑇𝑝 NOUN cana-611 251 40 ) ) PUNCT cana-611 251 41 = = SYM cana-611 251 42 1 1 NUM cana-611 251 43 486 486 NUM cana-611 251 44 𝑝2 𝑝2 NOUN cana-611 251 45 + + CCONJ cana-611 251 46 79 79 NUM cana-611 251 47 972 972 NUM cana-611 251 48 𝑝 𝑝 NOUN cana-611 251 49 + + CCONJ cana-611 251 50 7 7 NUM cana-611 251 51 24 24 NUM cana-611 251 52 ( ( PUNCT cana-611 251 53 74 74 NUM cana-611 251 54 ) ) PUNCT cana-611 251 55 proof proof NOUN cana-611 251 56 . . PUNCT cana-611 252 1 let let VERB cana-611 252 2 𝑇𝑝 𝑇𝑝 VERB cana-611 252 3 be be AUX cana-611 252 4 a a DET cana-611 252 5 triangular triangular NOUN cana-611 252 6 benzenoid benzenoid NOUN cana-611 252 7 where where SCONJ cana-611 252 8 𝑝shows 𝑝show VERB cana-611 252 9 the the DET cana-611 252 10 number number NOUN cana-611 252 11 of of ADP cana-611 252 12 hexagons hexagon NOUN cana-611 252 13 in in ADP cana-611 252 14 the the DET cana-611 252 15 base base NOUN cana-611 252 16 graphic graphic NOUN cana-611 252 17 and and CCONJ cana-611 252 18 total total ADJ cana-611 252 19 number number NOUN cana-611 252 20 of of ADP cana-611 252 21 hexagons hexagon NOUN cana-611 252 22 in in ADP cana-611 252 23 𝑇𝑝 𝑇𝑝 PROPN cana-611 252 24 is be AUX cana-611 252 25 𝑝(𝑝+1 𝑝(𝑝+1 NOUN cana-611 252 26 ) ) PUNCT cana-611 252 27 2 2 NUM cana-611 252 28 . . PUNCT cana-611 253 1 the the DET cana-611 253 2 edge edge NOUN cana-611 253 3 set set NOUN cana-611 253 4 of of ADP cana-611 253 5 𝑇𝑝 𝑇𝑝 PROPN cana-611 253 6 can can AUX cana-611 253 7 be be AUX cana-611 253 8 partitioned partition VERB cana-611 253 9 into into ADP cana-611 253 10 three three NUM cana-611 253 11 disjoint disjoint NOUN cana-611 253 12 sets set NOUN cana-611 253 13 πœ€2,2,πœ€2,3i πœ€2,2,πœ€2,3i PROPN cana-611 253 14 and and CCONJ cana-611 253 15 πœ€3,3 πœ€3,3 PROPN cana-611 253 16 where where SCONJ cana-611 253 17 πœ€(𝑇𝑝 πœ€(𝑇𝑝 NOUN cana-611 253 18 ) ) PUNCT cana-611 253 19 = = PUNCT cana-611 254 1 πœ€2,2βˆͺπœ€2,3βˆͺπœ€3,3 πœ€2,2βˆͺπœ€2,3βˆͺπœ€3,3 NOUN cana-611 254 2 . . PUNCT cana-611 255 1 further far ADV cana-611 255 2 , , PUNCT cana-611 255 3 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 255 4 = = SYM cana-611 255 5 6 6 NUM cana-611 255 6 , , PUNCT cana-611 255 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 255 8 = = SYM cana-611 255 9 6(π‘βˆ’ 6(π‘βˆ’ NUM cana-611 255 10 1 1 NUM cana-611 255 11 ) ) PUNCT cana-611 255 12 and and CCONJ cana-611 255 13 |πœ€33| |πœ€33| PROPN cana-611 255 14 = = SYM cana-611 255 15 3𝑝(π‘βˆ’1 3𝑝(π‘βˆ’1 NUM cana-611 255 16 ) ) PUNCT cana-611 255 17 2 2 NUM cana-611 255 18 . . PUNCT cana-611 256 1 such such ADJ cana-611 256 2 that that SCONJ cana-611 256 3 ∣(𝑇𝑝)∣ ∣(𝑇𝑝)∣ PROPN cana-611 256 4 = = SYM cana-611 256 5 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 256 6 + + NUM cana-611 256 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 256 8 + + CCONJ cana-611 256 9 βˆ£πœ€3,3∣. βˆ£πœ€3,3∣. PROPN cana-611 256 10 thus thus ADV cana-611 256 11 , , PUNCT cana-611 256 12 with with ADP cana-611 256 13 this this DET cana-611 256 14 background background NOUN cana-611 256 15 by by ADP cana-611 256 16 employing employ VERB cana-611 256 17 equations equation NOUN cana-611 256 18 ( ( PUNCT cana-611 256 19 1)-(5 1)-(5 NUM cana-611 256 20 ) ) PUNCT cana-611 256 21 and and CCONJ cana-611 256 22 ( ( PUNCT cana-611 256 23 6)-(10 6)-(10 NUM cana-611 256 24 ) ) PUNCT cana-611 256 25 , , PUNCT cana-611 256 26 we we PRON cana-611 256 27 get get VERB cana-611 256 28 the the DET cana-611 256 29 required require VERB cana-611 256 30 results result NOUN cana-611 256 31 . . PUNCT cana-611 257 1 theorem theorem NOUN cana-611 257 2 17 17 NUM cana-611 257 3 . . PUNCT cana-611 258 1 let let VERB cana-611 258 2 𝑋𝑝 𝑋𝑝 NOUN cana-611 258 3 be be AUX cana-611 258 4 a a DET cana-611 258 5 benzenoid benzenoid NOUN cana-611 258 6 hourglass hourglass NOUN cana-611 258 7 . . PUNCT cana-611 259 1 then then ADV cana-611 259 2 𝑅𝑆2(𝑋𝑝 𝑅𝑆2(𝑋𝑝 VERB cana-611 259 3 ) ) PUNCT cana-611 259 4 = = SYM cana-611 259 5 1 1 NUM cana-611 259 6 18 18 NUM cana-611 259 7 𝑝2 𝑝2 NOUN cana-611 259 8 + + CCONJ cana-611 259 9 185 185 NUM cana-611 259 10 558 558 NUM cana-611 259 11 𝑝 𝑝 NOUN cana-611 259 12 + + CCONJ cana-611 259 13 467 467 NUM cana-611 259 14 837 837 NUM cana-611 259 15 ( ( PUNCT cana-611 259 16 75 75 NUM cana-611 259 17 ) ) PUNCT cana-611 259 18 𝑅𝑆3(𝑋𝑝 𝑅𝑆3(𝑋𝑝 PROPN cana-611 259 19 ) ) PUNCT cana-611 259 20 = = SYM cana-611 260 1 1 1 NUM cana-611 260 2 18 18 NUM cana-611 260 3 𝑝2 𝑝2 NOUN cana-611 260 4 + + CCONJ cana-611 260 5 199 199 NUM cana-611 260 6 306 306 NUM cana-611 260 7 𝑝 𝑝 NOUN cana-611 260 8 + + CCONJ cana-611 260 9 61 61 NUM cana-611 260 10 459 459 NUM cana-611 260 11 ( ( PUNCT cana-611 260 12 76 76 NUM cana-611 260 13 ) ) PUNCT cana-611 260 14 𝑅𝑆4(𝑋𝑝 𝑅𝑆4(𝑋𝑝 PROPN cana-611 260 15 ) ) PUNCT cana-611 260 16 = = SYM cana-611 260 17 1 1 NUM cana-611 260 18 243 243 NUM cana-611 260 19 𝑝2 𝑝2 NOUN cana-611 260 20 + + CCONJ cana-611 260 21 26 26 NUM cana-611 260 22 243 243 NUM cana-611 260 23 𝑝 𝑝 NOUN cana-611 260 24 + + CCONJ cana-611 260 25 521 521 NUM cana-611 260 26 1458 1458 NUM cana-611 260 27 ( ( PUNCT cana-611 260 28 77 77 NUM cana-611 260 29 ) ) PUNCT cana-611 260 30 𝑅𝑆5(𝑋𝑝 𝑅𝑆5(𝑋𝑝 PROPN cana-611 260 31 ) ) PUNCT cana-611 260 32 = = SYM cana-611 260 33 1 1 NUM cana-611 260 34 243 243 NUM cana-611 260 35 𝑝2 𝑝2 NOUN cana-611 260 36 + + CCONJ cana-611 260 37 79 79 NUM cana-611 260 38 486 486 NUM cana-611 260 39 𝑝 𝑝 NOUN cana-611 260 40 + + CCONJ cana-611 260 41 413 413 NUM cana-611 260 42 1458 1458 NUM cana-611 260 43 ( ( PUNCT cana-611 260 44 78 78 NUM cana-611 260 45 ) ) PUNCT cana-611 260 46 proof proof NOUN cana-611 260 47 . . PUNCT cana-611 261 1 let let VERB cana-611 261 2 𝑋𝑝be 𝑋𝑝be PROPN cana-611 261 3 a a DET cana-611 261 4 benzenoid benzenoid NOUN cana-611 261 5 hourglass hourglass NOUN cana-611 261 6 . . PUNCT cana-611 262 1 the the DET cana-611 262 2 edge edge NOUN cana-611 262 3 set set NOUN cana-611 262 4 of of ADP cana-611 262 5 𝑋𝑝can 𝑋𝑝can PROPN cana-611 262 6 be be AUX cana-611 262 7 partitioned partition VERB cana-611 262 8 into into ADP cana-611 262 9 three three NUM cana-611 262 10 disjoint disjoint NOUN cana-611 262 11 sets set NOUN cana-611 262 12 πœ€2,2,πœ€2,3i πœ€2,2,πœ€2,3i PROPN cana-611 262 13 and and CCONJ cana-611 262 14 πœ€3,3i πœ€3,3i PROPN cana-611 262 15 where where SCONJ cana-611 262 16 πœ€(𝑋𝑝 πœ€(𝑋𝑝 VERB cana-611 262 17 ) ) PUNCT cana-611 263 1 = = SYM cana-611 263 2 πœ€2,2 πœ€2,2 PROPN cana-611 263 3 βˆͺπœ€2,3i βˆͺπœ€2,3i PROPN cana-611 263 4 βˆͺπœ€3,3 βˆͺπœ€3,3 NOUN cana-611 263 5 . . PUNCT cana-611 263 6 further far ADV cana-611 263 7 , , PUNCT cana-611 263 8 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 263 9 = = SYM cana-611 263 10 8 8 NUM cana-611 263 11 , , PUNCT cana-611 263 12 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ NOUN cana-611 263 13 = = SYM cana-611 263 14 4(3π‘βˆ’4 4(3π‘βˆ’4 NOUN cana-611 263 15 ) ) PUNCT cana-611 263 16 and and CCONJ cana-611 263 17 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 263 18 = = SYM cana-611 263 19 3𝑝2βˆ’3𝑝+4 3𝑝2βˆ’3𝑝+4 NOUN cana-611 263 20 . . PUNCT cana-611 264 1 such such ADJ cana-611 264 2 that that SCONJ cana-611 264 3 ∣(𝑋𝑝)∣ ∣(𝑋𝑝)∣ PROPN cana-611 264 4 = = SYM cana-611 264 5 βˆ£πœ€2,2∣+βˆ£πœ€2,3∣+βˆ£πœ€3,3∣. βˆ£πœ€2,2∣+βˆ£πœ€2,3∣+βˆ£πœ€3,3∣. PROPN cana-611 264 6 thus thus ADV cana-611 264 7 , , PUNCT cana-611 264 8 with with ADP cana-611 264 9 this this DET cana-611 264 10 background background NOUN cana-611 264 11 by by ADP cana-611 264 12 employing employ VERB cana-611 264 13 equations equation NOUN cana-611 264 14 ( ( PUNCT cana-611 264 15 1)-(5 1)-(5 NUM cana-611 264 16 ) ) PUNCT cana-611 264 17 and and CCONJ cana-611 264 18 ( ( PUNCT cana-611 264 19 6)-(10 6)-(10 NUM cana-611 264 20 ) ) PUNCT cana-611 264 21 , , PUNCT cana-611 264 22 we we PRON cana-611 264 23 get get VERB cana-611 264 24 the the DET cana-611 264 25 required require VERB cana-611 264 26 results result NOUN cana-611 264 27 . . PUNCT cana-611 265 1 theorem theorem NOUN cana-611 265 2 18 18 NUM cana-611 265 3 . . PUNCT cana-611 266 1 let let AUX cana-611 266 2 𝐡𝑝,be 𝐡𝑝,be PART cana-611 266 3 denote denote VERB cana-611 266 4 a a DET cana-611 266 5 jagged jagged ADJ cana-611 266 6 rectangle rectangle NOUN cana-611 266 7 benzenoid benzenoid NOUN cana-611 266 8 system system NOUN cana-611 266 9 for for ADP cana-611 266 10 all all DET cana-611 266 11 𝑝,π‘žβˆˆπ‘βˆ’1 𝑝,π‘žβˆˆπ‘βˆ’1 NOUN cana-611 266 12 . . PUNCT cana-611 267 1 then then ADV cana-611 267 2 𝑅𝑆2(𝐡𝑝,π‘ž 𝑅𝑆2(𝐡𝑝,π‘ž PRON cana-611 267 3 ) ) PUNCT cana-611 268 1 = = SYM cana-611 268 2 1 1 NUM cana-611 268 3 9 9 NUM cana-611 268 4 π‘π‘ž π‘π‘ž NOUN cana-611 268 5 + + CCONJ cana-611 268 6 247 247 NUM cana-611 268 7 1674 1674 NUM cana-611 268 8 𝑝 𝑝 NOUN cana-611 269 1 + + NUM cana-611 269 2 959 959 NUM cana-611 269 3 3348 3348 NUM cana-611 269 4 π‘ž π‘ž NOUN cana-611 269 5 + + NOUN cana-611 269 6 497 497 NUM cana-611 269 7 1674 1674 NUM cana-611 269 8 ( ( PUNCT cana-611 269 9 79 79 NUM cana-611 269 10 ) ) PUNCT cana-611 269 11 𝑅𝑆3(𝐡𝑝,π‘ž 𝑅𝑆3(𝐡𝑝,π‘ž NUM cana-611 269 12 ) ) PUNCT cana-611 269 13 = = SYM cana-611 269 14 1 1 NUM cana-611 269 15 9 9 NUM cana-611 269 16 π‘π‘ž π‘π‘ž NOUN cana-611 269 17 + + CCONJ cana-611 269 18 233 233 NUM cana-611 269 19 918 918 NUM cana-611 269 20 𝑝 𝑝 NOUN cana-611 270 1 + + NUM cana-611 270 2 721 721 NUM cana-611 270 3 1836 1836 NUM cana-611 270 4 π‘ž π‘ž X cana-611 270 5 + + NOUN cana-611 270 6 175 175 NUM cana-611 270 7 918 918 NUM cana-611 270 8 ( ( PUNCT cana-611 270 9 80 80 NUM cana-611 270 10 ) ) PUNCT cana-611 270 11 𝑅𝑆4(𝐡𝑝,π‘ž 𝑅𝑆4(𝐡𝑝,π‘ž NUM cana-611 270 12 ) ) PUNCT cana-611 270 13 = = SYM cana-611 270 14 2 2 NUM cana-611 270 15 243 243 NUM cana-611 270 16 π‘π‘ž π‘π‘ž NOUN cana-611 270 17 + + CCONJ cana-611 270 18 26 26 NUM cana-611 270 19 729 729 NUM cana-611 270 20 𝑝 𝑝 NOUN cana-611 271 1 + + CCONJ cana-611 271 2 905 905 NUM cana-611 271 3 5832 5832 NUM cana-611 271 4 π‘ž π‘ž NOUN cana-611 271 5 + + CCONJ cana-611 271 6 605 605 NUM cana-611 271 7 2916 2916 NUM cana-611 271 8 ( ( PUNCT cana-611 271 9 81 81 NUM cana-611 271 10 ) ) PUNCT cana-611 271 11 𝑅𝑆5(𝐡𝑝,π‘ž 𝑅𝑆5(𝐡𝑝,π‘ž NUM cana-611 271 12 ) ) PUNCT cana-611 271 13 = = SYM cana-611 271 14 2 2 NUM cana-611 271 15 243 243 NUM cana-611 271 16 π‘π‘ž π‘π‘ž NOUN cana-611 271 17 + + CCONJ cana-611 271 18 79 79 NUM cana-611 271 19 1458 1458 NUM cana-611 271 20 𝑝 𝑝 NOUN cana-611 271 21 + + NUM cana-611 271 22 1013 1013 NUM cana-611 271 23 5832 5832 NUM cana-611 271 24 π‘ž π‘ž X cana-611 272 1 + + CCONJ cana-611 273 1 551 551 NUM cana-611 273 2 2916 2916 NUM cana-611 273 3 ( ( PUNCT cana-611 273 4 82 82 NUM cana-611 273 5 ) ) PUNCT cana-611 273 6 proof proof NOUN cana-611 273 7 . . PUNCT cana-611 274 1 let let VERB cana-611 274 2 𝐡𝑝,be 𝐡𝑝,be PROPN cana-611 274 3 denotes denote VERB cana-611 274 4 a a DET cana-611 274 5 jagged jagged ADJ cana-611 274 6 rectangle rectangle NOUN cana-611 274 7 benzenoid benzenoid NOUN cana-611 274 8 system system NOUN cana-611 274 9 for for ADP cana-611 274 10 all all PRON cana-611 274 11 𝑝,π‘žβˆˆπ‘βˆ’ 𝑝,π‘žβˆˆπ‘βˆ’ PROPN cana-611 274 12 1 1 X cana-611 274 13 . . PUNCT cana-611 275 1 the the DET cana-611 275 2 edge edge NOUN cana-611 275 3 set set NOUN cana-611 275 4 of of ADP cana-611 275 5 𝐡𝑝,π‘žcan 𝐡𝑝,π‘žcan PROPN cana-611 275 6 be be AUX cana-611 275 7 partitioned partition VERB cana-611 275 8 into into ADP cana-611 275 9 three three NUM cana-611 275 10 disjoint disjoint NOUN cana-611 275 11 sets set NOUN cana-611 275 12 πœ€2,2,πœ€2,3 πœ€2,2,πœ€2,3 PUNCT cana-611 275 13 and and CCONJ cana-611 275 14 πœ€3,3 πœ€3,3 PROPN cana-611 275 15 where where SCONJ cana-611 275 16 πœ€(𝐡𝑝,π‘ž)= πœ€(𝐡𝑝,π‘ž)= NOUN cana-611 275 17 πœ€2,2 πœ€2,2 PROPN cana-611 275 18 βˆͺπœ€2,3 βˆͺπœ€2,3 PROPN cana-611 275 19 βˆͺπœ€3,3 βˆͺπœ€3,3 NOUN cana-611 275 20 . . PUNCT cana-611 276 1 further far ADV cana-611 276 2 , , PUNCT cana-611 276 3 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 276 4 = = SYM cana-611 276 5 2π‘ž+4 2π‘ž+4 NOUN cana-611 276 6 , , PUNCT cana-611 276 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ VERB cana-611 276 8 = = SYM cana-611 276 9 4𝑝+4π‘žβˆ’4 4𝑝+4π‘žβˆ’4 NUM cana-611 276 10 and and CCONJ cana-611 276 11 βˆ£πœ€3,3∣ βˆ£πœ€3,3∣ NOUN cana-611 276 12 = = NOUN cana-611 276 13 6π‘π‘ž+π‘βˆ’5π‘žβˆ’4 6π‘π‘ž+π‘βˆ’5π‘žβˆ’4 NUM cana-611 276 14 . . PUNCT cana-611 277 1 such such ADJ cana-611 277 2 that that DET cana-611 277 3 ∣(𝐡𝑝,π‘ž)∣ ∣(𝐡𝑝,π‘ž)∣ NOUN cana-611 277 4 = = SYM cana-611 277 5 βˆ£πœ€2,2∣ βˆ£πœ€2,2∣ NOUN cana-611 277 6 + + NUM cana-611 277 7 βˆ£πœ€2,3∣ βˆ£πœ€2,3∣ PROPN cana-611 277 8 + + CCONJ cana-611 277 9 βˆ£πœ€3,3∣. βˆ£πœ€3,3∣. PROPN cana-611 277 10 thus thus ADV cana-611 277 11 , , PUNCT cana-611 277 12 with with ADP cana-611 277 13 this this DET cana-611 277 14 background background NOUN cana-611 277 15 by by ADP cana-611 277 16 employing employ VERB cana-611 277 17 equations equation NOUN cana-611 277 18 ( ( PUNCT cana-611 277 19 1)-(5 1)-(5 NUM cana-611 277 20 ) ) PUNCT cana-611 277 21 and and CCONJ cana-611 277 22 ( ( PUNCT cana-611 277 23 6)(10 6)(10 NUM cana-611 277 24 ) ) PUNCT cana-611 277 25 , , PUNCT cana-611 277 26 we we PRON cana-611 277 27 get get VERB cana-611 277 28 the the DET cana-611 277 29 required require VERB cana-611 277 30 results result NOUN cana-611 277 31 . . PUNCT cana-611 278 1 communications communication NOUN cana-611 278 2 on on ADP cana-611 278 3 applied apply VERB cana-611 278 4 nonlinear nonlinear ADJ cana-611 278 5 analysis analysis NOUN cana-611 278 6 issn issn NOUN cana-611 278 7 : : PUNCT cana-611 278 8 1074 1074 NUM cana-611 278 9 - - PUNCT cana-611 278 10 133x 133x NUM cana-611 278 11 vol vol NOUN cana-611 278 12 31 31 NUM cana-611 278 13 no no NOUN cana-611 278 14 . . NOUN cana-611 278 15 2 2 NUM cana-611 278 16 ( ( PUNCT cana-611 278 17 2024 2024 NUM cana-611 278 18 ) ) PUNCT cana-611 278 19 428 428 NUM cana-611 279 1 https://internationalpubls.com https://internationalpubls.com X cana-611 279 2 conclusion conclusion NOUN cana-611 279 3 in in ADP cana-611 279 4 this this DET cana-611 279 5 paper paper NOUN cana-611 279 6 , , PUNCT cana-611 279 7 we we PRON cana-611 279 8 have have AUX cana-611 279 9 computed compute VERB cana-611 279 10 the the DET cana-611 279 11 topological topological ADJ cana-611 279 12 index index NOUN cana-611 279 13 values value NOUN cana-611 279 14 of of ADP cana-611 279 15 chemical chemical NOUN cana-611 279 16 structures structure NOUN cana-611 279 17 like like ADP cana-611 279 18 hexagonal hexagonal ADJ cana-611 279 19 parallelogram parallelogram NOUN cana-611 279 20 𝑃(π‘š 𝑃(π‘š PROPN cana-611 279 21 , , PUNCT cana-611 279 22 𝑛 𝑛 NOUN cana-611 279 23 ) ) PUNCT cana-611 279 24 nanotube nanotube NOUN cana-611 279 25 , , PUNCT cana-611 279 26 triangular triangular NOUN cana-611 279 27 benzenoid benzenoid ADJ cana-611 279 28 𝐺𝑛,zigzag 𝐺𝑛,zigzag NOUN cana-611 279 29 - - PUNCT cana-611 279 30 edge edge NOUN cana-611 279 31 coronoid coronoid PROPN cana-611 279 32 fused fuse VERB cana-611 279 33 with with ADP cana-611 279 34 starphene starphene NOUN cana-611 279 35 nanotubes nanotube NOUN cana-611 279 36 𝑍𝐢𝑆(π‘˜ 𝑍𝐢𝑆(π‘˜ NUM cana-611 279 37 , , PUNCT cana-611 279 38 𝑙 𝑙 NOUN cana-611 279 39 , , PUNCT cana-611 279 40 π‘š π‘š NOUN cana-611 279 41 ) ) PUNCT cana-611 279 42 , , PUNCT cana-611 279 43 dominating dominate VERB cana-611 279 44 derived derive VERB cana-611 279 45 networks network NOUN cana-611 279 46 𝐷1 𝐷1 NOUN cana-611 279 47 , , PUNCT cana-611 279 48 𝐷2 𝐷2 PROPN cana-611 279 49 , , PUNCT cana-611 279 50 𝐷3 𝐷3 PROPN cana-611 279 51 , , PUNCT cana-611 279 52 porphyrin porphyrin NOUN cana-611 279 53 dendrimer dendrimer NOUN cana-611 279 54 , , PUNCT cana-611 279 55 zinc zinc NOUN cana-611 279 56 - - PUNCT cana-611 279 57 porphyrin porphyrin NOUN cana-611 279 58 dendrimer dendrimer NOUN cana-611 279 59 , , PUNCT cana-611 279 60 propyl propyl ADJ cana-611 279 61 ether ether NOUN cana-611 279 62 imine imine NOUN cana-611 279 63 dendrimer dendrimer NOUN cana-611 279 64 , , PUNCT cana-611 279 65 poly(ethylene poly(ethylene ADJ cana-611 279 66 amido amido NOUN cana-611 279 67 amine amine ADJ cana-611 279 68 dendrimer dendrimer NOUN cana-611 279 69 , , PUNCT cana-611 279 70 pamam pamam NOUN cana-611 279 71 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1 PROPN cana-611 279 72 ) ) PUNCT cana-611 279 73 , , PUNCT cana-611 279 74 linear linear PROPN cana-611 279 75 polyomino polyomino PROPN cana-611 279 76 chain chain NOUN cana-611 279 77 𝐿𝑛 𝐿𝑛 PROPN cana-611 279 78 , , PUNCT cana-611 279 79 𝑍𝑛 𝑍𝑛 PROPN cana-611 279 80 , , PUNCT cana-611 279 81 𝐡𝑛 𝐡𝑛 PROPN cana-611 279 82 1(𝑛 1(𝑛 NUM cana-611 279 83 β‰₯ β‰₯ NOUN cana-611 279 84 3 3 NUM cana-611 279 85 ) ) PUNCT cana-611 279 86 , , PUNCT cana-611 279 87 𝐡𝑛 𝐡𝑛 PROPN cana-611 279 88 2(𝑛 2(𝑛 NUM cana-611 279 89 β‰₯ β‰₯ NOUN cana-611 279 90 3 3 NUM cana-611 279 91 ) ) PUNCT cana-611 279 92 and and CCONJ cana-611 279 93 triangular triangular NOUN cana-611 279 94 , , PUNCT cana-611 279 95 hourglass hourglass NOUN cana-611 279 96 , , PUNCT cana-611 279 97 and and CCONJ cana-611 279 98 jaggedrectangle jaggedrectangle VERB cana-611 279 99 benzenoid benzenoid NOUN cana-611 279 100 systems system NOUN cana-611 279 101 . . PUNCT cana-611 280 1 references reference NOUN cana-611 280 2 [ [ X cana-611 280 3 1 1 NUM cana-611 280 4 ] ] X cana-611 280 5 m.s m.s PROPN cana-611 280 6 . . PROPN cana-611 280 7 ahmad ahmad PROPN cana-611 280 8 , , PUNCT cana-611 280 9 w. w. PROPN cana-611 280 10 nazeer nazeer PROPN cana-611 280 11 , , PUNCT cana-611 280 12 s.m s.m PROPN cana-611 280 13 . . PROPN cana-611 280 14 kang kang PROPN cana-611 280 15 , , PUNCT cana-611 280 16 m. m. NOUN cana-611 280 17 imran imran PROPN cana-611 280 18 , , PUNCT cana-611 280 19 and and CCONJ cana-611 280 20 w. w. PROPN cana-611 280 21 gao gao PROPN cana-611 280 22 , , PUNCT cana-611 280 23 β€œ " PUNCT cana-611 280 24 calculating calculate VERB cana-611 280 25 degree degree NOUN cana-611 280 26 - - PUNCT cana-611 280 27 based base VERB cana-611 280 28 topological topological ADJ cana-611 280 29 indices index NOUN cana-611 280 30 of of ADP cana-611 280 31 dominating dominate VERB cana-611 280 32 david david PROPN cana-611 280 33 derived derive VERB cana-611 280 34 networks network NOUN cana-611 280 35 ” " PUNCT cana-611 280 36 , , PUNCT cana-611 280 37 open open ADJ cana-611 280 38 physics physics NOUN cana-611 280 39 , , PUNCT cana-611 280 40 vol vol NOUN cana-611 280 41 . . PROPN cana-611 280 42 15 15 NUM cana-611 280 43 , , PUNCT cana-611 280 44 no no INTJ cana-611 280 45 . . NOUN cana-611 280 46 1 1 NUM cana-611 280 47 , , PUNCT cana-611 280 48 pp pp ADJ cana-611 280 49 . . PUNCT cana-611 280 50 1015 1015 NUM cana-611 280 51 - - SYM cana-611 280 52 1021 1021 NUM cana-611 280 53 , , PUNCT cana-611 280 54 2017 2017 NUM cana-611 280 55 . . PUNCT cana-611 281 1 [ [ X cana-611 281 2 2 2 X cana-611 281 3 ] ] PUNCT cana-611 281 4 w. w. PROPN cana-611 281 5 gao gao PROPN cana-611 281 6 , , PUNCT cana-611 281 7 m.k m.k PROPN cana-611 281 8 . . PROPN cana-611 281 9 jamil jamil PROPN cana-611 281 10 , , PUNCT cana-611 281 11 a. a. PROPN cana-611 281 12 javed javed PROPN cana-611 281 13 , , PUNCT cana-611 281 14 m.r m.r PROPN cana-611 281 15 . . PROPN cana-611 281 16 farahani farahani PROPN cana-611 281 17 , , PUNCT cana-611 281 18 and and CCONJ cana-611 281 19 m. m. NOUN cana-611 281 20 imran imran PROPN cana-611 281 21 , , PUNCT cana-611 281 22 β€œ " PUNCT cana-611 281 23 inverse inverse NOUN cana-611 281 24 sum sum NOUN cana-611 281 25 indeg indeg NOUN cana-611 281 26 index index NOUN cana-611 281 27 of of ADP cana-611 281 28 the the DET cana-611 281 29 line line NOUN cana-611 281 30 graphs graph NOUN cana-611 281 31 of of ADP cana-611 281 32 subdivision subdivision NOUN cana-611 281 33 graphs graph NOUN cana-611 281 34 of of ADP cana-611 281 35 some some DET cana-611 281 36 chemical chemical ADJ cana-611 281 37 structures structure NOUN cana-611 281 38 ” " PUNCT cana-611 281 39 , , PUNCT cana-611 281 40 upb upb PROPN cana-611 281 41 sci sci PROPN cana-611 281 42 . . PUNCT cana-611 281 43 bulletin bulletin PROPN cana-611 281 44 b b PROPN cana-611 281 45 , , PUNCT cana-611 281 46 vol vol NOUN cana-611 281 47 . . PROPN cana-611 281 48 80 80 NUM cana-611 281 49 , , PUNCT cana-611 281 50 no no INTJ cana-611 281 51 . . NOUN cana-611 281 52 3 3 NUM cana-611 281 53 , , PUNCT cana-611 281 54 pp pp ADJ cana-611 281 55 . . PUNCT cana-611 281 56 97104 97104 NUM cana-611 281 57 , , PUNCT cana-611 281 58 2018 2018 NUM cana-611 281 59 . . PUNCT cana-611 282 1 [ [ X cana-611 282 2 3 3 X cana-611 282 3 ] ] X cana-611 282 4 s.m s.m PROPN cana-611 282 5 . . PROPN cana-611 282 6 hosamani hosamani PROPN cana-611 282 7 , , PUNCT cana-611 282 8 β€œ " PUNCT cana-611 282 9 computing compute VERB cana-611 282 10 sanskruti sanskruti NOUN cana-611 282 11 index index NOUN cana-611 282 12 of of ADP cana-611 282 13 certain certain ADJ cana-611 282 14 nanostructures nanostructure NOUN cana-611 282 15 ” " PUNCT cana-611 282 16 , , PUNCT cana-611 282 17 journal journal NOUN cana-611 282 18 of of ADP cana-611 282 19 applied apply VERB cana-611 282 20 mathematics mathematic NOUN cana-611 282 21 and and CCONJ cana-611 282 22 computing computing NOUN cana-611 282 23 , , PUNCT cana-611 282 24 vol vol NOUN cana-611 282 25 . . PROPN cana-611 282 26 54 54 NUM cana-611 282 27 , , PUNCT cana-611 282 28 pp pp ADJ cana-611 282 29 . . PUNCT cana-611 283 1 425 425 NUM cana-611 283 2 - - SYM cana-611 283 3 433 433 NUM cana-611 283 4 , , PUNCT cana-611 283 5 2017 2017 NUM cana-611 283 6 . . PUNCT cana-611 284 1 [ [ X cana-611 284 2 4 4 NUM cana-611 284 3 ] ] X cana-611 284 4 y.c y.c PROPN cana-611 284 5 . . PROPN cana-611 284 6 kwun kwun PROPN cana-611 284 7 , , PUNCT cana-611 284 8 a. a. PROPN cana-611 284 9 farooq farooq PROPN cana-611 284 10 , , PUNCT cana-611 284 11 w. w. PROPN cana-611 284 12 nazeer nazeer PROPN cana-611 284 13 , , PUNCT cana-611 284 14 z. z. PROPN cana-611 284 15 zahid zahid PROPN cana-611 284 16 , , PUNCT cana-611 284 17 s. s. PROPN cana-611 284 18 noreen noreen PROPN cana-611 284 19 , , PUNCT cana-611 284 20 and and CCONJ cana-611 284 21 s.m s.m PROPN cana-611 284 22 . . PROPN cana-611 284 23 kang kang PROPN cana-611 284 24 , , PUNCT cana-611 284 25 β€œ " PUNCT cana-611 284 26 computations computation NOUN cana-611 284 27 of of ADP cana-611 284 28 the the DET cana-611 284 29 mpolynomials mpolynomial NOUN cana-611 284 30 and and CCONJ cana-611 284 31 degree degree NOUN cana-611 284 32 - - PUNCT cana-611 284 33 based base VERB cana-611 284 34 topological topological ADJ cana-611 284 35 indices index NOUN cana-611 284 36 for for ADP cana-611 284 37 dendrimers dendrimer NOUN cana-611 284 38 and and CCONJ cana-611 284 39 polyomino polyomino NOUN cana-611 284 40 chains chain NOUN cana-611 284 41 ” " PUNCT cana-611 284 42 , , PUNCT cana-611 284 43 international international ADJ cana-611 284 44 journal journal NOUN cana-611 284 45 of of ADP cana-611 284 46 analytical analytical ADJ cana-611 284 47 chemistry chemistry NOUN cana-611 284 48 , , PUNCT cana-611 284 49 vol vol NOUN cana-611 284 50 . . PROPN cana-611 284 51 2018 2018 NUM cana-611 284 52 , , PUNCT cana-611 284 53 2018 2018 NUM cana-611 284 54 . . PUNCT cana-611 285 1 [ [ X cana-611 285 2 5 5 NUM cana-611 285 3 ] ] X cana-611 285 4 y.c y.c PROPN cana-611 285 5 . . PROPN cana-611 285 6 kwun kwun PROPN cana-611 285 7 , , PUNCT cana-611 285 8 a. a. PROPN cana-611 285 9 ali ali PROPN cana-611 285 10 , , PUNCT cana-611 285 11 w. w. PROPN cana-611 285 12 nazeer nazeer PROPN cana-611 285 13 , , PUNCT cana-611 285 14 m. m. PROPN cana-611 285 15 ahmad ahmad PROPN cana-611 285 16 chaudhary chaudhary PROPN cana-611 285 17 , , PUNCT cana-611 285 18 and and CCONJ cana-611 285 19 s.m s.m PROPN cana-611 285 20 . . PROPN cana-611 285 21 kang kang PROPN cana-611 285 22 , , PUNCT cana-611 285 23 m m NOUN cana-611 285 24 - - NOUN cana-611 285 25 polynomials polynomial NOUN cana-611 285 26 and and CCONJ cana-611 285 27 degreebased degreebase VERB cana-611 285 28 topological topological ADJ cana-611 285 29 indices index NOUN cana-611 285 30 of of ADP cana-611 285 31 triangular triangular NOUN cana-611 285 32 , , PUNCT cana-611 285 33 hourglass hourglass NOUN cana-611 285 34 , , PUNCT cana-611 285 35 and and CCONJ cana-611 285 36 jagged jagged ADJ cana-611 285 37 - - PUNCT cana-611 285 38 rectangle rectangle NOUN cana-611 285 39 benzenoid benzenoid NOUN cana-611 285 40 systems system NOUN cana-611 285 41 . . PUNCT cana-611 286 1 journal journal NOUN cana-611 286 2 of of ADP cana-611 286 3 chemistry chemistry NOUN cana-611 286 4 , , PUNCT cana-611 286 5 2018 2018 NUM cana-611 286 6 , , PUNCT cana-611 286 7 2018 2018 NUM cana-611 286 8 . . PUNCT cana-611 287 1 [ [ X cana-611 287 2 6 6 NUM cana-611 287 3 ] ] PUNCT cana-611 287 4 f. f. PROPN cana-611 287 5 harary harary PROPN cana-611 287 6 , , PUNCT cana-611 287 7 graph graph NOUN cana-611 287 8 theory theory NOUN cana-611 287 9 ( ( PUNCT cana-611 287 10 addison addison PROPN cana-611 287 11 - - PUNCT cana-611 287 12 weslay weslay ADJ cana-611 287 13 publ publ NOUN cana-611 287 14 . . PUNCT cana-611 288 1 comp comp NOUN cana-611 288 2 . . PUNCT cana-611 289 1 reading read VERB cana-611 289 2 massachusets massachusets PROPN cana-611 289 3 ) ) PUNCT cana-611 289 4 , , PUNCT cana-611 289 5 1969 1969 NUM cana-611 289 6 . . PUNCT cana-611 290 1 [ [ X cana-611 290 2 7 7 X cana-611 290 3 ] ] X cana-611 290 4 a.r a.r PROPN cana-611 290 5 . . PROPN cana-611 290 6 ashrafi ashrafi PROPN cana-611 290 7 , , PUNCT cana-611 290 8 t. t. NOUN cana-611 290 9 doΕ‘liΔ‡ doΕ‘liΔ‡ NOUN cana-611 290 10 , , PUNCT cana-611 290 11 and and CCONJ cana-611 290 12 a. a. NOUN cana-611 290 13 hamzeh hamzeh PROPN cana-611 290 14 , , PUNCT cana-611 290 15 β€œ " PUNCT cana-611 290 16 the the DET cana-611 290 17 zagreb zagreb PROPN cana-611 290 18 coindices coindice NOUN cana-611 290 19 of of ADP cana-611 290 20 graph graph NOUN cana-611 290 21 operations operation NOUN cana-611 290 22 ” " PUNCT cana-611 290 23 , , PUNCT cana-611 290 24 discrete discrete ADJ cana-611 290 25 applied apply VERB cana-611 290 26 mathematics mathematic NOUN cana-611 290 27 , , PUNCT cana-611 290 28 vol vol NOUN cana-611 290 29 . . PROPN cana-611 290 30 158 158 NUM cana-611 290 31 , , PUNCT cana-611 290 32 no no INTJ cana-611 290 33 . . NOUN cana-611 290 34 15 15 NUM cana-611 290 35 , , PUNCT cana-611 290 36 pp pp ADJ cana-611 290 37 . . PUNCT cana-611 291 1 1571 1571 NUM cana-611 291 2 - - SYM cana-611 291 3 1578 1578 NUM cana-611 291 4 , , PUNCT cana-611 291 5 2010 2010 NUM cana-611 291 6 . . PUNCT cana-611 292 1 [ [ X cana-611 292 2 8 8 NUM cana-611 292 3 ] ] X cana-611 292 4 b. b. PROPN cana-611 292 5 basavanagoud basavanagoud PROPN cana-611 292 6 , , PUNCT cana-611 292 7 i. i. PROPN cana-611 292 8 gutman gutman PROPN cana-611 292 9 , , PUNCT cana-611 292 10 and and CCONJ cana-611 292 11 c.s c.s PROPN cana-611 292 12 . . PROPN cana-611 292 13 gali gali PROPN cana-611 292 14 , , PUNCT cana-611 292 15 β€œ " PUNCT cana-611 292 16 on on ADP cana-611 292 17 second second ADJ cana-611 292 18 zagreb zagreb PROPN cana-611 292 19 index index NOUN cana-611 292 20 and and CCONJ cana-611 292 21 coindex coindex NOUN cana-611 292 22 of of ADP cana-611 292 23 some some DET cana-611 292 24 derived derive VERB cana-611 292 25 graphs graph NOUN cana-611 292 26 ” " PUNCT cana-611 292 27 , , PUNCT cana-611 292 28 kragujevac kragujevac PROPN cana-611 292 29 j. j. PROPN cana-611 292 30 sci sci PROPN cana-611 292 31 , , PUNCT cana-611 292 32 vol vol NOUN cana-611 292 33 . . PROPN cana-611 292 34 37 37 NUM cana-611 292 35 , , PUNCT cana-611 292 36 pp.113 pp.113 PROPN cana-611 292 37 - - PUNCT cana-611 292 38 121 121 NUM cana-611 292 39 , , PUNCT cana-611 292 40 2015 2015 NUM cana-611 292 41 . . PUNCT cana-611 293 1 [ [ X cana-611 293 2 9 9 NUM cana-611 293 3 ] ] PUNCT cana-611 293 4 furtula furtula NOUN cana-611 293 5 , , PUNCT cana-611 293 6 b. b. PROPN cana-611 293 7 and and CCONJ cana-611 293 8 gutman gutman PROPN cana-611 293 9 , , PUNCT cana-611 293 10 i. i. NOUN cana-611 293 11 , , PUNCT cana-611 293 12 2015 2015 NUM cana-611 293 13 . . PUNCT cana-611 294 1 a a DET cana-611 294 2 forgotten forget VERB cana-611 294 3 topological topological ADJ cana-611 294 4 index index NOUN cana-611 294 5 . . PUNCT cana-611 295 1 journal journal PROPN cana-611 295 2 of of ADP cana-611 295 3 mathematical mathematical ADJ cana-611 295 4 chemistry chemistry NOUN cana-611 295 5 , , PUNCT cana-611 295 6 53(4 53(4 NOUN cana-611 295 7 ) ) PUNCT cana-611 295 8 , , PUNCT cana-611 295 9 pp.1184 pp.1184 PROPN cana-611 295 10 - - NOUN cana-611 295 11 1190 1190 NUM cana-611 295 12 . . PUNCT cana-611 296 1 [ [ X cana-611 296 2 10 10 NUM cana-611 296 3 ] ] X cana-611 296 4 gutman gutman NOUN cana-611 296 5 , , PUNCT cana-611 296 6 i. i. PROPN cana-611 296 7 and and CCONJ cana-611 296 8 trinajstiΔ‡ trinajstiΔ‡ PROPN cana-611 296 9 , , PUNCT cana-611 296 10 n. n. NOUN cana-611 296 11 , , PUNCT cana-611 296 12 1972 1972 NUM cana-611 296 13 . . PUNCT cana-611 297 1 graph graph NOUN cana-611 297 2 theory theory NOUN cana-611 297 3 and and CCONJ cana-611 297 4 molecular molecular ADJ cana-611 297 5 orbitals orbital NOUN cana-611 297 6 . . PUNCT cana-611 298 1 total total ADJ cana-611 298 2 Ο† Ο† NUM cana-611 298 3 - - PUNCT cana-611 298 4 electron electron NOUN cana-611 298 5 energy energy NOUN cana-611 298 6 of of ADP cana-611 298 7 alternant alternant ADJ cana-611 298 8 hydrocarbons hydrocarbon NOUN cana-611 298 9 . . PUNCT cana-611 299 1 chemical chemical PROPN cana-611 299 2 physics physics PROPN cana-611 299 3 letters letter NOUN cana-611 299 4 , , PUNCT cana-611 299 5 17(4 17(4 NUM cana-611 299 6 ) ) PUNCT cana-611 299 7 , , PUNCT cana-611 299 8 pp.535 pp.535 PROPN cana-611 299 9 - - PUNCT cana-611 299 10 538 538 NUM cana-611 299 11 . . PUNCT cana-611 300 1 [ [ X cana-611 300 2 11 11 NUM cana-611 300 3 ] ] PUNCT cana-611 300 4 s. s. PROPN cana-611 300 5 m. m. PROPN cana-611 300 6 hosamani hosamani PROPN cana-611 300 7 , , PUNCT cana-611 300 8 v. v. PROPN cana-611 300 9 b. b. PROPN cana-611 300 10 awati awati PROPN cana-611 300 11 and and CCONJ cana-611 300 12 r. r. PROPN cana-611 300 13 s. s. PROPN cana-611 300 14 honamore honamore PROPN cana-611 300 15 , , PUNCT cana-611 300 16 β€œ " PUNCT cana-611 300 17 estimation estimation NOUN cana-611 300 18 of of ADP cana-611 300 19 numerical numerical ADJ cana-611 300 20 invariants invariant NOUN cana-611 300 21 associated associate VERB cana-611 300 22 with with ADP cana-611 300 23 nanostructures nanostructure NOUN cana-611 300 24 and and CCONJ cana-611 300 25 dendrimers dendrimer NOUN cana-611 300 26 via via ADP cana-611 300 27 degree degree NOUN cana-611 300 28 based base VERB cana-611 300 29 descriptors descriptor NOUN cana-611 300 30 ” " PUNCT cana-611 300 31 , , PUNCT cana-611 300 32 researchgate researchgate NOUN cana-611 300 33 https://www.researchgate.net/publication/337151723 https://www.researchgate.net/publication/337151723 NOUN cana-611 300 34 [ [ X cana-611 300 35 12 12 NUM cana-611 300 36 ] ] X cana-611 300 37 s.m s.m PROPN cana-611 300 38 . . PROPN cana-611 300 39 hosamani hosamani PROPN cana-611 300 40 , , PUNCT cana-611 300 41 and and CCONJ cana-611 300 42 i. i. PROPN cana-611 300 43 gutman gutman PROPN cana-611 300 44 , , PUNCT cana-611 300 45 β€œ " PUNCT cana-611 300 46 zagreb zagreb PROPN cana-611 300 47 indices index NOUN cana-611 300 48 of of ADP cana-611 300 49 transformation transformation NOUN cana-611 300 50 graphs graph NOUN cana-611 300 51 and and CCONJ cana-611 300 52 total total ADJ cana-611 300 53 transformation transformation NOUN cana-611 300 54 graphs graph NOUN cana-611 300 55 ” " PUNCT cana-611 300 56 , , PUNCT cana-611 300 57 applied apply VERB cana-611 300 58 mathematics mathematic NOUN cana-611 300 59 and and CCONJ cana-611 300 60 computation computation NOUN cana-611 300 61 , , PUNCT cana-611 300 62 vol vol NOUN cana-611 300 63 . . PROPN cana-611 300 64 247 247 NUM cana-611 300 65 , , PUNCT cana-611 300 66 pp.1156 pp.1156 PROPN cana-611 300 67 - - SYM cana-611 300 68 1160 1160 NUM cana-611 300 69 , , PUNCT cana-611 300 70 2014 2014 NUM cana-611 300 71 . . PUNCT cana-611 301 1 [ [ X cana-611 301 2 13 13 NUM cana-611 301 3 ] ] SYM cana-611 301 4 s.m s.m PROPN cana-611 301 5 . . PROPN cana-611 301 6 hosamani hosamani PROPN cana-611 301 7 , , PUNCT cana-611 301 8 and and CCONJ cana-611 301 9 b. b. PROPN cana-611 301 10 basavanagoud basavanagoud NOUN cana-611 301 11 , , PUNCT cana-611 301 12 β€œ " PUNCT cana-611 301 13 new new ADJ cana-611 301 14 upper upper ADJ cana-611 301 15 bounds bound NOUN cana-611 301 16 for for ADP cana-611 301 17 the the DET cana-611 301 18 first first ADJ cana-611 301 19 zagreb zagreb PROPN cana-611 301 20 index index PROPN cana-611 301 21 ” " PUNCT cana-611 301 22 , , PUNCT cana-611 301 23 match match VERB cana-611 301 24 commun commun PROPN cana-611 301 25 . . PUNCT cana-611 302 1 math math PROPN cana-611 302 2 . . PUNCT cana-611 303 1 comput comput NOUN cana-611 303 2 . . PUNCT cana-611 304 1 chem chem NOUN cana-611 304 2 , , PUNCT cana-611 304 3 vol vol NOUN cana-611 304 4 . . PROPN cana-611 305 1 74 74 NUM cana-611 305 2 , , PUNCT cana-611 305 3 no no INTJ cana-611 305 4 . . NOUN cana-611 305 5 1 1 NUM cana-611 305 6 , , PUNCT cana-611 305 7 pp pp ADJ cana-611 305 8 . . PUNCT cana-611 306 1 97 97 NUM cana-611 306 2 - - SYM cana-611 306 3 101 101 NUM cana-611 306 4 , , PUNCT cana-611 306 5 2015 2015 NUM cana-611 306 6 . . PUNCT cana-611 307 1 [ [ X cana-611 307 2 14 14 NUM cana-611 307 3 ] ] X cana-611 307 4 s.m s.m PROPN cana-611 307 5 . . PROPN cana-611 307 6 hosamani hosamani PROPN cana-611 307 7 , , PUNCT cana-611 307 8 s.h s.h PROPN cana-611 307 9 . . PROPN cana-611 307 10 malghan malghan PROPN cana-611 307 11 , , PUNCT cana-611 307 12 and and CCONJ cana-611 307 13 i.n i.n PROPN cana-611 307 14 . . PROPN cana-611 307 15 cangul cangul PROPN cana-611 307 16 , , PUNCT cana-611 307 17 β€œ " PUNCT cana-611 307 18 the the DET cana-611 307 19 first first ADJ cana-611 307 20 geometric geometric ADJ cana-611 307 21 - - PUNCT cana-611 307 22 arithmetic arithmetic ADJ cana-611 307 23 index index NOUN cana-611 307 24 of of ADP cana-611 307 25 graph graph NOUN cana-611 307 26 operations operation NOUN cana-611 307 27 ” " PUNCT cana-611 307 28 , , PUNCT cana-611 307 29 advances advance NOUN cana-611 307 30 and and CCONJ cana-611 307 31 applications application NOUN cana-611 307 32 in in ADP cana-611 307 33 mathematical mathematical ADJ cana-611 307 34 sciences science NOUN cana-611 307 35 , , PUNCT cana-611 307 36 vol vol NOUN cana-611 307 37 . . PROPN cana-611 307 38 14 14 NUM cana-611 307 39 , , PUNCT cana-611 307 40 no no INTJ cana-611 307 41 . . NOUN cana-611 307 42 6 6 NUM cana-611 307 43 , , PUNCT cana-611 307 44 p.155 p.155 NOUN cana-611 307 45 , , PUNCT cana-611 307 46 2015 2015 NUM cana-611 307 47 . . PUNCT cana-611 308 1 [ [ X cana-611 308 2 15 15 NUM cana-611 308 3 ] ] X cana-611 308 4 s.m s.m PROPN cana-611 308 5 . . PROPN cana-611 308 6 kang kang PROPN cana-611 308 7 , , PUNCT cana-611 308 8 m.a m.a PROPN cana-611 308 9 . . PROPN cana-611 308 10 zahid zahid PROPN cana-611 308 11 , , PUNCT cana-611 308 12 a.u.r a.u.r PROPN cana-611 308 13 . . PUNCT cana-611 308 14 virk virk PROPN cana-611 308 15 , , PUNCT cana-611 308 16 w. w. PROPN cana-611 308 17 nazeer nazeer PROPN cana-611 308 18 , , PUNCT cana-611 308 19 and and CCONJ cana-611 308 20 w. w. PROPN cana-611 308 21 gao gao PROPN cana-611 308 22 , , PUNCT cana-611 308 23 β€œ " PUNCT cana-611 308 24 calculating calculate VERB cana-611 308 25 the the DET cana-611 308 26 degree degree NOUN cana-611 308 27 - - PUNCT cana-611 308 28 based base VERB cana-611 308 29 topological topological ADJ cana-611 308 30 indices index NOUN cana-611 308 31 of of ADP cana-611 308 32 dendrimers dendrimer NOUN cana-611 308 33 ” " PUNCT cana-611 308 34 , , PUNCT cana-611 308 35 open open ADJ cana-611 308 36 chemistry chemistry NOUN cana-611 308 37 , , PUNCT cana-611 308 38 vol vol NOUN cana-611 308 39 . . PROPN cana-611 308 40 16 16 NUM cana-611 308 41 , , PUNCT cana-611 308 42 no no INTJ cana-611 308 43 . . NOUN cana-611 308 44 1 1 NUM cana-611 308 45 , , PUNCT cana-611 308 46 pp pp ADJ cana-611 308 47 . . PUNCT cana-611 309 1 681 681 NUM cana-611 309 2 - - SYM cana-611 309 3 688 688 NUM cana-611 309 4 , , PUNCT cana-611 309 5 2018 2018 NUM cana-611 309 6 . . PUNCT cana-611 310 1 [ [ X cana-611 310 2 16 16 NUM cana-611 310 3 ] ] PUNCT cana-611 310 4 x. x. NOUN cana-611 310 5 li li PROPN cana-611 310 6 , , PUNCT cana-611 310 7 and and CCONJ cana-611 310 8 j. j. PROPN cana-611 310 9 zheng zheng PROPN cana-611 310 10 , , PUNCT cana-611 310 11 β€œ " PUNCT cana-611 310 12 a a DET cana-611 310 13 unified unified ADJ cana-611 310 14 approach approach NOUN cana-611 310 15 to to ADP cana-611 310 16 the the DET cana-611 310 17 extremal extremal ADJ cana-611 310 18 trees tree NOUN cana-611 310 19 for for ADP cana-611 310 20 different different ADJ cana-611 310 21 indices index NOUN cana-611 310 22 ” " PUNCT cana-611 310 23 , , PUNCT cana-611 310 24 match match NOUN cana-611 310 25 commun commun PROPN cana-611 310 26 . . PUNCT cana-611 310 27 math math PROPN cana-611 310 28 . . PUNCT cana-611 311 1 comput comput NOUN cana-611 311 2 . . PUNCT cana-611 312 1 chem chem NOUN cana-611 312 2 , , PUNCT cana-611 312 3 vol vol NOUN cana-611 312 4 . . PROPN cana-611 312 5 54 54 NUM cana-611 312 6 , , PUNCT cana-611 312 7 no no INTJ cana-611 312 8 . . NOUN cana-611 312 9 1 1 NUM cana-611 312 10 , , PUNCT cana-611 312 11 pp pp ADJ cana-611 312 12 . . PUNCT cana-611 313 1 195 195 NUM cana-611 313 2 - - SYM cana-611 313 3 208 208 NUM cana-611 313 4 , , PUNCT cana-611 313 5 2005 2005 NUM cana-611 313 6 . . PUNCT cana-611 314 1 [ [ X cana-611 314 2 17 17 NUM cana-611 314 3 ] ] X cana-611 314 4 m.f m.f PROPN cana-611 314 5 . . PROPN cana-611 314 6 nadeem nadeem PROPN cana-611 314 7 , , PUNCT cana-611 314 8 s. s. PROPN cana-611 314 9 zafar zafar PROPN cana-611 314 10 and and CCONJ cana-611 314 11 z. z. PROPN cana-611 314 12 zahid zahid PROPN cana-611 314 13 , , PUNCT cana-611 314 14 β€œ " PUNCT cana-611 314 15 on on ADP cana-611 314 16 certain certain ADJ cana-611 314 17 topological topological ADJ cana-611 314 18 indices index NOUN cana-611 314 19 of of ADP cana-611 314 20 the the DET cana-611 314 21 line line NOUN cana-611 314 22 graph graph NOUN cana-611 314 23 of of ADP cana-611 314 24 subdivision subdivision NOUN cana-611 314 25 graphs graph NOUN cana-611 314 26 ” " PUNCT cana-611 314 27 , , PUNCT cana-611 314 28 applied apply VERB cana-611 314 29 mathematics mathematic NOUN cana-611 314 30 and and CCONJ cana-611 314 31 computation computation NOUN cana-611 314 32 , , PUNCT cana-611 314 33 vol vol NOUN cana-611 314 34 . . PROPN cana-611 314 35 271 271 NUM cana-611 314 36 , , PUNCT cana-611 314 37 pp pp ADJ cana-611 314 38 . . PUNCT cana-611 315 1 790 790 NUM cana-611 315 2 - - SYM cana-611 315 3 794 794 NUM cana-611 315 4 , , PUNCT cana-611 315 5 2015 2015 NUM cana-611 315 6 . . PUNCT cana-611 316 1 communications communication NOUN cana-611 316 2 on on ADP cana-611 316 3 applied apply VERB cana-611 316 4 nonlinear nonlinear ADJ cana-611 316 5 analysis analysis NOUN cana-611 316 6 issn issn NOUN cana-611 316 7 : : PUNCT cana-611 316 8 1074 1074 NUM cana-611 316 9 - - PUNCT cana-611 316 10 133x 133x NUM cana-611 316 11 vol vol NOUN cana-611 316 12 31 31 NUM cana-611 316 13 no no NOUN cana-611 316 14 . . NOUN cana-611 316 15 2 2 NUM cana-611 316 16 ( ( PUNCT cana-611 316 17 2024 2024 NUM cana-611 316 18 ) ) PUNCT cana-611 316 19 429 429 NUM cana-611 316 20 https://internationalpubls.com https://internationalpubls.com X cana-611 317 1 [ [ X cana-611 317 2 18 18 NUM cana-611 317 3 ] ] X cana-611 317 4 p.s p.s PROPN cana-611 317 5 . . PROPN cana-611 317 6 ranjini ranjini PROPN cana-611 317 7 , , PUNCT cana-611 317 8 v. v. ADP cana-611 317 9 lokesha lokesha PROPN cana-611 317 10 and and CCONJ cana-611 317 11 i.n i.n PROPN cana-611 317 12 . . PROPN cana-611 317 13 cangΓΌl cangΓΌl PROPN cana-611 317 14 , , PUNCT cana-611 317 15 β€œ " PUNCT cana-611 317 16 on on ADP cana-611 317 17 the the DET cana-611 317 18 zagreb zagreb PROPN cana-611 317 19 indices index NOUN cana-611 317 20 of of ADP cana-611 317 21 the the DET cana-611 317 22 line line NOUN cana-611 317 23 graphs graph NOUN cana-611 317 24 of of ADP cana-611 317 25 the the DET cana-611 317 26 subdivision subdivision NOUN cana-611 317 27 graphs graph NOUN cana-611 317 28 ” " PUNCT cana-611 317 29 , , PUNCT cana-611 317 30 applied apply VERB cana-611 317 31 mathematics mathematic NOUN cana-611 317 32 and and CCONJ cana-611 317 33 computation computation NOUN cana-611 317 34 , , PUNCT cana-611 317 35 vol vol NOUN cana-611 317 36 . . PROPN cana-611 317 37 218 218 NUM cana-611 317 38 , , PUNCT cana-611 317 39 no no INTJ cana-611 317 40 . . NOUN cana-611 317 41 3 3 NUM cana-611 317 42 , , PUNCT cana-611 317 43 pp pp ADJ cana-611 317 44 . . PUNCT cana-611 318 1 699 699 NUM cana-611 318 2 - - SYM cana-611 318 3 702 702 NUM cana-611 318 4 , , PUNCT cana-611 318 5 2011 2011 NUM cana-611 318 6 . . PUNCT cana-611 319 1 [ [ X cana-611 319 2 19 19 NUM cana-611 319 3 ] ] X cana-611 319 4 g. g. PROPN cana-611 319 5 su su PROPN cana-611 319 6 , , PUNCT cana-611 319 7 and and CCONJ cana-611 319 8 l. l. PROPN cana-611 319 9 xu xu PROPN cana-611 319 10 , , PUNCT cana-611 319 11 β€œ " PUNCT cana-611 319 12 topological topological ADJ cana-611 319 13 indices index NOUN cana-611 319 14 of of ADP cana-611 319 15 the the DET cana-611 319 16 line line NOUN cana-611 319 17 graph graph NOUN cana-611 319 18 of of ADP cana-611 319 19 subdivision subdivision NOUN cana-611 319 20 graphs graph NOUN cana-611 319 21 and and CCONJ cana-611 319 22 their their PRON cana-611 319 23 schurbounds schurbound NOUN cana-611 319 24 . . PUNCT cana-611 320 1 applied apply VERB cana-611 320 2 mathematics mathematic NOUN cana-611 320 3 and and CCONJ cana-611 320 4 computation computation NOUN cana-611 320 5 , , PUNCT cana-611 320 6 vol vol NOUN cana-611 320 7 . . PROPN cana-611 320 8 253 253 NUM cana-611 320 9 , , PUNCT cana-611 320 10 pp pp ADJ cana-611 320 11 . . PUNCT cana-611 321 1 395 395 NUM cana-611 321 2 - - SYM cana-611 321 3 401 401 NUM cana-611 321 4 , , PUNCT cana-611 321 5 2015 2015 NUM cana-611 321 6 . . PUNCT cana-611 322 1 [ [ X cana-611 322 2 20 20 NUM cana-611 322 3 ] ] X cana-611 322 4 h. h. PROPN cana-611 322 5 timmerman timmerman PROPN cana-611 322 6 , , PUNCT cana-611 322 7 r. r. PROPN cana-611 322 8 todeschini todeschini PROPN cana-611 322 9 , , PUNCT cana-611 322 10 v. v. PROPN cana-611 322 11 consonni consonni PROPN cana-611 322 12 , , PUNCT cana-611 322 13 r. r. PROPN cana-611 322 14 mannhold mannhold PROPN cana-611 322 15 , , PUNCT cana-611 322 16 and and CCONJ cana-611 322 17 h. h. PROPN cana-611 322 18 kubinyi kubinyi PROPN cana-611 322 19 , , PUNCT cana-611 322 20 β€œ " PUNCT cana-611 322 21 handbook handbook NOUN cana-611 322 22 of of ADP cana-611 322 23 molecular molecular ADJ cana-611 322 24 descriptors descriptor NOUN cana-611 322 25 ” " PUNCT cana-611 322 26 , , PUNCT cana-611 322 27 2002 2002 NUM cana-611 322 28 . . PUNCT cana-611 323 1 [ [ X cana-611 323 2 21 21 NUM cana-611 323 3 ] ] X cana-611 323 4 r. r. PROPN cana-611 323 5 todeschini todeschini PROPN cana-611 323 6 , , PUNCT cana-611 323 7 d. d. PROPN cana-611 323 8 ballabio ballabio PROPN cana-611 323 9 , , PUNCT cana-611 323 10 and and CCONJ cana-611 323 11 v. v. ADP cana-611 323 12 consonni consonni NOUN cana-611 323 13 , , PUNCT cana-611 323 14 β€œ " PUNCT cana-611 323 15 novel novel ADJ cana-611 323 16 molecular molecular ADJ cana-611 323 17 descriptors descriptor NOUN cana-611 323 18 based base VERB cana-611 323 19 on on ADP cana-611 323 20 functions function NOUN cana-611 323 21 of of ADP cana-611 323 22 new new ADJ cana-611 323 23 vertex vertex NOUN cana-611 323 24 degrees degree NOUN cana-611 323 25 ” " PUNCT cana-611 323 26 , , PUNCT cana-611 323 27 mathematical mathematical ADJ cana-611 323 28 chemistry chemistry NOUN cana-611 323 29 monographs monograph NOUN cana-611 323 30 , , PUNCT cana-611 323 31 pp pp ADJ cana-611 323 32 . . PUNCT cana-611 324 1 73 73 NUM cana-611 324 2 - - SYM cana-611 324 3 100 100 NUM cana-611 324 4 , , PUNCT cana-611 324 5 2010 2010 NUM cana-611 324 6 . . PUNCT cana-611 325 1 [ [ X cana-611 325 2 22 22 NUM cana-611 325 3 ] ] X cana-611 325 4 r. r. PROPN cana-611 325 5 todeschini todeschini PROPN cana-611 325 6 , , PUNCT cana-611 325 7 and and CCONJ cana-611 325 8 v. v. ADP cana-611 325 9 consonni consonni PROPN cana-611 325 10 , , PUNCT cana-611 325 11 new new ADJ cana-611 325 12 local local ADJ cana-611 325 13 vertex vertex NOUN cana-611 325 14 invariants invariant NOUN cana-611 325 15 and and CCONJ cana-611 325 16 molecular molecular ADJ cana-611 325 17 descriptors descriptor NOUN cana-611 325 18 based base VERB cana-611 325 19 on on ADP cana-611 325 20 functions function NOUN cana-611 325 21 of of ADP cana-611 325 22 the the DET cana-611 325 23 vertex vertex NOUN cana-611 325 24 degrees degree NOUN cana-611 325 25 . . PUNCT cana-611 326 1 match match PROPN cana-611 326 2 commun commun PROPN cana-611 326 3 . . PUNCT cana-611 327 1 math math PROPN cana-611 327 2 . . PUNCT cana-611 328 1 comput comput NOUN cana-611 328 2 . . PUNCT cana-611 329 1 chem chem NOUN cana-611 329 2 , , PUNCT cana-611 329 3 vol vol NOUN cana-611 329 4 . . PROPN cana-611 329 5 64 64 NUM cana-611 329 6 , , PUNCT cana-611 329 7 no no INTJ cana-611 329 8 . . NOUN cana-611 329 9 2 2 NUM cana-611 329 10 , , PUNCT cana-611 329 11 pp.359 pp.359 VERB cana-611 329 12 - - PUNCT cana-611 329 13 372 372 NUM cana-611 329 14 , , PUNCT cana-611 329 15 2010 2010 NUM cana-611 329 16 . . PUNCT