id sid tid token lemma pos iajs-3014 1 1 ihjpas ihjpas PROPN iajs-3014 1 2 . . PUNCT iajs-3014 2 1 36(1)2023 36(1)2023 NUM iajs-3014 2 2 284 284 NUM iajs-3014 2 3 this this DET iajs-3014 2 4 work work NOUN iajs-3014 2 5 is be AUX iajs-3014 2 6 licensed license VERB iajs-3014 2 7 under under ADP iajs-3014 2 8 a a DET iajs-3014 2 9 creative creative ADJ iajs-3014 2 10 commons common NOUN iajs-3014 2 11 attribution attribution NOUN iajs-3014 2 12 4.0 4.0 NUM iajs-3014 2 13 international international ADJ iajs-3014 2 14 license license NOUN iajs-3014 2 15 on on ADP iajs-3014 2 16 antimagic antimagic ADJ iajs-3014 2 17 labeling labeling NOUN iajs-3014 2 18 for for ADP iajs-3014 2 19 some some DET iajs-3014 2 20 families family NOUN iajs-3014 2 21 of of ADP iajs-3014 2 22 graphs graph NOUN iajs-3014 2 23 abstract abstract ADJ iajs-3014 2 24 antimagic antimagic ADJ iajs-3014 2 25 labeling labeling NOUN iajs-3014 2 26 of of ADP iajs-3014 2 27 a a DET iajs-3014 2 28 graph graph NOUN iajs-3014 2 29 𝐺 𝐺 NOUN iajs-3014 2 30 with with ADP iajs-3014 2 31 𝑝 𝑝 PROPN iajs-3014 2 32 vertices vertex NOUN iajs-3014 2 33 and and CCONJ iajs-3014 2 34 π‘ž π‘ž PRON iajs-3014 2 35 edges edge NOUN iajs-3014 2 36 is be AUX iajs-3014 2 37 assigned assign VERB iajs-3014 2 38 the the DET iajs-3014 2 39 labels label NOUN iajs-3014 2 40 for for ADP iajs-3014 2 41 its its PRON iajs-3014 2 42 edges edge NOUN iajs-3014 2 43 by by ADP iajs-3014 2 44 some some DET iajs-3014 2 45 integers integer NOUN iajs-3014 2 46 from from ADP iajs-3014 2 47 the the DET iajs-3014 2 48 set set NOUN iajs-3014 2 49 { { PUNCT iajs-3014 2 50 1,2 1,2 NUM iajs-3014 2 51 , , PUNCT iajs-3014 2 52 … … PUNCT iajs-3014 2 53 , , PUNCT iajs-3014 2 54 π‘ž π‘ž X iajs-3014 2 55 } } PUNCT iajs-3014 2 56 , , PUNCT iajs-3014 2 57 such such ADJ iajs-3014 2 58 that that SCONJ iajs-3014 2 59 no no DET iajs-3014 2 60 two two NUM iajs-3014 2 61 edges edge NOUN iajs-3014 2 62 received receive VERB iajs-3014 2 63 the the DET iajs-3014 2 64 same same ADJ iajs-3014 2 65 label label NOUN iajs-3014 2 66 , , PUNCT iajs-3014 2 67 and and CCONJ iajs-3014 2 68 the the DET iajs-3014 2 69 weights weight NOUN iajs-3014 2 70 of of ADP iajs-3014 2 71 vertices vertex NOUN iajs-3014 2 72 of of ADP iajs-3014 2 73 a a DET iajs-3014 2 74 graph graph NOUN iajs-3014 2 75 𝐺 𝐺 NOUN iajs-3014 2 76 are be AUX iajs-3014 2 77 pairwise pairwise NOUN iajs-3014 2 78 distinct distinct NOUN iajs-3014 2 79 . . PUNCT iajs-3014 3 1 where where SCONJ iajs-3014 3 2 the the DET iajs-3014 3 3 vertex vertex NOUN iajs-3014 3 4 - - PUNCT iajs-3014 3 5 weights weight NOUN iajs-3014 3 6 of of ADP iajs-3014 3 7 a a DET iajs-3014 3 8 vertex vertex NOUN iajs-3014 3 9 𝑣 𝑣 ADP iajs-3014 3 10 under under ADP iajs-3014 3 11 this this DET iajs-3014 3 12 labeling labeling NOUN iajs-3014 3 13 is be AUX iajs-3014 3 14 the the DET iajs-3014 3 15 sum sum NOUN iajs-3014 3 16 of of ADP iajs-3014 3 17 labels label NOUN iajs-3014 3 18 of of ADP iajs-3014 3 19 all all DET iajs-3014 3 20 edges edge NOUN iajs-3014 3 21 incident incident NOUN iajs-3014 3 22 to to ADP iajs-3014 3 23 this this DET iajs-3014 3 24 vertex vertex NOUN iajs-3014 3 25 , , PUNCT iajs-3014 3 26 in in ADP iajs-3014 3 27 this this DET iajs-3014 3 28 paper paper NOUN iajs-3014 3 29 , , PUNCT iajs-3014 3 30 we we PRON iajs-3014 3 31 deal deal VERB iajs-3014 3 32 with with ADP iajs-3014 3 33 the the DET iajs-3014 3 34 problem problem NOUN iajs-3014 3 35 of of ADP iajs-3014 3 36 finding find VERB iajs-3014 3 37 vertex vertex NOUN iajs-3014 3 38 antimagic antimagic ADJ iajs-3014 3 39 edge edge NOUN iajs-3014 3 40 labeling labeling NOUN iajs-3014 3 41 for for ADP iajs-3014 3 42 some some DET iajs-3014 3 43 special special ADJ iajs-3014 3 44 families family NOUN iajs-3014 3 45 of of ADP iajs-3014 3 46 graphs graph NOUN iajs-3014 3 47 called call VERB iajs-3014 3 48 strong strong ADJ iajs-3014 3 49 face face NOUN iajs-3014 3 50 graphs graph NOUN iajs-3014 3 51 . . PUNCT iajs-3014 4 1 we we PRON iajs-3014 4 2 prove prove VERB iajs-3014 4 3 that that SCONJ iajs-3014 4 4 vertex vertex NOUN iajs-3014 4 5 antimagic antimagic NOUN iajs-3014 4 6 , , PUNCT iajs-3014 4 7 edge edge NOUN iajs-3014 4 8 labeling labeling NOUN iajs-3014 4 9 for for ADP iajs-3014 4 10 strong strong ADJ iajs-3014 4 11 face face NOUN iajs-3014 4 12 ladder ladder NOUN iajs-3014 4 13 graph graph NOUN iajs-3014 4 14 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 4 15 βˆ— βˆ— NOUN iajs-3014 4 16 , , PUNCT iajs-3014 4 17 strong strong ADJ iajs-3014 4 18 face face NOUN iajs-3014 4 19 wheel wheel NOUN iajs-3014 4 20 graph graph NOUN iajs-3014 4 21 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 4 22 βˆ— βˆ— NOUN iajs-3014 4 23 , , PUNCT iajs-3014 4 24 strong strong ADJ iajs-3014 4 25 face face NOUN iajs-3014 4 26 fan fan NOUN iajs-3014 4 27 graph graph NOUN iajs-3014 4 28 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 4 29 βˆ— βˆ— NOUN iajs-3014 4 30 , , PUNCT iajs-3014 4 31 strong strong ADJ iajs-3014 4 32 face face NOUN iajs-3014 4 33 prism prism NOUN iajs-3014 4 34 graph graph NOUN iajs-3014 4 35 ( ( PUNCT iajs-3014 4 36 𝐢𝑛 𝐢𝑛 INTJ iajs-3014 4 37 Γ— Γ— NOUN iajs-3014 4 38 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 4 39 and and CCONJ iajs-3014 4 40 finally finally ADV iajs-3014 4 41 strong strong ADJ iajs-3014 4 42 face face VERB iajs-3014 4 43 friendship friendship NOUN iajs-3014 4 44 graph graph NOUN iajs-3014 4 45 ( ( PUNCT iajs-3014 4 46 𝑇𝑛)βˆ—. 𝑇𝑛)βˆ—. NOUN iajs-3014 4 47 keywords keyword NOUN iajs-3014 4 48 : : PUNCT iajs-3014 4 49 antimagic antimagic ADJ iajs-3014 4 50 graph graph NOUN iajs-3014 4 51 , , PUNCT iajs-3014 4 52 vertex vertex NOUN iajs-3014 4 53 antimagic antimagic ADJ iajs-3014 4 54 graph graph NOUN iajs-3014 4 55 , , PUNCT iajs-3014 4 56 edge edge NOUN iajs-3014 4 57 labeling labeling NOUN iajs-3014 4 58 , , PUNCT iajs-3014 4 59 strong strong ADJ iajs-3014 4 60 face face NOUN iajs-3014 4 61 graph graph NOUN iajs-3014 4 62 . . PUNCT iajs-3014 5 1 1 1 X iajs-3014 5 2 . . X iajs-3014 5 3 introduction introduction NOUN iajs-3014 5 4 let let VERB iajs-3014 5 5 a a DET iajs-3014 5 6 graph graph NOUN iajs-3014 5 7 g g NOUN iajs-3014 5 8 = = SYM iajs-3014 5 9 ( ( PUNCT iajs-3014 5 10 v v NOUN iajs-3014 5 11 , , PUNCT iajs-3014 5 12 e e NOUN iajs-3014 5 13 ) ) PUNCT iajs-3014 5 14 be be AUX iajs-3014 5 15 a a DET iajs-3014 5 16 finite finite NOUN iajs-3014 5 17 , , PUNCT iajs-3014 5 18 simple simple ADJ iajs-3014 5 19 and and CCONJ iajs-3014 5 20 undirected undirected ADJ iajs-3014 5 21 graph graph NOUN iajs-3014 5 22 , , PUNCT iajs-3014 5 23 where where SCONJ iajs-3014 5 24 v(g v(g ADJ iajs-3014 5 25 ) ) PUNCT iajs-3014 5 26 and and CCONJ iajs-3014 5 27 e(g e(g PROPN iajs-3014 5 28 ) ) PUNCT iajs-3014 5 29 are be AUX iajs-3014 5 30 the the DET iajs-3014 5 31 vertex vertex NOUN iajs-3014 5 32 set set NOUN iajs-3014 5 33 and and CCONJ iajs-3014 5 34 edge edge NOUN iajs-3014 5 35 set set VERB iajs-3014 5 36 respectively respectively ADV iajs-3014 5 37 . . PUNCT iajs-3014 6 1 an an DET iajs-3014 6 2 antimagic antimagic ADJ iajs-3014 6 3 labeling labeling NOUN iajs-3014 6 4 graphs graph NOUN iajs-3014 6 5 had have AUX iajs-3014 6 6 first first ADV iajs-3014 6 7 been be AUX iajs-3014 6 8 introduced introduce VERB iajs-3014 6 9 by by ADP iajs-3014 6 10 [ [ X iajs-3014 6 11 1 1 NUM iajs-3014 6 12 ] ] PUNCT iajs-3014 6 13 . . PUNCT iajs-3014 7 1 if if SCONJ iajs-3014 7 2 a a DET iajs-3014 7 3 graph graph NOUN iajs-3014 7 4 g g NOUN iajs-3014 7 5 with with ADP iajs-3014 7 6 p p NOUN iajs-3014 7 7 vertices vertex NOUN iajs-3014 7 8 and and CCONJ iajs-3014 7 9 q q NOUN iajs-3014 7 10 edges edge NOUN iajs-3014 7 11 can can AUX iajs-3014 7 12 have have VERB iajs-3014 7 13 its its PRON iajs-3014 7 14 edges edge NOUN iajs-3014 7 15 labeled label VERB iajs-3014 7 16 without without ADP iajs-3014 7 17 repetition repetition NOUN iajs-3014 7 18 and and CCONJ iajs-3014 7 19 the the DET iajs-3014 7 20 sums sum NOUN iajs-3014 7 21 of of ADP iajs-3014 7 22 the the DET iajs-3014 7 23 labels label NOUN iajs-3014 7 24 of of ADP iajs-3014 7 25 the the DET iajs-3014 7 26 edges edge NOUN iajs-3014 7 27 incident incident NOUN iajs-3014 7 28 to to ADP iajs-3014 7 29 each each DET iajs-3014 7 30 vertex vertex NOUN iajs-3014 7 31 are be AUX iajs-3014 7 32 pairwise pairwise NOUN iajs-3014 7 33 distinct distinct ADJ iajs-3014 7 34 , , PUNCT iajs-3014 7 35 the the DET iajs-3014 7 36 graph graph NOUN iajs-3014 7 37 is be AUX iajs-3014 7 38 said say VERB iajs-3014 7 39 to to PART iajs-3014 7 40 be be AUX iajs-3014 7 41 antimagic antimagic ADJ iajs-3014 7 42 [ [ X iajs-3014 7 43 2 2 NUM iajs-3014 7 44 ] ] PUNCT iajs-3014 7 45 . . PUNCT iajs-3014 8 1 in in ADP iajs-3014 8 2 addition addition NOUN iajs-3014 8 3 [ [ X iajs-3014 8 4 3 3 X iajs-3014 8 5 ] ] PUNCT iajs-3014 8 6 show show VERB iajs-3014 8 7 that that SCONJ iajs-3014 8 8 if if SCONJ iajs-3014 8 9 all all DET iajs-3014 8 10 vertex vertex NOUN iajs-3014 8 11 weights weight NOUN iajs-3014 8 12 are be AUX iajs-3014 8 13 distinct distinct ADJ iajs-3014 8 14 , , PUNCT iajs-3014 8 15 an an DET iajs-3014 8 16 edge edge NOUN iajs-3014 8 17 labeling labeling NOUN iajs-3014 8 18 of of ADP iajs-3014 8 19 a a DET iajs-3014 8 20 graph graph NOUN iajs-3014 8 21 g g NOUN iajs-3014 8 22 is be AUX iajs-3014 8 23 said say VERB iajs-3014 8 24 to to PART iajs-3014 8 25 have have VERB iajs-3014 8 26 a a DET iajs-3014 8 27 vertex vertex NOUN iajs-3014 8 28 antimagic antimagic ADJ iajs-3014 8 29 edge edge NOUN iajs-3014 8 30 labeling labeling NOUN iajs-3014 8 31 ( ( PUNCT iajs-3014 8 32 vae vae PROPN iajs-3014 8 33 labeling labeling PROPN iajs-3014 8 34 ) ) PUNCT iajs-3014 8 35 . . PUNCT iajs-3014 9 1 hartsfield hartsfield PROPN iajs-3014 9 2 and and CCONJ iajs-3014 9 3 ringel ringel NOUN iajs-3014 9 4 proved prove VERB iajs-3014 9 5 that that SCONJ iajs-3014 9 6 the the DET iajs-3014 9 7 path path NOUN iajs-3014 9 8 graphs graph VERB iajs-3014 9 9 𝑃𝑛 𝑃𝑛 NOUN iajs-3014 9 10 , , PUNCT iajs-3014 9 11 complete complete ADJ iajs-3014 9 12 graph graph NOUN iajs-3014 9 13 𝐾𝑛 𝐾𝑛 PROPN iajs-3014 9 14 , , PUNCT iajs-3014 9 15 𝑛 𝑛 DET iajs-3014 9 16 β‰₯ β‰₯ NOUN iajs-3014 9 17 3 3 NUM iajs-3014 9 18 , , PUNCT iajs-3014 9 19 wheels wheel NOUN iajs-3014 9 20 and and CCONJ iajs-3014 9 21 cycles cycle NOUN iajs-3014 9 22 graphs graph NOUN iajs-3014 9 23 are be AUX iajs-3014 9 24 antimagic antimagic ADJ iajs-3014 9 25 , , PUNCT iajs-3014 9 26 moreover moreover ADV iajs-3014 9 27 , , PUNCT iajs-3014 9 28 they they PRON iajs-3014 9 29 conjectured conjecture VERB iajs-3014 9 30 that that SCONJ iajs-3014 9 31 every every DET iajs-3014 9 32 tree tree NOUN iajs-3014 9 33 except except SCONJ iajs-3014 9 34 𝑃2 𝑃2 PROPN iajs-3014 9 35 is be AUX iajs-3014 9 36 antimagic antimagic ADJ iajs-3014 9 37 , , PUNCT iajs-3014 9 38 and and CCONJ iajs-3014 9 39 every every DET iajs-3014 9 40 connected connected ADJ iajs-3014 9 41 graph graph NOUN iajs-3014 9 42 except except SCONJ iajs-3014 9 43 𝑃2 𝑃2 PROPN iajs-3014 9 44 is be AUX iajs-3014 9 45 antimagic antimagic ADJ iajs-3014 9 46 , , PUNCT iajs-3014 9 47 both both CCONJ iajs-3014 9 48 these these DET iajs-3014 9 49 conjectures conjecture NOUN iajs-3014 9 50 are be AUX iajs-3014 9 51 still still ADV iajs-3014 9 52 open open ADJ iajs-3014 9 53 . . PUNCT iajs-3014 10 1 in in ADP iajs-3014 10 2 this this DET iajs-3014 10 3 paper paper NOUN iajs-3014 10 4 , , PUNCT iajs-3014 10 5 we we PRON iajs-3014 10 6 will will AUX iajs-3014 10 7 prove prove VERB iajs-3014 10 8 several several ADJ iajs-3014 10 9 graphs graph NOUN iajs-3014 10 10 derived derive VERB iajs-3014 10 11 from from ADP iajs-3014 10 12 a a DET iajs-3014 10 13 plane plane NOUN iajs-3014 10 14 graph graph NOUN iajs-3014 10 15 are be AUX iajs-3014 10 16 edge edge NOUN iajs-3014 10 17 labeling labeling NOUN iajs-3014 10 18 vertex vertex NOUN iajs-3014 10 19 antinagic antinagic NOUN iajs-3014 10 20 , , PUNCT iajs-3014 10 21 these these DET iajs-3014 10 22 graphs graph NOUN iajs-3014 10 23 are be AUX iajs-3014 10 24 called call VERB iajs-3014 10 25 strong strong ADJ iajs-3014 10 26 face face NOUN iajs-3014 10 27 graphs graph NOUN iajs-3014 10 28 . . PUNCT iajs-3014 11 1 the the DET iajs-3014 11 2 strong strong ADJ iajs-3014 11 3 face face NOUN iajs-3014 11 4 graph graph NOUN iajs-3014 11 5 , , PUNCT iajs-3014 11 6 first first ADV iajs-3014 11 7 introduced introduce VERB iajs-3014 11 8 by by ADP iajs-3014 11 9 [ [ X iajs-3014 11 10 4 4 NUM iajs-3014 11 11 ] ] PUNCT iajs-3014 11 12 . . PUNCT iajs-3014 12 1 where where SCONJ iajs-3014 12 2 they they PRON iajs-3014 12 3 proved prove VERB iajs-3014 12 4 that that PRON iajs-3014 12 5 face face VERB iajs-3014 12 6 antimagic antimagic ADJ iajs-3014 12 7 total total ADJ iajs-3014 12 8 labeling labeling NOUN iajs-3014 12 9 for for ADP iajs-3014 12 10 some some DET iajs-3014 12 11 families family NOUN iajs-3014 12 12 of of ADP iajs-3014 12 13 these these DET iajs-3014 12 14 graphs graph NOUN iajs-3014 12 15 . . PUNCT iajs-3014 13 1 in in ADP iajs-3014 13 2 our our PRON iajs-3014 13 3 study study NOUN iajs-3014 13 4 we we PRON iajs-3014 13 5 address address VERB iajs-3014 13 6 the the DET iajs-3014 13 7 problem problem NOUN iajs-3014 13 8 of of ADP iajs-3014 13 9 finding find VERB iajs-3014 13 10 vertex vertex NOUN iajs-3014 13 11 antimagic antimagic ADJ iajs-3014 13 12 edge edge NOUN iajs-3014 13 13 labeling labeling NOUN iajs-3014 13 14 for for ADP iajs-3014 13 15 this this DET iajs-3014 13 16 family family NOUN iajs-3014 13 17 of of ADP iajs-3014 13 18 graphs graph NOUN iajs-3014 13 19 . . PUNCT iajs-3014 14 1 doi.org/10.30526/36.1.3209 doi.org/10.30526/36.1.3209 NOUN iajs-3014 14 2 article article NOUN iajs-3014 14 3 history history NOUN iajs-3014 14 4 : : PUNCT iajs-3014 14 5 received receive VERB iajs-3014 14 6 15 15 NUM iajs-3014 14 7 september september PROPN iajs-3014 14 8 2022 2022 NUM iajs-3014 14 9 , , PUNCT iajs-3014 14 10 accepted accept VERB iajs-3014 14 11 9 9 NUM iajs-3014 14 12 october october NOUN iajs-3014 14 13 2022 2022 NUM iajs-3014 14 14 , , PUNCT iajs-3014 14 15 published publish VERB iajs-3014 14 16 in in ADP iajs-3014 14 17 january january PROPN iajs-3014 14 18 2023 2023 NUM iajs-3014 14 19 . . PUNCT iajs-3014 15 1 ibn ibn PROPN iajs-3014 15 2 al al PROPN iajs-3014 15 3 - - PUNCT iajs-3014 15 4 haitham haitham PROPN iajs-3014 15 5 journal journal PROPN iajs-3014 15 6 for for ADP iajs-3014 15 7 pure pure ADJ iajs-3014 15 8 and and CCONJ iajs-3014 15 9 applied apply VERB iajs-3014 15 10 sciences sciences PROPN iajs-3014 15 11 journal journal PROPN iajs-3014 15 12 homepagejih.uobaghdad.edu.iq homepagejih.uobaghdad.edu.iq PROPN iajs-3014 15 13 noor noor PROPN iajs-3014 15 14 khalil khalil PROPN iajs-3014 15 15 shawkat shawkat PROPN iajs-3014 15 16 department department PROPN iajs-3014 15 17 of of ADP iajs-3014 15 18 mathematics mathematics PROPN iajs-3014 15 19 , , PUNCT iajs-3014 15 20 college college NOUN iajs-3014 15 21 of of ADP iajs-3014 15 22 education education NOUN iajs-3014 15 23 for for ADP iajs-3014 15 24 pure pure ADJ iajs-3014 15 25 sciences science NOUN iajs-3014 15 26 , , PUNCT iajs-3014 15 27 ibn ibn PROPN iajs-3014 15 28 al al PROPN iajs-3014 15 29 – – PUNCT iajs-3014 15 30 haitham/ haitham/ NUM iajs-3014 15 31 university university NOUN iajs-3014 15 32 of of ADP iajs-3014 15 33 baghdad baghdad PROPN iajs-3014 15 34 , , PUNCT iajs-3014 15 35 baghdad baghdad PROPN iajs-3014 15 36 , , PUNCT iajs-3014 15 37 iraq iraq PROPN iajs-3014 15 38 . . PUNCT iajs-3014 16 1 nour.khaleel1203a@ihcoedu.uobaghdad.edu.iq nour.khaleel1203a@ihcoedu.uobaghdad.edu.iq PROPN iajs-3014 16 2 mohammed mohammed PROPN iajs-3014 16 3 ali ali PROPN iajs-3014 16 4 ahmed ahmed PROPN iajs-3014 16 5 department department PROPN iajs-3014 16 6 of of ADP iajs-3014 16 7 mathematics mathematics PROPN iajs-3014 16 8 , , PUNCT iajs-3014 16 9 college college NOUN iajs-3014 16 10 of of ADP iajs-3014 16 11 education education NOUN iajs-3014 16 12 for for ADP iajs-3014 16 13 pure pure ADJ iajs-3014 16 14 sciences science NOUN iajs-3014 16 15 , , PUNCT iajs-3014 16 16 ibn ibn PROPN iajs-3014 16 17 al al PROPN iajs-3014 16 18 – – PUNCT iajs-3014 16 19 haitham/ haitham/ NUM iajs-3014 16 20 university university NOUN iajs-3014 16 21 of of ADP iajs-3014 16 22 baghdad baghdad PROPN iajs-3014 16 23 , , PUNCT iajs-3014 16 24 baghdad baghdad PROPN iajs-3014 16 25 , , PUNCT iajs-3014 16 26 iraq iraq PROPN iajs-3014 16 27 . . PUNCT iajs-3014 17 1 mohammad.a.a@ihcoedu.uobaghdad.edu.iq mohammad.a.a@ihcoedu.uobaghdad.edu.iq PROPN iajs-3014 17 2 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3014 17 3 mailto:nour.khaleel1203a@ihcoedu.uobaghdad.edu.iq mailto:nour.khaleel1203a@ihcoedu.uobaghdad.edu.iq PROPN iajs-3014 17 4 mailto:mohammad.a.a@ihcoedu.uobaghdad.edu.iq mailto:mohammad.a.a@ihcoedu.uobaghdad.edu.iq PROPN iajs-3014 17 5 ihjpas ihjpas PROPN iajs-3014 17 6 . . PUNCT iajs-3014 18 1 36(1)2023 36(1)2023 NUM iajs-3014 18 2 285 285 NUM iajs-3014 18 3 the the DET iajs-3014 18 4 strong strong ADJ iajs-3014 18 5 face face NOUN iajs-3014 18 6 plane plane NOUN iajs-3014 18 7 graphs graph NOUN iajs-3014 18 8 are be AUX iajs-3014 18 9 obtained obtain VERB iajs-3014 18 10 from from ADP iajs-3014 18 11 a a DET iajs-3014 18 12 plane plane NOUN iajs-3014 18 13 graph graph NOUN iajs-3014 18 14 , , PUNCT iajs-3014 18 15 by by ADP iajs-3014 18 16 adding add VERB iajs-3014 18 17 a a DET iajs-3014 18 18 new new ADJ iajs-3014 18 19 vertex vertex NOUN iajs-3014 18 20 to to ADP iajs-3014 18 21 every every DET iajs-3014 18 22 face face NOUN iajs-3014 18 23 , , PUNCT iajs-3014 18 24 in in ADP iajs-3014 18 25 such such ADJ iajs-3014 18 26 way way NOUN iajs-3014 18 27 that that SCONJ iajs-3014 18 28 , , PUNCT iajs-3014 18 29 the the DET iajs-3014 18 30 results result NOUN iajs-3014 18 31 graphs graph NOUN iajs-3014 18 32 are be AUX iajs-3014 18 33 all all PRON iajs-3014 18 34 three three NUM iajs-3014 18 35 - - PUNCT iajs-3014 18 36 sided sided ADJ iajs-3014 18 37 faces face NOUN iajs-3014 18 38 , , PUNCT iajs-3014 18 39 moreover moreover ADV iajs-3014 18 40 if if SCONJ iajs-3014 18 41 the the DET iajs-3014 18 42 faces face NOUN iajs-3014 18 43 of of ADP iajs-3014 18 44 original original ADJ iajs-3014 18 45 plane plane NOUN iajs-3014 18 46 graphs graph NOUN iajs-3014 18 47 are be AUX iajs-3014 18 48 three three NUM iajs-3014 18 49 sided sided ADJ iajs-3014 18 50 faces face NOUN iajs-3014 18 51 , , PUNCT iajs-3014 18 52 then then ADV iajs-3014 18 53 the the DET iajs-3014 18 54 number number NOUN iajs-3014 18 55 of of ADP iajs-3014 18 56 faces face NOUN iajs-3014 18 57 will will AUX iajs-3014 18 58 be be AUX iajs-3014 18 59 increasing increase VERB iajs-3014 18 60 . . PUNCT iajs-3014 19 1 we we PRON iajs-3014 19 2 address address VERB iajs-3014 19 3 the the DET iajs-3014 19 4 problem problem NOUN iajs-3014 19 5 of of ADP iajs-3014 19 6 finding find VERB iajs-3014 19 7 vertex vertex NOUN iajs-3014 19 8 antimagic antimagic ADJ iajs-3014 19 9 edge edge NOUN iajs-3014 19 10 labeling labeling NOUN iajs-3014 19 11 , , PUNCT iajs-3014 19 12 for for ADP iajs-3014 19 13 short short ADJ iajs-3014 19 14 ( ( PUNCT iajs-3014 19 15 vael vael ADJ iajs-3014 19 16 ) ) PUNCT iajs-3014 19 17 , , PUNCT iajs-3014 19 18 for for ADP iajs-3014 19 19 the the DET iajs-3014 19 20 strong strong ADJ iajs-3014 19 21 face face NOUN iajs-3014 19 22 ladder ladder NOUN iajs-3014 19 23 graph graph NOUN iajs-3014 19 24 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 19 25 βˆ— βˆ— NOUN iajs-3014 19 26 , , PUNCT iajs-3014 19 27 the the DET iajs-3014 19 28 strong strong ADJ iajs-3014 19 29 face face NOUN iajs-3014 19 30 wheel wheel NOUN iajs-3014 19 31 graph graph NOUN iajs-3014 19 32 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 19 33 βˆ— βˆ— NOUN iajs-3014 19 34 , , PUNCT iajs-3014 19 35 the the DET iajs-3014 19 36 strong strong ADJ iajs-3014 19 37 face face NOUN iajs-3014 19 38 fan fan NOUN iajs-3014 19 39 graph graph NOUN iajs-3014 19 40 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 19 41 βˆ— βˆ— NOUN iajs-3014 19 42 , , PUNCT iajs-3014 19 43 the the DET iajs-3014 19 44 strong strong ADJ iajs-3014 19 45 face face NOUN iajs-3014 19 46 prism prism NOUN iajs-3014 19 47 graph graph NOUN iajs-3014 19 48 ( ( PUNCT iajs-3014 19 49 𝐢𝑛 𝐢𝑛 INTJ iajs-3014 19 50 Γ— Γ— PROPN iajs-3014 19 51 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 19 52 , , PUNCT iajs-3014 19 53 and and CCONJ iajs-3014 19 54 finally finally ADV iajs-3014 19 55 the the DET iajs-3014 19 56 strong strong ADJ iajs-3014 19 57 face face NOUN iajs-3014 19 58 friendship friendship NOUN iajs-3014 19 59 graph graph NOUN iajs-3014 19 60 ( ( PUNCT iajs-3014 19 61 𝑇𝑛)βˆ—. 𝑇𝑛)βˆ—. PROPN iajs-3014 19 62 2 2 NUM iajs-3014 19 63 . . PUNCT iajs-3014 19 64 main main ADJ iajs-3014 19 65 results result NOUN iajs-3014 19 66 theorem1 theorem1 NOUN iajs-3014 19 67 for for ADP iajs-3014 19 68 every every DET iajs-3014 19 69 𝑛 𝑛 PRON iajs-3014 19 70 β‰₯ β‰₯ NOUN iajs-3014 19 71 3 3 NUM iajs-3014 19 72 , , PUNCT iajs-3014 19 73 𝑛 𝑛 DET iajs-3014 19 74 β‰’ β‰’ NUM iajs-3014 19 75 2 2 NUM iajs-3014 19 76 , , PUNCT iajs-3014 19 77 ( ( PUNCT iajs-3014 19 78 mod mod NOUN iajs-3014 19 79 4 4 NUM iajs-3014 19 80 ) ) PUNCT iajs-3014 19 81 the the DET iajs-3014 19 82 strong strong ADJ iajs-3014 19 83 face face NOUN iajs-3014 19 84 ladder ladder NOUN iajs-3014 19 85 graph graph NOUN iajs-3014 19 86 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 19 87 βˆ— βˆ— NOUN iajs-3014 19 88 is be AUX iajs-3014 19 89 vael vael ADJ iajs-3014 19 90 . . PUNCT iajs-3014 20 1 proof proof NOUN iajs-3014 20 2 : : PUNCT iajs-3014 20 3 we we PRON iajs-3014 20 4 define define VERB iajs-3014 20 5 the the DET iajs-3014 20 6 vertex vertex NOUN iajs-3014 20 7 and and CCONJ iajs-3014 20 8 edge edge NOUN iajs-3014 20 9 sets set NOUN iajs-3014 20 10 of of ADP iajs-3014 20 11 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 20 12 βˆ— βˆ— NOUN iajs-3014 20 13 graph graph NOUN iajs-3014 20 14 as as SCONJ iajs-3014 20 15 follows follow VERB iajs-3014 20 16 : : PUNCT iajs-3014 20 17 𝑉 𝑉 PROPN iajs-3014 20 18 ( ( PUNCT iajs-3014 20 19 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 20 20 βˆ— βˆ— NOUN iajs-3014 20 21 ) ) PUNCT iajs-3014 20 22 = = PRON iajs-3014 20 23 { { PUNCT iajs-3014 20 24 𝑣𝑖 𝑣𝑖 ADP iajs-3014 20 25 : : PUNCT iajs-3014 21 1 𝑖 𝑖 SYM iajs-3014 21 2 = = SYM iajs-3014 21 3 1 1 NUM iajs-3014 21 4 , , PUNCT iajs-3014 21 5 2 2 NUM iajs-3014 21 6 , , PUNCT iajs-3014 21 7 … … PUNCT iajs-3014 21 8 , , PUNCT iajs-3014 21 9 3𝑛 3𝑛 NUM iajs-3014 21 10 βˆ’ βˆ’ NOUN iajs-3014 21 11 1 1 NUM iajs-3014 21 12 } } PUNCT iajs-3014 21 13 , , PUNCT iajs-3014 21 14 𝐸 𝐸 PROPN iajs-3014 21 15 ( ( PUNCT iajs-3014 21 16 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 21 17 βˆ— βˆ— NOUN iajs-3014 21 18 ) ) PUNCT iajs-3014 22 1 = = PRON iajs-3014 22 2 { { PUNCT iajs-3014 22 3 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN iajs-3014 22 4 , , PUNCT iajs-3014 22 5 𝑣𝑛+𝑖𝑣𝑛+𝑖+1 𝑣𝑛+𝑖𝑣𝑛+𝑖+1 NUM iajs-3014 22 6 , , PUNCT iajs-3014 22 7 𝑣𝑖𝑣2𝑛+𝑖 𝑣𝑖𝑣2𝑛+𝑖 PROPN iajs-3014 22 8 , , PUNCT iajs-3014 22 9 𝑣𝑖+1𝑣2𝑛+𝑖 𝑣𝑖+1𝑣2𝑛+𝑖 AUX iajs-3014 22 10 , , PUNCT iajs-3014 22 11 𝑣𝑛+𝑖𝑣2𝑛+𝑖 𝑣𝑛+𝑖𝑣2𝑛+𝑖 X iajs-3014 22 12 , , PUNCT iajs-3014 22 13 𝑣𝑛+𝑖+1𝑣2𝑛+𝑖 𝑣𝑛+𝑖+1𝑣2𝑛+𝑖 X iajs-3014 22 14 ∢ ∢ NOUN iajs-3014 22 15 𝑖 𝑖 X iajs-3014 22 16 = = SYM iajs-3014 22 17 1 1 NUM iajs-3014 22 18 , , PUNCT iajs-3014 22 19 2 2 NUM iajs-3014 22 20 , , PUNCT iajs-3014 22 21 … … PUNCT iajs-3014 22 22 , , PUNCT iajs-3014 22 23 𝑛 𝑛 DET iajs-3014 22 24 βˆ’ βˆ’ PROPN iajs-3014 22 25 1 1 NUM iajs-3014 22 26 } } PUNCT iajs-3014 22 27 βˆͺ βˆͺ X iajs-3014 22 28 { { PUNCT iajs-3014 22 29 𝑣𝑖𝑣𝑛+𝑖 𝑣𝑖𝑣𝑛+𝑖 NUM iajs-3014 22 30 : : PUNCT iajs-3014 22 31 𝑖 𝑖 SYM iajs-3014 22 32 = = SYM iajs-3014 22 33 1 1 NUM iajs-3014 22 34 , , PUNCT iajs-3014 22 35 2 2 NUM iajs-3014 22 36 … … PUNCT iajs-3014 22 37 , , PUNCT iajs-3014 22 38 𝑛 𝑛 NOUN iajs-3014 22 39 } } PUNCT iajs-3014 22 40 . . PUNCT iajs-3014 23 1 for for ADP iajs-3014 23 2 𝑛 𝑛 PROPN iajs-3014 23 3 β‰₯ β‰₯ NUM iajs-3014 23 4 3 3 NUM iajs-3014 23 5 , , PUNCT iajs-3014 23 6 𝑛 𝑛 DET iajs-3014 23 7 β‰’ β‰’ NUM iajs-3014 23 8 2 2 NUM iajs-3014 23 9 ( ( PUNCT iajs-3014 23 10 mod mod NOUN iajs-3014 23 11 4 4 X iajs-3014 23 12 ) ) PUNCT iajs-3014 23 13 we we PRON iajs-3014 23 14 define define VERB iajs-3014 23 15 the the DET iajs-3014 23 16 labeling labeling NOUN iajs-3014 23 17 of of ADP iajs-3014 23 18 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 23 19 βˆ— βˆ— NOUN iajs-3014 23 20 as as ADP iajs-3014 23 21 : : PUNCT iajs-3014 23 22 πœ‡1 πœ‡1 ADJ iajs-3014 23 23 : : PUNCT iajs-3014 23 24 𝐸 𝐸 PROPN iajs-3014 23 25 ( ( PUNCT iajs-3014 23 26 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 23 27 βˆ— βˆ— NOUN iajs-3014 23 28 ) ) PUNCT iajs-3014 23 29 β†’ β†’ SYM iajs-3014 23 30 { { PUNCT iajs-3014 23 31 1 1 NUM iajs-3014 23 32 , , PUNCT iajs-3014 23 33 2 2 NUM iajs-3014 23 34 , , PUNCT iajs-3014 23 35 … … PUNCT iajs-3014 23 36 , , PUNCT iajs-3014 23 37 7𝑛 7𝑛 NOUN iajs-3014 23 38 βˆ’ βˆ’ NOUN iajs-3014 23 39 6 6 NUM iajs-3014 23 40 } } PUNCT iajs-3014 23 41 . . PUNCT iajs-3014 24 1 such such ADJ iajs-3014 24 2 that that PRON iajs-3014 24 3 ; ; PUNCT iajs-3014 24 4 πœ‡1(𝑣𝑖𝑣𝑛+𝑖 πœ‡1(𝑣𝑖𝑣𝑛+𝑖 PROPN iajs-3014 24 5 ) ) PUNCT iajs-3014 24 6 = = SYM iajs-3014 25 1 7𝑖 7𝑖 ADJ iajs-3014 25 2 βˆ’ βˆ’ NOUN iajs-3014 25 3 6 6 NUM iajs-3014 25 4 for for ADP iajs-3014 25 5 𝑖 𝑖 NOUN iajs-3014 25 6 = = SYM iajs-3014 25 7 1 1 NUM iajs-3014 25 8 , , PUNCT iajs-3014 25 9 2 2 NUM iajs-3014 25 10 , , PUNCT iajs-3014 25 11 … … PUNCT iajs-3014 25 12 , , PUNCT iajs-3014 25 13 𝑛. 𝑛. NOUN iajs-3014 25 14 for for ADP iajs-3014 25 15 𝑖 𝑖 NOUN iajs-3014 25 16 = = SYM iajs-3014 25 17 1 1 NUM iajs-3014 25 18 , , PUNCT iajs-3014 25 19 2 2 NUM iajs-3014 25 20 , , PUNCT iajs-3014 25 21 … … PUNCT iajs-3014 25 22 , , PUNCT iajs-3014 26 1 𝑛 𝑛 PRON iajs-3014 26 2 βˆ’ βˆ’ PROPN iajs-3014 26 3 , , PUNCT iajs-3014 26 4 1 1 NUM iajs-3014 26 5 , , PUNCT iajs-3014 26 6 we we PRON iajs-3014 26 7 have have VERB iajs-3014 26 8 : : PUNCT iajs-3014 26 9 πœ‡1(𝑣𝑖𝑣𝑖+1 πœ‡1(𝑣𝑖𝑣𝑖+1 X iajs-3014 26 10 ) ) PUNCT iajs-3014 26 11 = = SYM iajs-3014 26 12 7𝑖 7𝑖 ADJ iajs-3014 27 1 βˆ’ βˆ’ NOUN iajs-3014 27 2 5 5 NUM iajs-3014 27 3 , , PUNCT iajs-3014 27 4 πœ‡1(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 πœ‡1(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 X iajs-3014 27 5 ) ) PUNCT iajs-3014 27 6 = = SYM iajs-3014 27 7 7𝑖 7𝑖 ADJ iajs-3014 27 8 βˆ’ βˆ’ NOUN iajs-3014 27 9 2 2 NUM iajs-3014 27 10 , , PUNCT iajs-3014 27 11 πœ‡1(𝑣𝑖𝑣2𝑛+𝑖 πœ‡1(𝑣𝑖𝑣2𝑛+𝑖 NOUN iajs-3014 27 12 ) ) PUNCT iajs-3014 27 13 = = SYM iajs-3014 28 1 7𝑖 7𝑖 ADJ iajs-3014 28 2 βˆ’ βˆ’ NOUN iajs-3014 28 3 4 4 NUM iajs-3014 28 4 , , PUNCT iajs-3014 28 5 πœ‡1(𝑣𝑖+1𝑣2𝑛+𝑖 πœ‡1(𝑣𝑖+1𝑣2𝑛+𝑖 NOUN iajs-3014 28 6 ) ) PUNCT iajs-3014 28 7 = = SYM iajs-3014 29 1 7𝑖 7𝑖 ADJ iajs-3014 29 2 βˆ’ βˆ’ NOUN iajs-3014 29 3 1 1 NUM iajs-3014 29 4 , , PUNCT iajs-3014 29 5 πœ‡1(𝑣𝑛+𝑖𝑣2𝑛+𝑖 πœ‡1(𝑣𝑛+𝑖𝑣2𝑛+𝑖 ADV iajs-3014 29 6 ) ) PUNCT iajs-3014 29 7 = = SYM iajs-3014 30 1 7𝑖 7𝑖 ADJ iajs-3014 30 2 βˆ’ βˆ’ NOUN iajs-3014 30 3 3 3 NUM iajs-3014 30 4 , , PUNCT iajs-3014 30 5 πœ‡1(𝑣𝑛+𝑖+1𝑣2𝑛+𝑖 πœ‡1(𝑣𝑛+𝑖+1𝑣2𝑛+𝑖 PROPN iajs-3014 30 6 ) ) PUNCT iajs-3014 30 7 = = SYM iajs-3014 31 1 7𝑖. 7𝑖. NOUN iajs-3014 31 2 for for ADP iajs-3014 31 3 the the DET iajs-3014 31 4 vertex vertex NOUN iajs-3014 31 5 - - PUNCT iajs-3014 31 6 weights weight NOUN iajs-3014 31 7 we we PRON iajs-3014 31 8 get get VERB iajs-3014 31 9 : : PUNCT iajs-3014 31 10 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 31 11 ( ( PUNCT iajs-3014 31 12 𝑣1 𝑣1 PROPN iajs-3014 31 13 ) ) PUNCT iajs-3014 31 14 = = SYM iajs-3014 31 15 πœ‡1(𝑣1𝑣2 πœ‡1(𝑣1𝑣2 PROPN iajs-3014 31 16 ) ) PUNCT iajs-3014 32 1 + + NUM iajs-3014 32 2 πœ‡1(𝑣1𝑣𝑛+1 πœ‡1(𝑣1𝑣𝑛+1 NOUN iajs-3014 32 3 ) ) PUNCT iajs-3014 32 4 + + PUNCT iajs-3014 33 1 πœ‡1(𝑣1𝑣2𝑛+1 πœ‡1(𝑣1𝑣2𝑛+1 PROPN iajs-3014 33 2 ) ) PUNCT iajs-3014 33 3 = = SYM iajs-3014 33 4 6 6 NUM iajs-3014 33 5 , , PUNCT iajs-3014 33 6 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 33 7 ( ( PUNCT iajs-3014 33 8 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 33 9 ) ) PUNCT iajs-3014 33 10 = = SYM iajs-3014 33 11 πœ‡1(𝑣𝑖𝑣𝑖+1 πœ‡1(𝑣𝑖𝑣𝑖+1 PROPN iajs-3014 33 12 ) ) PUNCT iajs-3014 33 13 + + ADJ iajs-3014 33 14 πœ‡1(π‘£π‘–π‘£π‘–βˆ’1 πœ‡1(π‘£π‘–π‘£π‘–βˆ’1 NOUN iajs-3014 33 15 ) ) PUNCT iajs-3014 33 16 + + CCONJ iajs-3014 33 17 πœ‡1(𝑣1𝑣2𝑛+𝑖 πœ‡1(𝑣1𝑣2𝑛+𝑖 PROPN iajs-3014 33 18 ) ) PUNCT iajs-3014 33 19 + + CCONJ iajs-3014 33 20 πœ‡1(𝑣𝑖𝑣2𝑛+π‘–βˆ’1 πœ‡1(𝑣𝑖𝑣2𝑛+π‘–βˆ’1 NOUN iajs-3014 33 21 ) ) PUNCT iajs-3014 33 22 + + PUNCT iajs-3014 33 23 πœ‡1(𝑣𝑖𝑣𝑛+𝑖 πœ‡1(𝑣𝑖𝑣𝑛+𝑖 PROPN iajs-3014 33 24 ) ) PUNCT iajs-3014 33 25 for for ADP iajs-3014 33 26 𝑖 𝑖 NOUN iajs-3014 33 27 = = SYM iajs-3014 33 28 2 2 NUM iajs-3014 33 29 , , PUNCT iajs-3014 33 30 3 3 NUM iajs-3014 33 31 … … PUNCT iajs-3014 33 32 , , PUNCT iajs-3014 33 33 𝑛 𝑛 DET iajs-3014 33 34 βˆ’ βˆ’ PROPN iajs-3014 33 35 1 1 NUM iajs-3014 33 36 = = SYM iajs-3014 33 37 ( ( PUNCT iajs-3014 33 38 7𝑖 7𝑖 ADJ iajs-3014 33 39 βˆ’ βˆ’ NOUN iajs-3014 33 40 5 5 NUM iajs-3014 33 41 ) ) PUNCT iajs-3014 33 42 + + CCONJ iajs-3014 33 43 ( ( PUNCT iajs-3014 33 44 7𝑖 7𝑖 ADJ iajs-3014 33 45 βˆ’ βˆ’ NOUN iajs-3014 33 46 12 12 NUM iajs-3014 33 47 ) ) PUNCT iajs-3014 33 48 + + CCONJ iajs-3014 33 49 ( ( PUNCT iajs-3014 33 50 7𝑖 7𝑖 ADJ iajs-3014 33 51 βˆ’ βˆ’ NOUN iajs-3014 33 52 4 4 NUM iajs-3014 33 53 ) ) PUNCT iajs-3014 33 54 + + CCONJ iajs-3014 33 55 ( ( PUNCT iajs-3014 33 56 7𝑖 7𝑖 ADJ iajs-3014 33 57 βˆ’ βˆ’ PROPN iajs-3014 33 58 8) 8) NUM iajs-3014 33 59 + + CCONJ iajs-3014 33 60 ( ( PUNCT iajs-3014 33 61 7𝑖 7𝑖 ADJ iajs-3014 33 62 βˆ’ βˆ’ NOUN iajs-3014 33 63 6 6 NUM iajs-3014 33 64 ) ) PUNCT iajs-3014 33 65 = = NOUN iajs-3014 33 66 35𝑖 35𝑖 NOUN iajs-3014 33 67 βˆ’ βˆ’ PROPN iajs-3014 33 68 35 35 NUM iajs-3014 33 69 for𝑖 for𝑖 NOUN iajs-3014 33 70 = = SYM iajs-3014 33 71 2 2 NUM iajs-3014 33 72 , , PUNCT iajs-3014 33 73 3 3 NUM iajs-3014 33 74 … … PUNCT iajs-3014 33 75 , , PUNCT iajs-3014 33 76 𝑛 𝑛 DET iajs-3014 33 77 βˆ’ βˆ’ PROPN iajs-3014 33 78 1 1 NUM iajs-3014 33 79 , , PUNCT iajs-3014 33 80 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 33 81 ( ( PUNCT iajs-3014 33 82 𝑣𝑛 𝑣𝑛 NOUN iajs-3014 33 83 ) ) PUNCT iajs-3014 33 84 = = SYM iajs-3014 33 85 πœ‡1(π‘£π‘›π‘£π‘›βˆ’1 πœ‡1(π‘£π‘›π‘£π‘›βˆ’1 PROPN iajs-3014 33 86 ) ) PUNCT iajs-3014 33 87 + + X iajs-3014 34 1 πœ‡1(𝑣𝑛𝑣2𝑛 πœ‡1(𝑣𝑛𝑣2𝑛 ADJ iajs-3014 34 2 ) ) PUNCT iajs-3014 34 3 + + CCONJ iajs-3014 34 4 πœ‡1(𝑣𝑛𝑣3π‘›βˆ’1 πœ‡1(𝑣𝑛𝑣3π‘›βˆ’1 NOUN iajs-3014 34 5 ) ) PUNCT iajs-3014 34 6 = = PUNCT iajs-3014 34 7 ( ( PUNCT iajs-3014 34 8 7𝑛 7𝑛 NUM iajs-3014 34 9 βˆ’ βˆ’ NUM iajs-3014 34 10 12 12 NUM iajs-3014 34 11 ) ) PUNCT iajs-3014 34 12 + + CCONJ iajs-3014 34 13 ( ( PUNCT iajs-3014 34 14 7𝑛 7𝑛 NUM iajs-3014 34 15 βˆ’ βˆ’ NUM iajs-3014 34 16 6 6 NUM iajs-3014 34 17 ) ) PUNCT iajs-3014 34 18 + + CCONJ iajs-3014 34 19 ( ( PUNCT iajs-3014 34 20 7𝑛 7𝑛 NUM iajs-3014 34 21 βˆ’ βˆ’ PROPN iajs-3014 34 22 8) 8) NUM iajs-3014 34 23 = = NOUN iajs-3014 34 24 21𝑛 21𝑛 NOUN iajs-3014 34 25 βˆ’ βˆ’ PROPN iajs-3014 34 26 26 26 NUM iajs-3014 34 27 , , PUNCT iajs-3014 34 28 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 34 29 ( ( PUNCT iajs-3014 34 30 𝑣𝑛+1 𝑣𝑛+1 NOUN iajs-3014 34 31 ) ) PUNCT iajs-3014 34 32 = = SYM iajs-3014 35 1 πœ‡1(𝑣𝑛+1𝑣1 πœ‡1(𝑣𝑛+1𝑣1 PROPN iajs-3014 35 2 ) ) PUNCT iajs-3014 35 3 + + NUM iajs-3014 35 4 πœ‡1(𝑣𝑛+1𝑣𝑛+2 πœ‡1(𝑣𝑛+1𝑣𝑛+2 X iajs-3014 35 5 ) ) PUNCT iajs-3014 35 6 + + X iajs-3014 35 7 πœ‡1(𝑣𝑛+1𝑣2𝑛+1 πœ‡1(𝑣𝑛+1𝑣2𝑛+1 NUM iajs-3014 35 8 ) ) PUNCT iajs-3014 35 9 = = SYM iajs-3014 36 1 1 1 NUM iajs-3014 36 2 + + NUM iajs-3014 36 3 5 5 NUM iajs-3014 36 4 + + SYM iajs-3014 36 5 4 4 NUM iajs-3014 36 6 = = SYM iajs-3014 36 7 10 10 NUM iajs-3014 36 8 , , PUNCT iajs-3014 36 9 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 NOUN iajs-3014 36 10 ( ( PUNCT iajs-3014 36 11 𝑣𝑛+𝑖 𝑣𝑛+𝑖 X iajs-3014 36 12 ) ) PUNCT iajs-3014 36 13 = = SYM iajs-3014 36 14 πœ‡1(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 πœ‡1(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 PROPN iajs-3014 36 15 ) ) PUNCT iajs-3014 36 16 + + NUM iajs-3014 36 17 πœ‡1(𝑣𝑛+𝑖𝑣𝑛+π‘–βˆ’1 πœ‡1(𝑣𝑛+𝑖𝑣𝑛+π‘–βˆ’1 NOUN iajs-3014 36 18 ) ) PUNCT iajs-3014 37 1 + + CCONJ iajs-3014 37 2 πœ‡1(𝑣𝑛+𝑖𝑣2𝑛+𝑖 πœ‡1(𝑣𝑛+𝑖𝑣2𝑛+𝑖 NOUN iajs-3014 37 3 ) ) PUNCT iajs-3014 37 4 + + NUM iajs-3014 37 5 πœ‡1(𝑣𝑛+𝑖𝑣2𝑛+π‘–βˆ’1 πœ‡1(𝑣𝑛+𝑖𝑣2𝑛+π‘–βˆ’1 NOUN iajs-3014 37 6 ) ) PUNCT iajs-3014 37 7 + + NOUN iajs-3014 37 8 πœ‡1(𝑣𝑛+𝑖𝑣𝑖 πœ‡1(𝑣𝑛+𝑖𝑣𝑖 NOUN iajs-3014 37 9 ) ) PUNCT iajs-3014 37 10 for for ADP iajs-3014 37 11 𝑖 𝑖 NOUN iajs-3014 37 12 = = SYM iajs-3014 37 13 2 2 NUM iajs-3014 37 14 , , PUNCT iajs-3014 37 15 3 3 NUM iajs-3014 37 16 … … SYM iajs-3014 37 17 𝑛 𝑛 NOUN iajs-3014 37 18 , , PUNCT iajs-3014 37 19 βˆ’1 βˆ’1 NOUN iajs-3014 37 20 , , PUNCT iajs-3014 37 21 = = SYM iajs-3014 37 22 ( ( PUNCT iajs-3014 37 23 7𝑖 7𝑖 ADJ iajs-3014 37 24 βˆ’ βˆ’ NOUN iajs-3014 37 25 2 2 NUM iajs-3014 37 26 ) ) PUNCT iajs-3014 37 27 + + CCONJ iajs-3014 37 28 ( ( PUNCT iajs-3014 37 29 7𝑖 7𝑖 ADJ iajs-3014 37 30 βˆ’ βˆ’ NOUN iajs-3014 37 31 9 9 NUM iajs-3014 37 32 ) ) PUNCT iajs-3014 37 33 + + CCONJ iajs-3014 37 34 ( ( PUNCT iajs-3014 37 35 7𝑖 7𝑖 ADJ iajs-3014 37 36 βˆ’ βˆ’ NOUN iajs-3014 37 37 3 3 NUM iajs-3014 37 38 ) ) PUNCT iajs-3014 37 39 + + CCONJ iajs-3014 37 40 ( ( PUNCT iajs-3014 37 41 7𝑖 7𝑖 ADJ iajs-3014 37 42 βˆ’ βˆ’ NOUN iajs-3014 37 43 7 7 NUM iajs-3014 37 44 ) ) PUNCT iajs-3014 37 45 + + CCONJ iajs-3014 37 46 ( ( PUNCT iajs-3014 37 47 7𝑖 7𝑖 ADJ iajs-3014 37 48 βˆ’ βˆ’ NOUN iajs-3014 37 49 6 6 NUM iajs-3014 37 50 ) ) PUNCT iajs-3014 37 51 for𝑖 for𝑖 NOUN iajs-3014 37 52 = = SYM iajs-3014 37 53 2 2 NUM iajs-3014 37 54 , , PUNCT iajs-3014 37 55 3 3 NUM iajs-3014 37 56 … … PUNCT iajs-3014 37 57 , , PUNCT iajs-3014 37 58 𝑛 𝑛 DET iajs-3014 37 59 βˆ’ βˆ’ PROPN iajs-3014 37 60 1 1 NUM iajs-3014 37 61 = = SYM iajs-3014 37 62 35𝑖 35𝑖 NOUN iajs-3014 37 63 βˆ’ βˆ’ PROPN iajs-3014 37 64 27 27 NUM iajs-3014 37 65 for for ADP iajs-3014 37 66 𝑖 𝑖 NOUN iajs-3014 37 67 = = SYM iajs-3014 37 68 2 2 NUM iajs-3014 37 69 , , PUNCT iajs-3014 37 70 3 3 NUM iajs-3014 37 71 … … PUNCT iajs-3014 37 72 , , PUNCT iajs-3014 37 73 𝑛 𝑛 DET iajs-3014 37 74 βˆ’ βˆ’ PROPN iajs-3014 37 75 1 1 NUM iajs-3014 37 76 , , PUNCT iajs-3014 37 77 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 37 78 ( ( PUNCT iajs-3014 37 79 𝑣2𝑛 𝑣2𝑛 PROPN iajs-3014 37 80 ) ) PUNCT iajs-3014 37 81 = = SYM iajs-3014 37 82 πœ‡1(𝑣2𝑛𝑣2π‘›βˆ’1 πœ‡1(𝑣2𝑛𝑣2π‘›βˆ’1 X iajs-3014 37 83 ) ) PUNCT iajs-3014 38 1 + + CCONJ iajs-3014 38 2 πœ‡1(𝑣2𝑛𝑣𝑛 πœ‡1(𝑣2𝑛𝑣𝑛 X iajs-3014 38 3 ) ) PUNCT iajs-3014 38 4 + + CCONJ iajs-3014 38 5 πœ‡1(𝑣2𝑛𝑣3π‘›βˆ’1 πœ‡1(𝑣2𝑛𝑣3π‘›βˆ’1 NUM iajs-3014 38 6 ) ) PUNCT iajs-3014 38 7 = = NOUN iajs-3014 38 8 ( ( PUNCT iajs-3014 38 9 7𝑛 7𝑛 NUM iajs-3014 38 10 βˆ’ βˆ’ NUM iajs-3014 38 11 9 9 NUM iajs-3014 38 12 ) ) PUNCT iajs-3014 38 13 + + CCONJ iajs-3014 38 14 ( ( PUNCT iajs-3014 38 15 7𝑛 7𝑛 NUM iajs-3014 38 16 βˆ’ βˆ’ NUM iajs-3014 38 17 6 6 NUM iajs-3014 38 18 ) ) PUNCT iajs-3014 38 19 + + CCONJ iajs-3014 38 20 ( ( PUNCT iajs-3014 38 21 7𝑛 7𝑛 NUM iajs-3014 38 22 βˆ’ βˆ’ NOUN iajs-3014 38 23 7 7 NUM iajs-3014 38 24 ) ) PUNCT iajs-3014 38 25 = = NOUN iajs-3014 38 26 21𝑛 21𝑛 NOUN iajs-3014 38 27 βˆ’ βˆ’ PROPN iajs-3014 38 28 22 22 NUM iajs-3014 38 29 . . PUNCT iajs-3014 39 1 finally finally ADV iajs-3014 39 2 : : PUNCT iajs-3014 39 3 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 39 4 ( ( PUNCT iajs-3014 39 5 𝑣2𝑛+𝑖 𝑣2𝑛+𝑖 PROPN iajs-3014 39 6 ) ) PUNCT iajs-3014 39 7 = = SYM iajs-3014 39 8 πœ‡1(𝑣2𝑛+𝑖𝑣𝑖 πœ‡1(𝑣2𝑛+𝑖𝑣𝑖 NOUN iajs-3014 39 9 ) ) PUNCT iajs-3014 39 10 + + X iajs-3014 39 11 πœ‡1(𝑣2𝑛+𝑖𝑣𝑖+1 πœ‡1(𝑣2𝑛+𝑖𝑣𝑖+1 NOUN iajs-3014 39 12 ) ) PUNCT iajs-3014 39 13 + + NUM iajs-3014 39 14 πœ‡1(𝑣2𝑛+𝑖𝑣𝑛+𝑖 πœ‡1(𝑣2𝑛+𝑖𝑣𝑛+𝑖 X iajs-3014 39 15 ) ) PUNCT iajs-3014 39 16 + + X iajs-3014 39 17 πœ‡1(𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 πœ‡1(𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 NOUN iajs-3014 39 18 ) ) PUNCT iajs-3014 39 19 for for ADP iajs-3014 39 20 𝑖 𝑖 PRON iajs-3014 39 21 = = SYM iajs-3014 39 22 1,2 1,2 NUM iajs-3014 39 23 , , PUNCT iajs-3014 39 24 … … PUNCT iajs-3014 39 25 , , PUNCT iajs-3014 39 26 𝑛 𝑛 DET iajs-3014 39 27 βˆ’ βˆ’ PROPN iajs-3014 39 28 1 1 NUM iajs-3014 39 29 , , PUNCT iajs-3014 39 30 = = PRON iajs-3014 39 31 ( ( PUNCT iajs-3014 39 32 7𝑖 7𝑖 ADJ iajs-3014 39 33 βˆ’ βˆ’ NOUN iajs-3014 39 34 4 4 NUM iajs-3014 39 35 ) ) PUNCT iajs-3014 39 36 + + CCONJ iajs-3014 39 37 ( ( PUNCT iajs-3014 39 38 7𝑖 7𝑖 ADJ iajs-3014 39 39 βˆ’ βˆ’ NOUN iajs-3014 39 40 1 1 NUM iajs-3014 39 41 ) ) PUNCT iajs-3014 39 42 + + CCONJ iajs-3014 39 43 ( ( PUNCT iajs-3014 39 44 7𝑖 7𝑖 ADJ iajs-3014 39 45 βˆ’ βˆ’ NOUN iajs-3014 39 46 3 3 NUM iajs-3014 39 47 ) ) PUNCT iajs-3014 39 48 + + CCONJ iajs-3014 39 49 ( ( PUNCT iajs-3014 39 50 7𝑖 7𝑖 ADJ iajs-3014 39 51 ) ) PUNCT iajs-3014 39 52 for for ADP iajs-3014 39 53 𝑖 𝑖 PRON iajs-3014 39 54 = = SYM iajs-3014 39 55 1,2 1,2 NUM iajs-3014 39 56 , , PUNCT iajs-3014 39 57 … … PUNCT iajs-3014 39 58 , , PUNCT iajs-3014 39 59 𝑛 𝑛 DET iajs-3014 39 60 βˆ’ βˆ’ PROPN iajs-3014 39 61 1 1 NUM iajs-3014 39 62 , , PUNCT iajs-3014 39 63 = = NOUN iajs-3014 39 64 28𝑖 28𝑖 NOUN iajs-3014 39 65 βˆ’ βˆ’ NOUN iajs-3014 39 66 8 8 NUM iajs-3014 39 67 for for ADP iajs-3014 39 68 𝑖 𝑖 NOUN iajs-3014 39 69 = = SYM iajs-3014 39 70 1 1 NUM iajs-3014 39 71 , , PUNCT iajs-3014 39 72 2 2 NUM iajs-3014 39 73 , , PUNCT iajs-3014 39 74 … … PUNCT iajs-3014 39 75 , , PUNCT iajs-3014 39 76 𝑛 𝑛 DET iajs-3014 39 77 βˆ’ βˆ’ NOUN iajs-3014 39 78 1 1 NUM iajs-3014 39 79 . . PUNCT iajs-3014 39 80 ihjpas ihjpas PROPN iajs-3014 39 81 . . PUNCT iajs-3014 40 1 36(1)2023 36(1)2023 NUM iajs-3014 40 2 286 286 NUM iajs-3014 40 3 based base VERB iajs-3014 40 4 on on ADP iajs-3014 40 5 the the DET iajs-3014 40 6 vertex vertex NOUN iajs-3014 40 7 weight weight NOUN iajs-3014 40 8 of of ADP iajs-3014 40 9 the the DET iajs-3014 40 10 previous previous ADJ iajs-3014 40 11 we we PRON iajs-3014 40 12 conclude conclude VERB iajs-3014 40 13 that that SCONJ iajs-3014 40 14 they they PRON iajs-3014 40 15 are be AUX iajs-3014 40 16 all all ADV iajs-3014 40 17 distinct distinct ADJ iajs-3014 40 18 and and CCONJ iajs-3014 40 19 the the DET iajs-3014 40 20 following follow VERB iajs-3014 40 21 notes note NOUN iajs-3014 40 22 are be AUX iajs-3014 40 23 observed observe VERB iajs-3014 40 24 : : PUNCT iajs-3014 40 25 1π‘€π‘‘πœ‡1 1π‘€π‘‘πœ‡1 NUM iajs-3014 40 26 ( ( PUNCT iajs-3014 40 27 𝑣1 𝑣1 PROPN iajs-3014 40 28 ) ) PUNCT iajs-3014 40 29 < < X iajs-3014 40 30 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 40 31 ( ( PUNCT iajs-3014 40 32 𝑣𝑛+1 𝑣𝑛+1 NOUN iajs-3014 40 33 ) ) PUNCT iajs-3014 40 34 < < X iajs-3014 40 35 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 40 36 ( ( PUNCT iajs-3014 40 37 𝑣𝑛 𝑣𝑛 NOUN iajs-3014 40 38 ) ) PUNCT iajs-3014 40 39 < < X iajs-3014 40 40 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 40 41 ( ( PUNCT iajs-3014 40 42 𝑣2𝑛 𝑣2𝑛 PROPN iajs-3014 40 43 ) ) PUNCT iajs-3014 40 44 , , PUNCT iajs-3014 40 45 2π‘€π‘‘πœ‡1 2π‘€π‘‘πœ‡1 NUM iajs-3014 40 46 ( ( PUNCT iajs-3014 40 47 𝑣2𝑛+𝑖 𝑣2𝑛+𝑖 PROPN iajs-3014 40 48 ) ) PUNCT iajs-3014 40 49 < < X iajs-3014 40 50 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 40 51 ( ( PUNCT iajs-3014 40 52 𝑣2𝑛+𝑖+1 𝑣2𝑛+𝑖+1 NUM iajs-3014 40 53 ) ) PUNCT iajs-3014 40 54 for for ADP iajs-3014 40 55 𝑖 𝑖 PRON iajs-3014 40 56 = = SYM iajs-3014 40 57 1,2 1,2 NUM iajs-3014 40 58 , , PUNCT iajs-3014 40 59 … … PUNCT iajs-3014 40 60 , , PUNCT iajs-3014 40 61 𝑛 𝑛 DET iajs-3014 40 62 βˆ’ βˆ’ PROPN iajs-3014 40 63 2 2 NUM iajs-3014 40 64 , , PUNCT iajs-3014 40 65 3π‘€π‘‘πœ‡1 3π‘€π‘‘πœ‡1 NUM iajs-3014 40 66 ( ( PUNCT iajs-3014 40 67 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 40 68 ) ) PUNCT iajs-3014 40 69 < < X iajs-3014 40 70 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 40 71 ( ( PUNCT iajs-3014 40 72 𝑣𝑛+𝑖 𝑣𝑛+𝑖 PROPN iajs-3014 40 73 ) ) PUNCT iajs-3014 40 74 < < X iajs-3014 40 75 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 40 76 ( ( PUNCT iajs-3014 40 77 𝑣𝑖+1 𝑣𝑖+1 PROPN iajs-3014 40 78 ) ) PUNCT iajs-3014 40 79 < < X iajs-3014 40 80 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 40 81 ( ( PUNCT iajs-3014 40 82 𝑣𝑛+𝑖+1 𝑣𝑛+𝑖+1 NOUN iajs-3014 40 83 ) ) PUNCT iajs-3014 40 84 for for ADP iajs-3014 40 85 𝑖 𝑖 NOUN iajs-3014 40 86 = = SYM iajs-3014 40 87 2,3 2,3 NUM iajs-3014 40 88 , , PUNCT iajs-3014 40 89 … … PUNCT iajs-3014 40 90 , , PUNCT iajs-3014 40 91 𝑛 𝑛 DET iajs-3014 40 92 βˆ’ βˆ’ PROPN iajs-3014 40 93 2 2 NUM iajs-3014 40 94 . . PUNCT iajs-3014 41 1 thus thus ADV iajs-3014 41 2 , , PUNCT iajs-3014 41 3 the the DET iajs-3014 41 4 strong strong ADJ iajs-3014 41 5 face face NOUN iajs-3014 41 6 ladder ladder NOUN iajs-3014 41 7 graph graph NOUN iajs-3014 41 8 𝐿𝑛 𝐿𝑛 PROPN iajs-3014 41 9 βˆ— βˆ— NOUN iajs-3014 41 10 is be AUX iajs-3014 41 11 vertex vertex NOUN iajs-3014 41 12 antimagic antimagic ADJ iajs-3014 41 13 edge edge NOUN iajs-3014 41 14 labeling labeling NOUN iajs-3014 41 15 for for ADP iajs-3014 41 16 every every DET iajs-3014 41 17 𝑛 𝑛 PRON iajs-3014 41 18 β‰₯ β‰₯ NOUN iajs-3014 41 19 3 3 NUM iajs-3014 41 20 , , PUNCT iajs-3014 41 21 𝑛 𝑛 DET iajs-3014 41 22 β‰’ β‰’ NUM iajs-3014 41 23 2 2 NUM iajs-3014 41 24 ( ( PUNCT iajs-3014 41 25 mod mod NOUN iajs-3014 41 26 4 4 NUM iajs-3014 41 27 ) ) PUNCT iajs-3014 41 28 . . PUNCT iajs-3014 42 1 however however ADV iajs-3014 42 2 , , PUNCT iajs-3014 42 3 when when SCONJ iajs-3014 42 4 𝑛 𝑛 PROPN iajs-3014 42 5 = = SYM iajs-3014 42 6 6 6 NUM iajs-3014 42 7 , , PUNCT iajs-3014 42 8 we we PRON iajs-3014 42 9 can can AUX iajs-3014 42 10 easily easily ADV iajs-3014 42 11 show show VERB iajs-3014 42 12 that that SCONJ iajs-3014 42 13 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 42 14 ( ( PUNCT iajs-3014 42 15 𝑣12 𝑣12 NOUN iajs-3014 42 16 ) ) PUNCT iajs-3014 42 17 = = NOUN iajs-3014 42 18 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 42 19 ( ( PUNCT iajs-3014 42 20 𝑣16 𝑣16 NOUN iajs-3014 42 21 ) ) PUNCT iajs-3014 42 22 and and CCONJ iajs-3014 42 23 similarly similarly ADV iajs-3014 42 24 when when SCONJ iajs-3014 42 25 𝑛 𝑛 PROPN iajs-3014 42 26 = = SYM iajs-3014 42 27 10 10 NUM iajs-3014 42 28 , , PUNCT iajs-3014 42 29 then then ADV iajs-3014 42 30 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 42 31 ( ( PUNCT iajs-3014 42 32 𝑣20 𝑣20 NUM iajs-3014 42 33 ) ) PUNCT iajs-3014 42 34 = = NOUN iajs-3014 42 35 π‘€π‘‘πœ‡1 π‘€π‘‘πœ‡1 PROPN iajs-3014 42 36 ( ( PUNCT iajs-3014 42 37 𝑣27 𝑣27 NOUN iajs-3014 42 38 ) ) PUNCT iajs-3014 42 39 and and CCONJ iajs-3014 42 40 so so ADV iajs-3014 42 41 on on ADV iajs-3014 42 42 . . PUNCT iajs-3014 43 1 theorem theorem VERB iajs-3014 43 2 2 2 NUM iajs-3014 43 3 the the DET iajs-3014 43 4 strong strong ADJ iajs-3014 43 5 face face NOUN iajs-3014 43 6 wheel wheel NOUN iajs-3014 43 7 graph graph NOUN iajs-3014 43 8 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 43 9 βˆ— βˆ— NOUN iajs-3014 43 10 is be AUX iajs-3014 43 11 vael vael ADJ iajs-3014 43 12 , , PUNCT iajs-3014 43 13 for for ADP iajs-3014 43 14 every every DET iajs-3014 43 15 𝑛 𝑛 PRON iajs-3014 43 16 β‰₯ β‰₯ NOUN iajs-3014 43 17 3 3 NUM iajs-3014 43 18 , , PUNCT iajs-3014 43 19 𝑛 𝑛 DET iajs-3014 43 20 β‰’ β‰’ NUM iajs-3014 43 21 0 0 NUM iajs-3014 43 22 ( ( PUNCT iajs-3014 43 23 mod mod PROPN iajs-3014 43 24 3 3 NUM iajs-3014 43 25 ) ) PUNCT iajs-3014 43 26 . . PUNCT iajs-3014 44 1 proof proof NOUN iajs-3014 44 2 : : PUNCT iajs-3014 44 3 we we PRON iajs-3014 44 4 define define VERB iajs-3014 44 5 the the DET iajs-3014 44 6 vertex vertex NOUN iajs-3014 44 7 and and CCONJ iajs-3014 44 8 edge edge NOUN iajs-3014 44 9 sets set NOUN iajs-3014 44 10 of of ADP iajs-3014 44 11 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 44 12 graph graph NOUN iajs-3014 44 13 as as SCONJ iajs-3014 44 14 follows follow VERB iajs-3014 44 15 : : PUNCT iajs-3014 44 16 𝑉 𝑉 PROPN iajs-3014 44 17 ( ( PUNCT iajs-3014 44 18 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 44 19 βˆ— βˆ— NOUN iajs-3014 44 20 ) ) PUNCT iajs-3014 44 21 = = PRON iajs-3014 44 22 { { PUNCT iajs-3014 44 23 𝑣𝑖 𝑣𝑖 ADP iajs-3014 44 24 : : PUNCT iajs-3014 44 25 𝑖 𝑖 SYM iajs-3014 44 26 = = SYM iajs-3014 44 27 1 1 NUM iajs-3014 44 28 , , PUNCT iajs-3014 44 29 2 2 NUM iajs-3014 44 30 , , PUNCT iajs-3014 44 31 … … PUNCT iajs-3014 44 32 , , PUNCT iajs-3014 44 33 2𝑛 2𝑛 PROPN iajs-3014 44 34 + + CCONJ iajs-3014 44 35 1 1 NUM iajs-3014 44 36 } } PUNCT iajs-3014 44 37 , , PUNCT iajs-3014 44 38 𝐸 𝐸 PROPN iajs-3014 44 39 ( ( PUNCT iajs-3014 44 40 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 44 41 βˆ— βˆ— NOUN iajs-3014 44 42 ) ) PUNCT iajs-3014 44 43 = = PRON iajs-3014 44 44 { { PUNCT iajs-3014 44 45 𝑣𝑖𝑣2𝑛+1 𝑣𝑖𝑣2𝑛+1 NOUN iajs-3014 44 46 , , PUNCT iajs-3014 44 47 𝑣2𝑛+1𝑣𝑛+𝑖 𝑣2𝑛+1𝑣𝑛+𝑖 PROPN iajs-3014 44 48 , , PUNCT iajs-3014 44 49 𝑣𝑛+𝑖𝑣𝑖 𝑣𝑛+𝑖𝑣𝑖 X iajs-3014 45 1 ∢ ∢ NOUN iajs-3014 45 2 𝑖 𝑖 X iajs-3014 45 3 = = SYM iajs-3014 45 4 1 1 NUM iajs-3014 45 5 , , PUNCT iajs-3014 45 6 2 2 NUM iajs-3014 45 7 , , PUNCT iajs-3014 45 8 … … PUNCT iajs-3014 45 9 , , PUNCT iajs-3014 45 10 𝑛 𝑛 ADJ iajs-3014 45 11 } } PUNCT iajs-3014 45 12 βˆͺ βˆͺ X iajs-3014 45 13 { { PUNCT iajs-3014 45 14 𝑣𝑛+𝑖𝑣𝑖+1,𝑣𝑖𝑣𝑖+1 𝑣𝑛+𝑖𝑣𝑖+1,𝑣𝑖𝑣𝑖+1 PRON iajs-3014 45 15 ∢ ∢ NOUN iajs-3014 45 16 𝑖 𝑖 NOUN iajs-3014 46 1 = = SYM iajs-3014 46 2 1 1 NUM iajs-3014 46 3 , , PUNCT iajs-3014 46 4 2 2 NUM iajs-3014 46 5 … … PUNCT iajs-3014 46 6 𝑛 𝑛 PRON iajs-3014 46 7 βˆ’ βˆ’ PROPN iajs-3014 46 8 1 1 NUM iajs-3014 46 9 } } PUNCT iajs-3014 46 10 βˆͺ βˆͺ ADJ iajs-3014 46 11 { { PUNCT iajs-3014 46 12 𝑣1𝑣𝑛 𝑣1𝑣𝑛 PROPN iajs-3014 46 13 , , PUNCT iajs-3014 46 14 𝑣1𝑣2𝑛 𝑣1𝑣2𝑛 NUM iajs-3014 46 15 } } PUNCT iajs-3014 46 16 . . PUNCT iajs-3014 47 1 for for ADP iajs-3014 47 2 𝑛 𝑛 PROPN iajs-3014 47 3 β‰₯ β‰₯ NUM iajs-3014 47 4 3 3 NUM iajs-3014 47 5 , , PUNCT iajs-3014 47 6 𝑛 𝑛 DET iajs-3014 47 7 β‰’ β‰’ NUM iajs-3014 47 8 0 0 NUM iajs-3014 47 9 ( ( PUNCT iajs-3014 47 10 mod mod NOUN iajs-3014 47 11 3 3 X iajs-3014 47 12 ) ) PUNCT iajs-3014 47 13 we we PRON iajs-3014 47 14 define define VERB iajs-3014 47 15 the the DET iajs-3014 47 16 labeling labeling NOUN iajs-3014 47 17 of of ADP iajs-3014 47 18 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 47 19 βˆ— βˆ— NOUN iajs-3014 47 20 as as ADP iajs-3014 47 21 : : PUNCT iajs-3014 47 22 πœ‡2 πœ‡2 NOUN iajs-3014 47 23 : : PUNCT iajs-3014 47 24 𝐸 𝐸 PROPN iajs-3014 47 25 ( ( PUNCT iajs-3014 47 26 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 47 27 βˆ— βˆ— NOUN iajs-3014 47 28 ) ) PUNCT iajs-3014 47 29 β†’ β†’ SYM iajs-3014 47 30 { { PUNCT iajs-3014 47 31 1 1 NUM iajs-3014 47 32 , , PUNCT iajs-3014 47 33 2 2 NUM iajs-3014 47 34 , , PUNCT iajs-3014 47 35 … … PUNCT iajs-3014 47 36 , , PUNCT iajs-3014 47 37 5𝑛 5𝑛 NUM iajs-3014 47 38 } } PUNCT iajs-3014 47 39 . . PUNCT iajs-3014 48 1 such such ADJ iajs-3014 48 2 that that PRON iajs-3014 48 3 : : PUNCT iajs-3014 48 4 for for ADP iajs-3014 48 5 𝑖 𝑖 PRON iajs-3014 48 6 = = SYM iajs-3014 48 7 1,2 1,2 NUM iajs-3014 48 8 , , PUNCT iajs-3014 48 9 … … PUNCT iajs-3014 48 10 , , PUNCT iajs-3014 48 11 𝑛 𝑛 PROPN iajs-3014 48 12 , , PUNCT iajs-3014 48 13 πœ‡2(𝑣2𝑛+1𝑣𝑛+𝑖 πœ‡2(𝑣2𝑛+1𝑣𝑛+𝑖 ADJ iajs-3014 48 14 ) ) PUNCT iajs-3014 48 15 = = SYM iajs-3014 48 16 4𝑖 4𝑖 NOUN iajs-3014 48 17 , , PUNCT iajs-3014 48 18 πœ‡2(𝑣2𝑛+1𝑣𝑖 πœ‡2(𝑣2𝑛+1𝑣𝑖 PROPN iajs-3014 48 19 ) ) PUNCT iajs-3014 48 20 = = SYM iajs-3014 49 1 4𝑖 4𝑖 NOUN iajs-3014 49 2 βˆ’ βˆ’ PROPN iajs-3014 49 3 2 2 NUM iajs-3014 49 4 , , PUNCT iajs-3014 49 5 πœ‡2(𝑣𝑖𝑣𝑛+𝑖 πœ‡2(𝑣𝑖𝑣𝑛+𝑖 PROPN iajs-3014 49 6 ) ) PUNCT iajs-3014 49 7 = = SYM iajs-3014 49 8 4𝑖 4𝑖 NOUN iajs-3014 49 9 βˆ’ βˆ’ NOUN iajs-3014 50 1 1 1 NUM iajs-3014 50 2 . . PUNCT iajs-3014 51 1 and and CCONJ iajs-3014 51 2 for for ADP iajs-3014 51 3 𝑖 𝑖 PRON iajs-3014 51 4 = = SYM iajs-3014 51 5 1,2 1,2 NUM iajs-3014 51 6 , , PUNCT iajs-3014 51 7 … … PUNCT iajs-3014 51 8 , , PUNCT iajs-3014 51 9 𝑛 𝑛 DET iajs-3014 51 10 βˆ’ βˆ’ PROPN iajs-3014 51 11 1 1 NUM iajs-3014 51 12 , , PUNCT iajs-3014 51 13 πœ‡2(𝑣𝑖𝑣𝑖+1 πœ‡2(𝑣𝑖𝑣𝑖+1 NUM iajs-3014 51 14 ) ) PUNCT iajs-3014 51 15 = = SYM iajs-3014 51 16 4𝑛 4𝑛 NOUN iajs-3014 51 17 + + CCONJ iajs-3014 51 18 𝑖 𝑖 X iajs-3014 51 19 , , PUNCT iajs-3014 51 20 πœ‡2(𝑣𝑛+𝑖𝑣𝑖+1 πœ‡2(𝑣𝑛+𝑖𝑣𝑖+1 PROPN iajs-3014 51 21 ) ) PUNCT iajs-3014 51 22 = = SYM iajs-3014 51 23 4𝑖 4𝑖 NOUN iajs-3014 52 1 + + NOUN iajs-3014 52 2 1 1 X iajs-3014 52 3 . . PUNCT iajs-3014 52 4 finally finally ADV iajs-3014 52 5 , , PUNCT iajs-3014 52 6 πœ‡2(𝑣𝑛𝑣1 πœ‡2(𝑣𝑛𝑣1 PROPN iajs-3014 52 7 ) ) PUNCT iajs-3014 52 8 = = SYM iajs-3014 53 1 5𝑛 5𝑛 NUM iajs-3014 53 2 , , PUNCT iajs-3014 53 3 πœ‡2(𝑣2𝑛𝑣1 πœ‡2(𝑣2𝑛𝑣1 NOUN iajs-3014 53 4 ) ) PUNCT iajs-3014 53 5 = = SYM iajs-3014 54 1 1 1 X iajs-3014 54 2 . . X iajs-3014 54 3 for for ADP iajs-3014 54 4 the the DET iajs-3014 54 5 vertex vertex NOUN iajs-3014 54 6 - - PUNCT iajs-3014 54 7 weights weight NOUN iajs-3014 54 8 we we PRON iajs-3014 54 9 get get VERB iajs-3014 54 10 : : PUNCT iajs-3014 54 11 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 54 12 ( ( PUNCT iajs-3014 54 13 𝑣1 𝑣1 PROPN iajs-3014 54 14 ) ) PUNCT iajs-3014 54 15 = = SYM iajs-3014 54 16 πœ‡2(𝑣1𝑣2 πœ‡2(𝑣1𝑣2 NOUN iajs-3014 54 17 ) ) PUNCT iajs-3014 54 18 + + NUM iajs-3014 54 19 πœ‡2(𝑣1𝑣𝑛 πœ‡2(𝑣1𝑣𝑛 NOUN iajs-3014 54 20 ) ) PUNCT iajs-3014 55 1 + + CCONJ iajs-3014 55 2 πœ‡2(𝑣1𝑣𝑛+1 πœ‡2(𝑣1𝑣𝑛+1 ADJ iajs-3014 55 3 ) ) PUNCT iajs-3014 55 4 + + CCONJ iajs-3014 55 5 πœ‡2(𝑣1𝑣2𝑛+1 πœ‡2(𝑣1𝑣2𝑛+1 PROPN iajs-3014 55 6 ) ) PUNCT iajs-3014 55 7 + + SYM iajs-3014 55 8 πœ‡2(𝑣1𝑣2𝑛 πœ‡2(𝑣1𝑣2𝑛 ADJ iajs-3014 55 9 ) ) PUNCT iajs-3014 55 10 = = PUNCT iajs-3014 55 11 ( ( PUNCT iajs-3014 55 12 4𝑛 4𝑛 NOUN iajs-3014 55 13 + + CCONJ iajs-3014 55 14 1 1 X iajs-3014 55 15 ) ) PUNCT iajs-3014 55 16 + + CCONJ iajs-3014 55 17 ( ( PUNCT iajs-3014 55 18 5𝑛 5𝑛 NUM iajs-3014 55 19 ) ) PUNCT iajs-3014 55 20 + + CCONJ iajs-3014 55 21 3 3 NUM iajs-3014 55 22 + + SYM iajs-3014 55 23 2 2 NUM iajs-3014 55 24 + + SYM iajs-3014 55 25 1 1 NUM iajs-3014 55 26 = = NOUN iajs-3014 55 27 9𝑛 9𝑛 VERB iajs-3014 55 28 + + CCONJ iajs-3014 55 29 7 7 NUM iajs-3014 55 30 , , PUNCT iajs-3014 55 31 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 55 32 ( ( PUNCT iajs-3014 55 33 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 55 34 ) ) PUNCT iajs-3014 55 35 = = SYM iajs-3014 55 36 πœ‡2(𝑣𝑖𝑣𝑖+1 πœ‡2(𝑣𝑖𝑣𝑖+1 X iajs-3014 55 37 ) ) PUNCT iajs-3014 55 38 + + PUNCT iajs-3014 55 39 πœ‡2(π‘£π‘–π‘£π‘–βˆ’1 πœ‡2(π‘£π‘–π‘£π‘–βˆ’1 NOUN iajs-3014 55 40 ) ) PUNCT iajs-3014 55 41 + + NUM iajs-3014 55 42 πœ‡2(𝑣𝑖𝑣𝑛+𝑖 πœ‡2(𝑣𝑖𝑣𝑛+𝑖 X iajs-3014 55 43 ) ) PUNCT iajs-3014 55 44 + + SYM iajs-3014 55 45 πœ‡2(𝑣𝑖𝑣2𝑛+1 πœ‡2(𝑣𝑖𝑣2𝑛+1 NOUN iajs-3014 55 46 ) ) PUNCT iajs-3014 55 47 + + CCONJ iajs-3014 55 48 πœ‡2(𝑣𝑖𝑣𝑛+π‘–βˆ’1 πœ‡2(𝑣𝑖𝑣𝑛+π‘–βˆ’1 NOUN iajs-3014 55 49 ) ) PUNCT iajs-3014 55 50 for for ADP iajs-3014 55 51 𝑖 𝑖 NOUN iajs-3014 55 52 = = SYM iajs-3014 55 53 2 2 NUM iajs-3014 55 54 , , PUNCT iajs-3014 55 55 3 3 NUM iajs-3014 55 56 … … PUNCT iajs-3014 55 57 𝑛 𝑛 PRON iajs-3014 55 58 βˆ’ βˆ’ PROPN iajs-3014 55 59 1 1 NUM iajs-3014 55 60 , , PUNCT iajs-3014 55 61 = = PRON iajs-3014 55 62 ( ( PUNCT iajs-3014 55 63 4𝑛 4𝑛 NOUN iajs-3014 55 64 + + CCONJ iajs-3014 55 65 𝑖 𝑖 X iajs-3014 55 66 ) ) PUNCT iajs-3014 55 67 + + CCONJ iajs-3014 55 68 ( ( PUNCT iajs-3014 55 69 4𝑛 4𝑛 NOUN iajs-3014 55 70 βˆ’ βˆ’ NOUN iajs-3014 55 71 1 1 NUM iajs-3014 55 72 + + NUM iajs-3014 55 73 𝑖 𝑖 X iajs-3014 55 74 ) ) PUNCT iajs-3014 55 75 + + CCONJ iajs-3014 55 76 ( ( PUNCT iajs-3014 55 77 4𝑖 4𝑖 INTJ iajs-3014 55 78 βˆ’ βˆ’ NOUN iajs-3014 55 79 1 1 NUM iajs-3014 55 80 ) ) PUNCT iajs-3014 55 81 + + CCONJ iajs-3014 55 82 ( ( PUNCT iajs-3014 55 83 4𝑖 4𝑖 INTJ iajs-3014 55 84 βˆ’ βˆ’ NOUN iajs-3014 55 85 2 2 NUM iajs-3014 55 86 ) ) PUNCT iajs-3014 55 87 + + CCONJ iajs-3014 55 88 ( ( PUNCT iajs-3014 55 89 4𝑖 4𝑖 INTJ iajs-3014 55 90 βˆ’ βˆ’ NOUN iajs-3014 55 91 3 3 NUM iajs-3014 55 92 ) ) PUNCT iajs-3014 55 93 = = NOUN iajs-3014 55 94 8𝑛 8𝑛 NOUN iajs-3014 55 95 + + CCONJ iajs-3014 55 96 14𝑖 14𝑖 NOUN iajs-3014 55 97 βˆ’ βˆ’ ADP iajs-3014 55 98 7 7 NUM iajs-3014 55 99 for for ADP iajs-3014 55 100 𝑖 𝑖 NOUN iajs-3014 55 101 = = SYM iajs-3014 55 102 2 2 NUM iajs-3014 55 103 , , PUNCT iajs-3014 55 104 3 3 NUM iajs-3014 55 105 , , PUNCT iajs-3014 55 106 … … PUNCT iajs-3014 55 107 , , PUNCT iajs-3014 55 108 𝑛 𝑛 DET iajs-3014 55 109 βˆ’ βˆ’ PROPN iajs-3014 55 110 1 1 NUM iajs-3014 55 111 , , PUNCT iajs-3014 55 112 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 55 113 ( ( PUNCT iajs-3014 55 114 𝑣𝑛 𝑣𝑛 NOUN iajs-3014 55 115 ) ) PUNCT iajs-3014 55 116 = = PUNCT iajs-3014 55 117 πœ‡2(𝑣𝑛𝑣1 πœ‡2(𝑣𝑛𝑣1 PROPN iajs-3014 55 118 ) ) PUNCT iajs-3014 55 119 + + PUNCT iajs-3014 55 120 πœ‡2(π‘£π‘›π‘£π‘›βˆ’1 πœ‡2(π‘£π‘›π‘£π‘›βˆ’1 NOUN iajs-3014 55 121 ) ) PUNCT iajs-3014 55 122 + + CCONJ iajs-3014 55 123 πœ‡2(𝑣𝑛𝑣2𝑛 πœ‡2(𝑣𝑛𝑣2𝑛 ADJ iajs-3014 55 124 ) ) PUNCT iajs-3014 55 125 + + CCONJ iajs-3014 55 126 πœ‡2(𝑣𝑛𝑣2𝑛+1 πœ‡2(𝑣𝑛𝑣2𝑛+1 ADJ iajs-3014 55 127 ) ) PUNCT iajs-3014 55 128 + + NUM iajs-3014 55 129 πœ‡2(𝑣𝑛𝑣2π‘›βˆ’1 πœ‡2(𝑣𝑛𝑣2π‘›βˆ’1 NOUN iajs-3014 55 130 ) ) PUNCT iajs-3014 55 131 = = PUNCT iajs-3014 55 132 ( ( PUNCT iajs-3014 55 133 5𝑛 5𝑛 NUM iajs-3014 55 134 ) ) PUNCT iajs-3014 56 1 + + CCONJ iajs-3014 56 2 ( ( PUNCT iajs-3014 56 3 5𝑛 5𝑛 NUM iajs-3014 56 4 βˆ’ βˆ’ PROPN iajs-3014 56 5 1 1 NUM iajs-3014 56 6 ) ) PUNCT iajs-3014 56 7 + + CCONJ iajs-3014 56 8 ( ( PUNCT iajs-3014 56 9 4𝑛 4𝑛 NOUN iajs-3014 56 10 βˆ’ βˆ’ NOUN iajs-3014 56 11 1 1 NUM iajs-3014 56 12 ) ) PUNCT iajs-3014 56 13 + + CCONJ iajs-3014 56 14 ( ( PUNCT iajs-3014 56 15 4𝑛 4𝑛 NOUN iajs-3014 56 16 βˆ’ βˆ’ NOUN iajs-3014 56 17 2 2 NUM iajs-3014 56 18 ) ) PUNCT iajs-3014 56 19 + + CCONJ iajs-3014 56 20 ( ( PUNCT iajs-3014 56 21 4𝑛 4𝑛 NOUN iajs-3014 56 22 βˆ’ βˆ’ NOUN iajs-3014 57 1 3 3 NUM iajs-3014 57 2 ) ) PUNCT iajs-3014 57 3 = = NOUN iajs-3014 57 4 22𝑛 22𝑛 NUM iajs-3014 57 5 βˆ’ βˆ’ PROPN iajs-3014 57 6 7 7 NUM iajs-3014 57 7 , , PUNCT iajs-3014 57 8 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 57 9 ( ( PUNCT iajs-3014 57 10 𝑣𝑛+𝑖 𝑣𝑛+𝑖 NOUN iajs-3014 57 11 ) ) PUNCT iajs-3014 57 12 = = SYM iajs-3014 57 13 πœ‡2(𝑣𝑛+𝑖𝑣𝑖 πœ‡2(𝑣𝑛+𝑖𝑣𝑖 NOUN iajs-3014 57 14 ) ) PUNCT iajs-3014 58 1 + + CCONJ iajs-3014 58 2 πœ‡2(𝑣𝑛+𝑖𝑣𝑖+1 πœ‡2(𝑣𝑛+𝑖𝑣𝑖+1 NOUN iajs-3014 58 3 ) ) PUNCT iajs-3014 58 4 + + NUM iajs-3014 58 5 πœ‡2(𝑣𝑛+𝑖𝑣2𝑛+1 πœ‡2(𝑣𝑛+𝑖𝑣2𝑛+1 ADJ iajs-3014 58 6 ) ) PUNCT iajs-3014 58 7 for for ADP iajs-3014 58 8 𝑖 𝑖 NOUN iajs-3014 58 9 = = SYM iajs-3014 58 10 1 1 NUM iajs-3014 58 11 , , PUNCT iajs-3014 58 12 2 2 NUM iajs-3014 58 13 , , PUNCT iajs-3014 58 14 … … PUNCT iajs-3014 58 15 𝑛 𝑛 PRON iajs-3014 58 16 βˆ’ βˆ’ PROPN iajs-3014 58 17 1 1 NUM iajs-3014 58 18 , , PUNCT iajs-3014 58 19 = = PRON iajs-3014 58 20 ( ( PUNCT iajs-3014 58 21 4𝑖 4𝑖 NOUN iajs-3014 58 22 βˆ’ βˆ’ PROPN iajs-3014 58 23 1 1 NUM iajs-3014 58 24 ) ) PUNCT iajs-3014 58 25 + + CCONJ iajs-3014 58 26 ( ( PUNCT iajs-3014 58 27 4𝑖 4𝑖 NOUN iajs-3014 58 28 + + NOUN iajs-3014 58 29 1 1 NUM iajs-3014 58 30 ) ) PUNCT iajs-3014 58 31 + + CCONJ iajs-3014 58 32 ( ( PUNCT iajs-3014 58 33 4𝑖 4𝑖 NOUN iajs-3014 58 34 ) ) PUNCT iajs-3014 58 35 = = NOUN iajs-3014 58 36 12𝑖 12𝑖 NOUN iajs-3014 58 37 for for ADP iajs-3014 58 38 𝑖 𝑖 NOUN iajs-3014 58 39 = = SYM iajs-3014 58 40 1 1 NUM iajs-3014 58 41 , , PUNCT iajs-3014 58 42 2 2 NUM iajs-3014 58 43 , , PUNCT iajs-3014 58 44 … … PUNCT iajs-3014 58 45 , , PUNCT iajs-3014 58 46 𝑛 𝑛 DET iajs-3014 58 47 βˆ’ βˆ’ PROPN iajs-3014 58 48 1 1 NUM iajs-3014 58 49 , , PUNCT iajs-3014 58 50 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 58 51 ( ( PUNCT iajs-3014 58 52 𝑣2𝑛 𝑣2𝑛 ADJ iajs-3014 58 53 ) ) PUNCT iajs-3014 58 54 = = PUNCT iajs-3014 58 55 πœ‡2(𝑣2𝑛𝑣1 πœ‡2(𝑣2𝑛𝑣1 PROPN iajs-3014 58 56 ) ) PUNCT iajs-3014 58 57 + + NUM iajs-3014 58 58 πœ‡2(𝑣2𝑛𝑣𝑛 πœ‡2(𝑣2𝑛𝑣𝑛 X iajs-3014 58 59 ) ) PUNCT iajs-3014 58 60 + + CCONJ iajs-3014 58 61 πœ‡2(𝑣2𝑛𝑣2𝑛+1 πœ‡2(𝑣2𝑛𝑣2𝑛+1 X iajs-3014 58 62 ) ) PUNCT iajs-3014 58 63 = = SYM iajs-3014 59 1 1 1 NUM iajs-3014 59 2 + + CCONJ iajs-3014 59 3 ( ( PUNCT iajs-3014 59 4 4𝑛 4𝑛 NOUN iajs-3014 59 5 βˆ’ βˆ’ NOUN iajs-3014 59 6 1 1 NUM iajs-3014 59 7 ) ) PUNCT iajs-3014 60 1 + + CCONJ iajs-3014 60 2 ( ( PUNCT iajs-3014 60 3 4𝑛 4𝑛 NUM iajs-3014 60 4 ) ) PUNCT iajs-3014 61 1 = = SYM iajs-3014 61 2 8𝑛. 8𝑛. NOUN iajs-3014 61 3 finally finally ADV iajs-3014 61 4 : : PUNCT iajs-3014 61 5 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 61 6 ( ( PUNCT iajs-3014 61 7 𝑣2𝑛+1 𝑣2𝑛+1 NOUN iajs-3014 61 8 ) ) PUNCT iajs-3014 61 9 = = SYM iajs-3014 61 10 βˆ‘ βˆ‘ PUNCT iajs-3014 61 11 πœ‡2(𝑣2𝑛+1𝑣𝑖 πœ‡2(𝑣2𝑛+1𝑣𝑖 PROPN iajs-3014 61 12 ) ) PUNCT iajs-3014 61 13 𝑛 𝑛 PROPN iajs-3014 61 14 𝑖=1 𝑖=1 PROPN iajs-3014 62 1 + + CCONJ iajs-3014 62 2 βˆ‘ βˆ‘ PROPN iajs-3014 62 3 πœ‡2(𝑣2𝑛+1𝑣𝑛+𝑖 πœ‡2(𝑣2𝑛+1𝑣𝑛+𝑖 PROPN iajs-3014 62 4 ) ) PUNCT iajs-3014 62 5 𝑛 𝑛 PRON iajs-3014 62 6 𝑖=1 𝑖=1 PROPN iajs-3014 62 7 ihjpas ihjpas PROPN iajs-3014 62 8 . . PUNCT iajs-3014 63 1 36(1)2023 36(1)2023 NUM iajs-3014 63 2 287 287 NUM iajs-3014 63 3 = = NOUN iajs-3014 63 4 βˆ‘(4𝑖 βˆ‘(4𝑖 NOUN iajs-3014 63 5 βˆ’ βˆ’ NUM iajs-3014 63 6 2 2 NUM iajs-3014 63 7 ) ) PUNCT iajs-3014 63 8 𝑛 𝑛 PRON iajs-3014 63 9 𝑖=1 𝑖=1 PROPN iajs-3014 63 10 + + CCONJ iajs-3014 63 11 βˆ‘(4𝑖 βˆ‘(4𝑖 PROPN iajs-3014 63 12 ) ) PUNCT iajs-3014 63 13 𝑛 𝑛 PROPN iajs-3014 63 14 𝑖=1 𝑖=1 PUNCT iajs-3014 63 15 = = SYM iajs-3014 63 16 2𝑛(𝑛 2𝑛(𝑛 PROPN iajs-3014 63 17 + + CCONJ iajs-3014 63 18 1 1 X iajs-3014 63 19 ) ) PUNCT iajs-3014 64 1 βˆ’ βˆ’ PROPN iajs-3014 64 2 2𝑛 2𝑛 PROPN iajs-3014 64 3 + + CCONJ iajs-3014 64 4 2𝑛(𝑛 2𝑛(𝑛 NUM iajs-3014 64 5 + + CCONJ iajs-3014 64 6 1 1 X iajs-3014 64 7 ) ) PUNCT iajs-3014 64 8 = = SYM iajs-3014 64 9 4𝑛2 4𝑛2 NUM iajs-3014 65 1 + + CCONJ iajs-3014 65 2 2𝑛. 2𝑛. NOUN iajs-3014 65 3 which which PRON iajs-3014 65 4 means mean VERB iajs-3014 65 5 that that SCONJ iajs-3014 65 6 the the DET iajs-3014 65 7 vertex vertex NOUN iajs-3014 65 8 weights weight NOUN iajs-3014 65 9 are be AUX iajs-3014 65 10 all all ADV iajs-3014 65 11 distinct distinct ADJ iajs-3014 65 12 and and CCONJ iajs-3014 65 13 the the DET iajs-3014 65 14 following follow VERB iajs-3014 65 15 notes note NOUN iajs-3014 65 16 are be AUX iajs-3014 65 17 observed observe VERB iajs-3014 65 18 : : PUNCT iajs-3014 65 19 1π‘€π‘‘πœ‡2 1π‘€π‘‘πœ‡2 NUM iajs-3014 65 20 ( ( PUNCT iajs-3014 65 21 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 65 22 ) ) PUNCT iajs-3014 65 23 < < X iajs-3014 65 24 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 PROPN iajs-3014 65 25 ( ( PUNCT iajs-3014 65 26 𝑣𝑖+1 𝑣𝑖+1 PROPN iajs-3014 65 27 ) ) PUNCT iajs-3014 65 28 for for ADP iajs-3014 65 29 𝑖 𝑖 PRON iajs-3014 65 30 = = SYM iajs-3014 65 31 1,2 1,2 NUM iajs-3014 65 32 , , PUNCT iajs-3014 65 33 … … PUNCT iajs-3014 65 34 , , PUNCT iajs-3014 65 35 𝑛 𝑛 PRON iajs-3014 65 36 βˆ’ βˆ’ PROPN iajs-3014 65 37 1 1 NUM iajs-3014 65 38 , , PUNCT iajs-3014 65 39 2π‘€π‘‘πœ‡2 2π‘€π‘‘πœ‡2 NUM iajs-3014 65 40 ( ( PUNCT iajs-3014 65 41 𝑣𝑛+𝑖 𝑣𝑛+𝑖 NUM iajs-3014 65 42 ) ) PUNCT iajs-3014 65 43 < < X iajs-3014 65 44 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 65 45 ( ( PUNCT iajs-3014 65 46 𝑣𝑛+𝑖+1 𝑣𝑛+𝑖+1 NOUN iajs-3014 65 47 ) ) PUNCT iajs-3014 65 48 for for ADP iajs-3014 65 49 𝑖 𝑖 PRON iajs-3014 65 50 = = SYM iajs-3014 65 51 1,2 1,2 NUM iajs-3014 65 52 , , PUNCT iajs-3014 65 53 … … PUNCT iajs-3014 65 54 , , PUNCT iajs-3014 65 55 𝑛 𝑛 PRON iajs-3014 65 56 βˆ’ βˆ’ PROPN iajs-3014 65 57 1 1 NUM iajs-3014 65 58 , , PUNCT iajs-3014 65 59 3the 3the PRON iajs-3014 65 60 weight weight NOUN iajs-3014 65 61 of of ADP iajs-3014 65 62 vertex vertex NOUN iajs-3014 65 63 ( ( PUNCT iajs-3014 65 64 𝑣2𝑛+1 𝑣2𝑛+1 NOUN iajs-3014 65 65 ) ) PUNCT iajs-3014 65 66 is be AUX iajs-3014 65 67 greater great ADJ iajs-3014 65 68 than than ADP iajs-3014 65 69 the the DET iajs-3014 65 70 weight weight NOUN iajs-3014 65 71 of of ADP iajs-3014 65 72 any any DET iajs-3014 65 73 other other ADJ iajs-3014 65 74 vertex vertex NOUN iajs-3014 65 75 in in ADP iajs-3014 65 76 the the DET iajs-3014 65 77 strong strong ADJ iajs-3014 65 78 face face NOUN iajs-3014 65 79 wheel wheel NOUN iajs-3014 65 80 graph graph NOUN iajs-3014 66 1 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 66 2 βˆ—. βˆ—. PROPN iajs-3014 66 3 thus thus ADV iajs-3014 66 4 , , PUNCT iajs-3014 66 5 the the DET iajs-3014 66 6 strong strong ADJ iajs-3014 66 7 face face NOUN iajs-3014 66 8 wheel wheel NOUN iajs-3014 66 9 graph graph NOUN iajs-3014 66 10 π‘Šπ‘› π‘Šπ‘› PROPN iajs-3014 66 11 βˆ— βˆ— NOUN iajs-3014 66 12 is be AUX iajs-3014 66 13 vertex vertex NOUN iajs-3014 66 14 antimagic antimagic ADJ iajs-3014 66 15 edge edge NOUN iajs-3014 66 16 labeling labeling NOUN iajs-3014 66 17 , , PUNCT iajs-3014 66 18 for for ADP iajs-3014 66 19 every every DET iajs-3014 66 20 𝑛 𝑛 PRON iajs-3014 66 21 β‰₯ β‰₯ NOUN iajs-3014 66 22 3 3 NUM iajs-3014 66 23 , , PUNCT iajs-3014 66 24 𝑛 𝑛 DET iajs-3014 66 25 β‰’ β‰’ NUM iajs-3014 66 26 0 0 NUM iajs-3014 66 27 ( ( PUNCT iajs-3014 66 28 mod mod PROPN iajs-3014 66 29 3 3 NUM iajs-3014 66 30 ) ) PUNCT iajs-3014 66 31 . . PUNCT iajs-3014 67 1 however however ADV iajs-3014 67 2 , , PUNCT iajs-3014 67 3 when when SCONJ iajs-3014 67 4 𝑛 𝑛 PROPN iajs-3014 67 5 = = SYM iajs-3014 67 6 3 3 NUM iajs-3014 67 7 , , PUNCT iajs-3014 67 8 we we PRON iajs-3014 67 9 can can AUX iajs-3014 67 10 easily easily ADV iajs-3014 67 11 show show VERB iajs-3014 67 12 that that DET iajs-3014 67 13 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 67 14 ( ( PUNCT iajs-3014 67 15 𝑣5 𝑣5 NOUN iajs-3014 67 16 ) ) PUNCT iajs-3014 67 17 = = SYM iajs-3014 68 1 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 68 2 ( ( PUNCT iajs-3014 68 3 𝑣6 𝑣6 NOUN iajs-3014 68 4 ) ) PUNCT iajs-3014 68 5 and and CCONJ iajs-3014 68 6 similarly similarly ADV iajs-3014 68 7 when when SCONJ iajs-3014 68 8 𝑛 𝑛 PROPN iajs-3014 68 9 = = SYM iajs-3014 68 10 6 6 NUM iajs-3014 68 11 , , PUNCT iajs-3014 68 12 π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’π‘› NOUN iajs-3014 68 13 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 68 14 ( ( PUNCT iajs-3014 68 15 𝑣10 𝑣10 NOUN iajs-3014 68 16 ) ) PUNCT iajs-3014 68 17 = = SYM iajs-3014 68 18 π‘€π‘‘πœ‡2 π‘€π‘‘πœ‡2 NOUN iajs-3014 68 19 ( ( PUNCT iajs-3014 68 20 𝑣12 𝑣12 PROPN iajs-3014 68 21 ) ) PUNCT iajs-3014 68 22 and and CCONJ iajs-3014 68 23 so so ADV iajs-3014 68 24 on on ADV iajs-3014 68 25 . . PUNCT iajs-3014 68 26 theorem theorem VERB iajs-3014 68 27 3 3 NUM iajs-3014 68 28 the the DET iajs-3014 68 29 strong strong ADJ iajs-3014 68 30 face face NOUN iajs-3014 68 31 fan fan NOUN iajs-3014 68 32 graph graph NOUN iajs-3014 68 33 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 68 34 βˆ— βˆ— NOUN iajs-3014 68 35 , , PUNCT iajs-3014 68 36 is be AUX iajs-3014 68 37 vael vael ADJ iajs-3014 68 38 , , PUNCT iajs-3014 68 39 for for ADP iajs-3014 68 40 every every DET iajs-3014 68 41 𝑛 𝑛 PRON iajs-3014 68 42 β‰₯ β‰₯ NOUN iajs-3014 68 43 4 4 NUM iajs-3014 68 44 , , PUNCT iajs-3014 68 45 𝑛 𝑛 DET iajs-3014 68 46 β‰’ β‰’ NUM iajs-3014 68 47 3 3 NUM iajs-3014 68 48 ( ( PUNCT iajs-3014 68 49 mod mod NOUN iajs-3014 68 50 5 5 NUM iajs-3014 68 51 ) ) PUNCT iajs-3014 68 52 . . PUNCT iajs-3014 69 1 proof proof NOUN iajs-3014 69 2 : : PUNCT iajs-3014 69 3 we we PRON iajs-3014 69 4 define define VERB iajs-3014 69 5 the the DET iajs-3014 69 6 vertex vertex NOUN iajs-3014 69 7 and and CCONJ iajs-3014 69 8 edge edge NOUN iajs-3014 69 9 sets set NOUN iajs-3014 69 10 of of ADP iajs-3014 69 11 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 69 12 βˆ— βˆ— NOUN iajs-3014 69 13 graph graph NOUN iajs-3014 69 14 as as SCONJ iajs-3014 69 15 follows follow VERB iajs-3014 69 16 : : PUNCT iajs-3014 69 17 𝑉 𝑉 PROPN iajs-3014 69 18 ( ( PUNCT iajs-3014 69 19 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 69 20 βˆ— βˆ— NOUN iajs-3014 69 21 ) ) PUNCT iajs-3014 69 22 = = PRON iajs-3014 69 23 { { PUNCT iajs-3014 69 24 𝑣 𝑣 X iajs-3014 69 25 , , PUNCT iajs-3014 69 26 𝑣𝑖 𝑣𝑖 ADV iajs-3014 69 27 : : PUNCT iajs-3014 70 1 𝑖 𝑖 SYM iajs-3014 70 2 = = SYM iajs-3014 70 3 1 1 NUM iajs-3014 70 4 , , PUNCT iajs-3014 70 5 2 2 NUM iajs-3014 70 6 , , PUNCT iajs-3014 70 7 … … PUNCT iajs-3014 70 8 2𝑛 2𝑛 NOUN iajs-3014 70 9 βˆ’ βˆ’ NOUN iajs-3014 70 10 1 1 NUM iajs-3014 70 11 } } PUNCT iajs-3014 70 12 , , PUNCT iajs-3014 70 13 𝐸 𝐸 PROPN iajs-3014 70 14 ( ( PUNCT iajs-3014 70 15 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 70 16 βˆ— βˆ— VERB iajs-3014 70 17 ) ) PUNCT iajs-3014 71 1 = = SYM iajs-3014 71 2 { { PUNCT iajs-3014 71 3 𝑣𝑣𝑖 𝑣𝑣𝑖 NOUN iajs-3014 71 4 ∢ ∢ NOUN iajs-3014 71 5 𝑖 𝑖 NOUN iajs-3014 71 6 = = SYM iajs-3014 71 7 1 1 NUM iajs-3014 71 8 , , PUNCT iajs-3014 71 9 2 2 NUM iajs-3014 71 10 , , PUNCT iajs-3014 71 11 … … PUNCT iajs-3014 71 12 , , PUNCT iajs-3014 71 13 𝑛 𝑛 ADJ iajs-3014 71 14 } } PUNCT iajs-3014 71 15 βˆͺ βˆͺ ADJ iajs-3014 71 16 { { PUNCT iajs-3014 71 17 𝑣𝑖𝑣𝑖+1,𝑣 𝑣𝑖𝑣𝑖+1,𝑣 NOUN iajs-3014 71 18 𝑣𝑛+𝑖 𝑣𝑛+𝑖 PROPN iajs-3014 71 19 , , PUNCT iajs-3014 71 20 𝑣𝑖𝑣𝑛+𝑖 𝑣𝑖𝑣𝑛+𝑖 PRON iajs-3014 71 21 , , PUNCT iajs-3014 71 22 𝑣𝑖+1𝑣𝑛+𝑖 𝑣𝑖+1𝑣𝑛+𝑖 NOUN iajs-3014 71 23 ∢ ∢ VERB iajs-3014 71 24 𝑖 𝑖 NOUN iajs-3014 71 25 = = SYM iajs-3014 71 26 1 1 NUM iajs-3014 71 27 , , PUNCT iajs-3014 71 28 2 2 NUM iajs-3014 71 29 … … PUNCT iajs-3014 71 30 , , PUNCT iajs-3014 71 31 𝑛 𝑛 DET iajs-3014 71 32 βˆ’ βˆ’ NOUN iajs-3014 71 33 1 1 NUM iajs-3014 71 34 } } PUNCT iajs-3014 71 35 . . PUNCT iajs-3014 72 1 for for ADP iajs-3014 72 2 𝑛 𝑛 PRON iajs-3014 72 3 β‰₯ β‰₯ NUM iajs-3014 72 4 4 4 NUM iajs-3014 72 5 , , PUNCT iajs-3014 72 6 𝑛 𝑛 DET iajs-3014 72 7 β‰’ β‰’ NUM iajs-3014 72 8 3 3 NUM iajs-3014 72 9 ( ( PUNCT iajs-3014 72 10 mod mod NOUN iajs-3014 72 11 5 5 NUM iajs-3014 72 12 ) ) PUNCT iajs-3014 72 13 we we PRON iajs-3014 72 14 define define VERB iajs-3014 72 15 the the DET iajs-3014 72 16 labeling labeling NOUN iajs-3014 72 17 of of ADP iajs-3014 72 18 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 72 19 βˆ— βˆ— NOUN iajs-3014 72 20 , , PUNCT iajs-3014 72 21 as as ADP iajs-3014 72 22 : : PUNCT iajs-3014 72 23 πœ‡3 πœ‡3 NOUN iajs-3014 72 24 : : PUNCT iajs-3014 72 25 𝐸 𝐸 PROPN iajs-3014 72 26 ( ( PUNCT iajs-3014 72 27 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 72 28 βˆ— βˆ— NOUN iajs-3014 72 29 ) ) PUNCT iajs-3014 72 30 β†’ β†’ SYM iajs-3014 72 31 { { PUNCT iajs-3014 72 32 1 1 NUM iajs-3014 72 33 , , PUNCT iajs-3014 72 34 2 2 NUM iajs-3014 72 35 , , PUNCT iajs-3014 72 36 … … PUNCT iajs-3014 72 37 , , PUNCT iajs-3014 72 38 5𝑛 5𝑛 NUM iajs-3014 72 39 βˆ’ βˆ’ NOUN iajs-3014 72 40 4 4 NUM iajs-3014 72 41 } } PUNCT iajs-3014 72 42 . . PUNCT iajs-3014 73 1 such such ADJ iajs-3014 73 2 that that PRON iajs-3014 73 3 : : PUNCT iajs-3014 73 4 πœ‡3(𝑣𝑣𝑖 πœ‡3(𝑣𝑣𝑖 X iajs-3014 73 5 ) ) PUNCT iajs-3014 73 6 = = NOUN iajs-3014 74 1 5𝑖 5𝑖 NOUN iajs-3014 74 2 βˆ’ βˆ’ NOUN iajs-3014 74 3 4 4 NUM iajs-3014 74 4 for for ADP iajs-3014 74 5 𝑖 𝑖 NOUN iajs-3014 74 6 = = SYM iajs-3014 74 7 1 1 NUM iajs-3014 74 8 , , PUNCT iajs-3014 74 9 2 2 NUM iajs-3014 74 10 , , PUNCT iajs-3014 74 11 … … PUNCT iajs-3014 74 12 , , PUNCT iajs-3014 74 13 𝑛 𝑛 NOUN iajs-3014 74 14 , , PUNCT iajs-3014 74 15 and and CCONJ iajs-3014 74 16 for for ADP iajs-3014 74 17 𝑖 𝑖 PRON iajs-3014 74 18 = = SYM iajs-3014 74 19 1 1 NUM iajs-3014 74 20 , , PUNCT iajs-3014 74 21 2 2 NUM iajs-3014 74 22 , , PUNCT iajs-3014 74 23 … … PUNCT iajs-3014 74 24 , , PUNCT iajs-3014 74 25 𝑛 𝑛 DET iajs-3014 74 26 βˆ’ βˆ’ PROPN iajs-3014 74 27 1 1 NUM iajs-3014 74 28 , , PUNCT iajs-3014 74 29 πœ‡3(𝑣𝑖𝑣𝑖+1 πœ‡3(𝑣𝑖𝑣𝑖+1 NOUN iajs-3014 74 30 ) ) PUNCT iajs-3014 74 31 = = PUNCT iajs-3014 75 1 5𝑖 5𝑖 NOUN iajs-3014 75 2 βˆ’ βˆ’ NOUN iajs-3014 75 3 3 3 NUM iajs-3014 75 4 , , PUNCT iajs-3014 75 5 πœ‡3(𝑣𝑖𝑣𝑛+𝑖 πœ‡3(𝑣𝑖𝑣𝑛+𝑖 PROPN iajs-3014 75 6 ) ) PUNCT iajs-3014 75 7 = = NOUN iajs-3014 76 1 5𝑖 5𝑖 NOUN iajs-3014 76 2 βˆ’ βˆ’ NOUN iajs-3014 76 3 2 2 NUM iajs-3014 76 4 , , PUNCT iajs-3014 76 5 πœ‡3(𝑣𝑖+1𝑣𝑛+𝑖 πœ‡3(𝑣𝑖+1𝑣𝑛+𝑖 NOUN iajs-3014 76 6 ) ) PUNCT iajs-3014 76 7 = = SYM iajs-3014 76 8 5𝑖 5𝑖 NUM iajs-3014 76 9 , , PUNCT iajs-3014 76 10 πœ‡3(𝑣𝑣𝑛+𝑖 πœ‡3(𝑣𝑣𝑛+𝑖 NOUN iajs-3014 76 11 ) ) PUNCT iajs-3014 76 12 = = PUNCT iajs-3014 77 1 5𝑖 5𝑖 NOUN iajs-3014 77 2 βˆ’ βˆ’ NOUN iajs-3014 78 1 1 1 X iajs-3014 78 2 . . X iajs-3014 79 1 for for ADP iajs-3014 79 2 the the DET iajs-3014 79 3 vertex vertex NOUN iajs-3014 79 4 - - PUNCT iajs-3014 79 5 weights weight NOUN iajs-3014 79 6 we we PRON iajs-3014 79 7 get get VERB iajs-3014 79 8 : : PUNCT iajs-3014 79 9 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 79 10 ( ( PUNCT iajs-3014 79 11 𝑣1 𝑣1 NOUN iajs-3014 79 12 ) ) PUNCT iajs-3014 79 13 = = SYM iajs-3014 79 14 πœ‡3(𝑣1𝑣 πœ‡3(𝑣1𝑣 PROPN iajs-3014 79 15 ) ) PUNCT iajs-3014 80 1 + + CCONJ iajs-3014 80 2 πœ‡3(𝑣1𝑣2 πœ‡3(𝑣1𝑣2 PROPN iajs-3014 80 3 ) ) PUNCT iajs-3014 81 1 + + CCONJ iajs-3014 81 2 πœ‡3(𝑣1𝑣𝑛+1 πœ‡3(𝑣1𝑣𝑛+1 ADJ iajs-3014 81 3 ) ) PUNCT iajs-3014 81 4 = = SYM iajs-3014 82 1 1 1 NUM iajs-3014 82 2 + + NUM iajs-3014 82 3 2 2 NUM iajs-3014 82 4 + + SYM iajs-3014 82 5 3 3 NUM iajs-3014 82 6 = = SYM iajs-3014 82 7 6 6 NUM iajs-3014 82 8 , , PUNCT iajs-3014 82 9 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 82 10 ( ( PUNCT iajs-3014 82 11 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 82 12 ) ) PUNCT iajs-3014 82 13 = = SYM iajs-3014 82 14 πœ‡3(𝑣𝑖𝑣 πœ‡3(𝑣𝑖𝑣 NOUN iajs-3014 82 15 ) ) PUNCT iajs-3014 82 16 + + CCONJ iajs-3014 82 17 πœ‡3(𝑣𝑖𝑣𝑖+1 πœ‡3(𝑣𝑖𝑣𝑖+1 NOUN iajs-3014 82 18 ) ) PUNCT iajs-3014 82 19 + + X iajs-3014 82 20 πœ‡3(π‘£π‘–π‘£π‘–βˆ’1 πœ‡3(π‘£π‘–π‘£π‘–βˆ’1 X iajs-3014 82 21 ) ) PUNCT iajs-3014 82 22 + + NUM iajs-3014 82 23 πœ‡3(𝑣𝑖𝑣𝑛+π‘–βˆ’1 πœ‡3(𝑣𝑖𝑣𝑛+π‘–βˆ’1 X iajs-3014 82 24 ) ) PUNCT iajs-3014 82 25 + + PUNCT iajs-3014 82 26 πœ‡3(𝑣𝑖𝑣𝑛+𝑖 πœ‡3(𝑣𝑖𝑣𝑛+𝑖 PROPN iajs-3014 82 27 ) ) PUNCT iajs-3014 82 28 for for ADP iajs-3014 82 29 𝑖 𝑖 NOUN iajs-3014 82 30 = = SYM iajs-3014 82 31 2 2 NUM iajs-3014 82 32 , , PUNCT iajs-3014 82 33 3 3 NUM iajs-3014 82 34 … … PUNCT iajs-3014 82 35 , , PUNCT iajs-3014 82 36 𝑛 𝑛 DET iajs-3014 82 37 βˆ’ βˆ’ PROPN iajs-3014 82 38 1 1 NUM iajs-3014 82 39 , , PUNCT iajs-3014 82 40 = = PRON iajs-3014 82 41 ( ( PUNCT iajs-3014 82 42 5𝑖 5𝑖 NOUN iajs-3014 82 43 βˆ’ βˆ’ NOUN iajs-3014 82 44 4 4 NUM iajs-3014 82 45 ) ) PUNCT iajs-3014 82 46 + + CCONJ iajs-3014 82 47 ( ( PUNCT iajs-3014 82 48 5𝑖 5𝑖 NUM iajs-3014 82 49 βˆ’ βˆ’ NOUN iajs-3014 82 50 3 3 NUM iajs-3014 82 51 ) ) PUNCT iajs-3014 82 52 + + CCONJ iajs-3014 82 53 ( ( PUNCT iajs-3014 82 54 5𝑖 5𝑖 NUM iajs-3014 82 55 βˆ’ βˆ’ PROPN iajs-3014 82 56 8) 8) NUM iajs-3014 82 57 + + CCONJ iajs-3014 82 58 ( ( PUNCT iajs-3014 82 59 5𝑖 5𝑖 NOUN iajs-3014 82 60 βˆ’ βˆ’ PROPN iajs-3014 82 61 5 5 NUM iajs-3014 82 62 ) ) PUNCT iajs-3014 82 63 + + CCONJ iajs-3014 82 64 ( ( PUNCT iajs-3014 82 65 5𝑖 5𝑖 NUM iajs-3014 82 66 βˆ’ βˆ’ NOUN iajs-3014 82 67 2 2 X iajs-3014 82 68 ) ) PUNCT iajs-3014 82 69 = = NOUN iajs-3014 82 70 25𝑖 25𝑖 NOUN iajs-3014 82 71 βˆ’ βˆ’ PROPN iajs-3014 82 72 22 22 NUM iajs-3014 82 73 for for ADP iajs-3014 82 74 𝑖 𝑖 NOUN iajs-3014 82 75 = = SYM iajs-3014 82 76 2 2 NUM iajs-3014 82 77 , , PUNCT iajs-3014 82 78 3 3 NUM iajs-3014 82 79 , , PUNCT iajs-3014 82 80 … … PUNCT iajs-3014 82 81 , , PUNCT iajs-3014 82 82 𝑛 𝑛 DET iajs-3014 82 83 βˆ’ βˆ’ PROPN iajs-3014 82 84 1 1 NUM iajs-3014 82 85 , , PUNCT iajs-3014 82 86 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 82 87 ( ( PUNCT iajs-3014 82 88 𝑣𝑛 𝑣𝑛 NOUN iajs-3014 82 89 ) ) PUNCT iajs-3014 82 90 = = SYM iajs-3014 82 91 πœ‡3(𝑣𝑛𝑣 πœ‡3(𝑣𝑛𝑣 NOUN iajs-3014 82 92 ) ) PUNCT iajs-3014 82 93 + + NUM iajs-3014 82 94 πœ‡3(π‘£π‘›π‘£π‘›βˆ’1 πœ‡3(π‘£π‘›π‘£π‘›βˆ’1 NOUN iajs-3014 82 95 ) ) PUNCT iajs-3014 82 96 + + CCONJ iajs-3014 82 97 πœ‡3(𝑣𝑛𝑣2π‘›βˆ’1 πœ‡3(𝑣𝑛𝑣2π‘›βˆ’1 NOUN iajs-3014 82 98 ) ) PUNCT iajs-3014 82 99 = = PUNCT iajs-3014 83 1 ( ( PUNCT iajs-3014 83 2 5𝑛 5𝑛 NUM iajs-3014 83 3 βˆ’ βˆ’ NOUN iajs-3014 83 4 4 4 NUM iajs-3014 83 5 ) ) PUNCT iajs-3014 83 6 + + CCONJ iajs-3014 83 7 ( ( PUNCT iajs-3014 83 8 5𝑛 5𝑛 NUM iajs-3014 83 9 βˆ’ βˆ’ PROPN iajs-3014 83 10 8) 8) NUM iajs-3014 83 11 + + CCONJ iajs-3014 83 12 ( ( PUNCT iajs-3014 83 13 5𝑛 5𝑛 NUM iajs-3014 83 14 βˆ’ βˆ’ NOUN iajs-3014 83 15 5 5 NUM iajs-3014 83 16 ) ) PUNCT iajs-3014 83 17 = = NOUN iajs-3014 83 18 15𝑛 15𝑛 NUM iajs-3014 83 19 βˆ’ βˆ’ PROPN iajs-3014 83 20 17 17 NUM iajs-3014 83 21 , , PUNCT iajs-3014 83 22 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 83 23 ( ( PUNCT iajs-3014 83 24 𝑣𝑛+𝑖 𝑣𝑛+𝑖 NUM iajs-3014 83 25 ) ) PUNCT iajs-3014 83 26 = = SYM iajs-3014 83 27 πœ‡3(𝑣𝑛+𝑖𝑣𝑖 πœ‡3(𝑣𝑛+𝑖𝑣𝑖 NOUN iajs-3014 83 28 ) ) PUNCT iajs-3014 84 1 + + CCONJ iajs-3014 84 2 πœ‡3(𝑣𝑛+𝑖𝑣𝑖+1 πœ‡3(𝑣𝑛+𝑖𝑣𝑖+1 NOUN iajs-3014 84 3 ) ) PUNCT iajs-3014 85 1 + + NOUN iajs-3014 85 2 πœ‡3(𝑣𝑛+𝑖𝑣 πœ‡3(𝑣𝑛+𝑖𝑣 NUM iajs-3014 85 3 ) ) PUNCT iajs-3014 85 4 for for ADP iajs-3014 85 5 𝑖 𝑖 PRON iajs-3014 85 6 = = SYM iajs-3014 85 7 1,2 1,2 NUM iajs-3014 85 8 , , PUNCT iajs-3014 85 9 … … PUNCT iajs-3014 85 10 , , PUNCT iajs-3014 85 11 𝑛 𝑛 DET iajs-3014 85 12 βˆ’ βˆ’ PROPN iajs-3014 85 13 1 1 NUM iajs-3014 85 14 , , PUNCT iajs-3014 85 15 = = PRON iajs-3014 85 16 ( ( PUNCT iajs-3014 85 17 5𝑖 5𝑖 NOUN iajs-3014 85 18 βˆ’ βˆ’ NOUN iajs-3014 85 19 2 2 NUM iajs-3014 85 20 ) ) PUNCT iajs-3014 85 21 + + CCONJ iajs-3014 85 22 ( ( PUNCT iajs-3014 85 23 5𝑖 5𝑖 NUM iajs-3014 85 24 ) ) PUNCT iajs-3014 86 1 + + CCONJ iajs-3014 86 2 ( ( PUNCT iajs-3014 86 3 5𝑖 5𝑖 NUM iajs-3014 86 4 βˆ’ βˆ’ NOUN iajs-3014 86 5 1 1 NUM iajs-3014 86 6 ) ) PUNCT iajs-3014 86 7 for for ADP iajs-3014 86 8 𝑖 𝑖 PRON iajs-3014 86 9 = = SYM iajs-3014 86 10 1,2 1,2 NUM iajs-3014 86 11 , , PUNCT iajs-3014 86 12 … … PUNCT iajs-3014 86 13 , , PUNCT iajs-3014 86 14 𝑛 𝑛 DET iajs-3014 86 15 βˆ’ βˆ’ PROPN iajs-3014 86 16 1 1 NUM iajs-3014 86 17 , , PUNCT iajs-3014 86 18 = = SYM iajs-3014 86 19 15𝑖 15𝑖 X iajs-3014 86 20 βˆ’ βˆ’ NOUN iajs-3014 86 21 3 3 NUM iajs-3014 86 22 for for ADP iajs-3014 86 23 𝑖 𝑖 PRON iajs-3014 86 24 = = SYM iajs-3014 86 25 1,2 1,2 NUM iajs-3014 86 26 , , PUNCT iajs-3014 86 27 … … PUNCT iajs-3014 86 28 , , PUNCT iajs-3014 86 29 𝑛 𝑛 DET iajs-3014 86 30 βˆ’ βˆ’ NOUN iajs-3014 86 31 1 1 NUM iajs-3014 86 32 . . PUNCT iajs-3014 86 33 finally finally ADV iajs-3014 86 34 : : PUNCT iajs-3014 86 35 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 86 36 ( ( PUNCT iajs-3014 86 37 𝑣 𝑣 X iajs-3014 86 38 ) ) PUNCT iajs-3014 86 39 = = SYM iajs-3014 86 40 βˆ‘ βˆ‘ PUNCT iajs-3014 86 41 πœ‡3(𝑣𝑣𝑖 πœ‡3(𝑣𝑣𝑖 PROPN iajs-3014 86 42 ) ) PUNCT iajs-3014 86 43 𝑛 𝑛 PRON iajs-3014 86 44 𝑖=1 𝑖=1 PROPN iajs-3014 86 45 + + CCONJ iajs-3014 86 46 βˆ‘ βˆ‘ ADV iajs-3014 86 47 πœ‡3(𝑣𝑣𝑛+𝑖 πœ‡3(𝑣𝑣𝑛+𝑖 NOUN iajs-3014 86 48 ) ) PUNCT iajs-3014 86 49 π‘›βˆ’1 π‘›βˆ’1 PROPN iajs-3014 86 50 𝑖=1 𝑖=1 PUNCT iajs-3014 86 51 = = SYM iajs-3014 86 52 βˆ‘(5𝑖 βˆ‘(5𝑖 PROPN iajs-3014 86 53 βˆ’ βˆ’ PROPN iajs-3014 86 54 4 4 NUM iajs-3014 86 55 ) ) PUNCT iajs-3014 86 56 𝑛 𝑛 PRON iajs-3014 86 57 𝑖=1 𝑖=1 PROPN iajs-3014 86 58 + + CCONJ iajs-3014 86 59 βˆ‘(5𝑖 βˆ‘(5𝑖 PROPN iajs-3014 86 60 βˆ’ βˆ’ PROPN iajs-3014 86 61 1 1 NUM iajs-3014 86 62 ) ) PUNCT iajs-3014 86 63 π‘›βˆ’1 π‘›βˆ’1 NUM iajs-3014 86 64 𝑖=1 𝑖=1 PUNCT iajs-3014 86 65 = = SYM iajs-3014 86 66 5 5 NUM iajs-3014 86 67 ( ( PUNCT iajs-3014 86 68 𝑛2 𝑛2 NOUN iajs-3014 86 69 + + CCONJ iajs-3014 86 70 𝑛 𝑛 DET iajs-3014 86 71 2 2 NUM iajs-3014 86 72 ) ) PUNCT iajs-3014 86 73 βˆ’ βˆ’ NOUN iajs-3014 86 74 4𝑛 4𝑛 NOUN iajs-3014 87 1 + + CCONJ iajs-3014 87 2 5 5 NUM iajs-3014 87 3 ( ( PUNCT iajs-3014 87 4 𝑛2 𝑛2 NOUN iajs-3014 87 5 βˆ’ βˆ’ PROPN iajs-3014 87 6 𝑛 𝑛 PROPN iajs-3014 87 7 2 2 NUM iajs-3014 87 8 ) ) PUNCT iajs-3014 87 9 βˆ’ βˆ’ NOUN iajs-3014 87 10 𝑛 𝑛 DET iajs-3014 88 1 + + NOUN iajs-3014 88 2 1 1 NUM iajs-3014 88 3 ihjpas ihjpa NOUN iajs-3014 88 4 . . PUNCT iajs-3014 89 1 36(1)2023 36(1)2023 NUM iajs-3014 89 2 288 288 NUM iajs-3014 89 3 = = SYM iajs-3014 89 4 5𝑛2 5𝑛2 NOUN iajs-3014 89 5 βˆ’ βˆ’ NUM iajs-3014 89 6 5𝑛 5𝑛 NUM iajs-3014 89 7 + + X iajs-3014 89 8 1 1 X iajs-3014 89 9 . . PUNCT iajs-3014 90 1 so so ADV iajs-3014 90 2 , , PUNCT iajs-3014 90 3 the the DET iajs-3014 90 4 weights weight NOUN iajs-3014 90 5 of of ADP iajs-3014 90 6 all all DET iajs-3014 90 7 vertices vertex NOUN iajs-3014 90 8 are be AUX iajs-3014 90 9 distinct distinct ADJ iajs-3014 90 10 and and CCONJ iajs-3014 90 11 the the DET iajs-3014 90 12 following follow VERB iajs-3014 90 13 notes note NOUN iajs-3014 90 14 are be AUX iajs-3014 90 15 observed observe VERB iajs-3014 90 16 : : PUNCT iajs-3014 90 17 1π‘€π‘‘πœ‡3 1π‘€π‘‘πœ‡3 NUM iajs-3014 90 18 ( ( PUNCT iajs-3014 90 19 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 90 20 ) ) PUNCT iajs-3014 90 21 < < X iajs-3014 90 22 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 PROPN iajs-3014 90 23 ( ( PUNCT iajs-3014 90 24 𝑣𝑖+1 𝑣𝑖+1 NUM iajs-3014 90 25 ) ) PUNCT iajs-3014 90 26 for for ADP iajs-3014 90 27 𝑖 𝑖 PRON iajs-3014 90 28 = = SYM iajs-3014 90 29 1,2 1,2 NUM iajs-3014 90 30 , , PUNCT iajs-3014 90 31 … … PUNCT iajs-3014 90 32 , , PUNCT iajs-3014 90 33 𝑛 𝑛 DET iajs-3014 90 34 βˆ’ βˆ’ PROPN iajs-3014 90 35 1 1 NUM iajs-3014 90 36 , , PUNCT iajs-3014 90 37 2π‘€π‘‘πœ‡3 2π‘€π‘‘πœ‡3 NUM iajs-3014 90 38 ( ( PUNCT iajs-3014 90 39 𝑣𝑛+𝑖 𝑣𝑛+𝑖 NUM iajs-3014 90 40 ) ) PUNCT iajs-3014 90 41 < < X iajs-3014 90 42 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 PROPN iajs-3014 90 43 ( ( PUNCT iajs-3014 90 44 𝑣𝑛+𝑖+1 𝑣𝑛+𝑖+1 NOUN iajs-3014 90 45 ) ) PUNCT iajs-3014 90 46 for for ADP iajs-3014 90 47 𝑖 𝑖 PRON iajs-3014 90 48 = = SYM iajs-3014 90 49 1,2 1,2 NUM iajs-3014 90 50 , , PUNCT iajs-3014 90 51 … … PUNCT iajs-3014 90 52 , , PUNCT iajs-3014 90 53 𝑛 𝑛 DET iajs-3014 90 54 βˆ’ βˆ’ PROPN iajs-3014 90 55 2 2 NUM iajs-3014 90 56 . . PUNCT iajs-3014 91 1 thus thus ADV iajs-3014 91 2 , , PUNCT iajs-3014 91 3 the the DET iajs-3014 91 4 strong strong ADJ iajs-3014 91 5 face face NOUN iajs-3014 91 6 fan fan NOUN iajs-3014 91 7 graph graph NOUN iajs-3014 91 8 𝐹𝑛 𝐹𝑛 PROPN iajs-3014 91 9 βˆ— βˆ— NOUN iajs-3014 91 10 is be AUX iajs-3014 91 11 vertex vertex NOUN iajs-3014 91 12 antimagic antimagic ADJ iajs-3014 91 13 edge edge NOUN iajs-3014 91 14 labeling labeling NOUN iajs-3014 91 15 , , PUNCT iajs-3014 91 16 for for ADP iajs-3014 91 17 every every DET iajs-3014 91 18 𝑛 𝑛 PROPN iajs-3014 91 19 , , PUNCT iajs-3014 91 20 𝑛 𝑛 DET iajs-3014 91 21 β‰₯ β‰₯ NOUN iajs-3014 91 22 4 4 NUM iajs-3014 91 23 , , PUNCT iajs-3014 91 24 𝑛 𝑛 DET iajs-3014 91 25 β‰’ β‰’ NUM iajs-3014 91 26 3 3 NUM iajs-3014 91 27 ( ( PUNCT iajs-3014 91 28 mod mod NOUN iajs-3014 91 29 5 5 NUM iajs-3014 91 30 ) ) PUNCT iajs-3014 91 31 . . PUNCT iajs-3014 92 1 however however ADV iajs-3014 92 2 , , PUNCT iajs-3014 92 3 when when SCONJ iajs-3014 92 4 𝑛 𝑛 PROPN iajs-3014 92 5 = = SYM iajs-3014 92 6 8 8 NUM iajs-3014 92 7 , , PUNCT iajs-3014 92 8 we we PRON iajs-3014 92 9 can can AUX iajs-3014 92 10 easily easily ADV iajs-3014 92 11 show show VERB iajs-3014 92 12 that that SCONJ iajs-3014 92 13 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 92 14 ( ( PUNCT iajs-3014 92 15 𝑣5 𝑣5 NOUN iajs-3014 92 16 ) ) PUNCT iajs-3014 92 17 = = SYM iajs-3014 92 18 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 92 19 ( ( PUNCT iajs-3014 92 20 𝑣8 𝑣8 NOUN iajs-3014 92 21 ) ) PUNCT iajs-3014 92 22 and and CCONJ iajs-3014 92 23 similarly similarly ADV iajs-3014 92 24 when when SCONJ iajs-3014 92 25 𝑛 𝑛 PROPN iajs-3014 92 26 = = SYM iajs-3014 92 27 13 13 NUM iajs-3014 92 28 , , PUNCT iajs-3014 92 29 π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’π‘› VERB iajs-3014 92 30 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 92 31 ( ( PUNCT iajs-3014 92 32 𝑣8 𝑣8 NOUN iajs-3014 92 33 ) ) PUNCT iajs-3014 92 34 = = SYM iajs-3014 92 35 π‘€π‘‘πœ‡3 π‘€π‘‘πœ‡3 NOUN iajs-3014 92 36 ( ( PUNCT iajs-3014 92 37 𝑣13 𝑣13 PROPN iajs-3014 92 38 ) ) PUNCT iajs-3014 92 39 and and CCONJ iajs-3014 92 40 so so ADV iajs-3014 92 41 on on ADV iajs-3014 92 42 . . PUNCT iajs-3014 93 1 theorem4 theorem4 X iajs-3014 94 1 the the DET iajs-3014 94 2 strong strong ADJ iajs-3014 94 3 face face NOUN iajs-3014 94 4 prism prism NOUN iajs-3014 94 5 graph graph NOUN iajs-3014 94 6 ( ( PUNCT iajs-3014 94 7 𝐢𝑛 𝐢𝑛 INTJ iajs-3014 94 8 Γ— Γ— NOUN iajs-3014 94 9 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 94 10 is be AUX iajs-3014 94 11 vael vael ADJ iajs-3014 94 12 , , PUNCT iajs-3014 94 13 for for ADP iajs-3014 94 14 every every DET iajs-3014 94 15 𝑛 𝑛 DET iajs-3014 94 16 β‰₯ β‰₯ NOUN iajs-3014 94 17 3 3 NUM iajs-3014 94 18 , , PUNCT iajs-3014 94 19 𝑛 𝑛 DET iajs-3014 94 20 β‰’ β‰’ NUM iajs-3014 94 21 0 0 NUM iajs-3014 94 22 ( ( PUNCT iajs-3014 94 23 mod mod PROPN iajs-3014 94 24 5 5 NUM iajs-3014 94 25 ) ) PUNCT iajs-3014 94 26 . . PUNCT iajs-3014 95 1 proof proof NOUN iajs-3014 95 2 : : PUNCT iajs-3014 95 3 we we PRON iajs-3014 95 4 define define VERB iajs-3014 95 5 the the DET iajs-3014 95 6 vertex vertex NOUN iajs-3014 95 7 and and CCONJ iajs-3014 95 8 edge edge NOUN iajs-3014 95 9 sets set NOUN iajs-3014 95 10 of of ADP iajs-3014 95 11 ( ( PUNCT iajs-3014 95 12 𝐢𝑛 𝐢𝑛 INTJ iajs-3014 95 13 Γ— Γ— NOUN iajs-3014 95 14 𝑃2)βˆ—graph 𝑃2)βˆ—graph NOUN iajs-3014 95 15 as as SCONJ iajs-3014 95 16 follows follow VERB iajs-3014 95 17 ; ; PUNCT iajs-3014 95 18 𝑉((𝐢𝑛 𝑉((𝐢𝑛 NOUN iajs-3014 95 19 Γ— Γ— PROPN iajs-3014 95 20 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 95 21 ) ) PUNCT iajs-3014 95 22 = = PRON iajs-3014 95 23 { { PUNCT iajs-3014 95 24 𝑣𝑖 𝑣𝑖 ADP iajs-3014 95 25 : : PUNCT iajs-3014 96 1 𝑖 𝑖 SYM iajs-3014 96 2 = = SYM iajs-3014 96 3 1 1 NUM iajs-3014 96 4 , , PUNCT iajs-3014 96 5 2 2 NUM iajs-3014 96 6 , , PUNCT iajs-3014 96 7 … … PUNCT iajs-3014 96 8 , , PUNCT iajs-3014 96 9 3𝑛 3𝑛 NUM iajs-3014 96 10 + + CCONJ iajs-3014 96 11 1 1 NUM iajs-3014 96 12 } } PUNCT iajs-3014 96 13 , , PUNCT iajs-3014 96 14 𝐸 𝐸 PROPN iajs-3014 96 15 ( ( PUNCT iajs-3014 96 16 ( ( PUNCT iajs-3014 96 17 𝐢𝑛 𝐢𝑛 INTJ iajs-3014 96 18 Γ— Γ— NOUN iajs-3014 96 19 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 96 20 ) ) PUNCT iajs-3014 96 21 = = PRON iajs-3014 96 22 { { PUNCT iajs-3014 96 23 𝑣𝑖𝑣𝑖+1 𝑣𝑖𝑣𝑖+1 PROPN iajs-3014 96 24 , , PUNCT iajs-3014 97 1 𝑣𝑛+𝑖𝑣𝑛+𝑖+1 𝑣𝑛+𝑖𝑣𝑛+𝑖+1 NUM iajs-3014 97 2 ∢ ∢ NOUN iajs-3014 97 3 𝑖 𝑖 X iajs-3014 97 4 = = SYM iajs-3014 97 5 1 1 NUM iajs-3014 97 6 , , PUNCT iajs-3014 97 7 2 2 NUM iajs-3014 97 8 , , PUNCT iajs-3014 97 9 … … PUNCT iajs-3014 97 10 𝑛 𝑛 PRON iajs-3014 97 11 βˆ’ βˆ’ NUM iajs-3014 97 12 1 1 NUM iajs-3014 97 13 } } PUNCT iajs-3014 97 14 βˆͺ βˆͺ ADJ iajs-3014 97 15 { { PUNCT iajs-3014 97 16 𝑣𝑛𝑣1 𝑣𝑛𝑣1 NOUN iajs-3014 97 17 , , PUNCT iajs-3014 97 18 𝑣2𝑛𝑣𝑛+1 𝑣2𝑛𝑣𝑛+1 NOUN iajs-3014 97 19 , , PUNCT iajs-3014 97 20 𝑣3𝑛𝑣1 𝑣3𝑛𝑣1 NOUN iajs-3014 97 21 , , PUNCT iajs-3014 97 22 𝑣3𝑛𝑣𝑛+1 𝑣3𝑛𝑣𝑛+1 NOUN iajs-3014 97 23 } } PUNCT iajs-3014 97 24 βˆͺ βˆͺ NOUN iajs-3014 97 25 { { PUNCT iajs-3014 97 26 𝑣3𝑛+1𝑣𝑛+𝑖 𝑣3𝑛+1𝑣𝑛+𝑖 PROPN iajs-3014 97 27 , , PUNCT iajs-3014 97 28 𝑣𝑛+𝑖𝑣𝑖 𝑣𝑛+𝑖𝑣𝑖 X iajs-3014 97 29 ∢ ∢ NOUN iajs-3014 97 30 𝑖 𝑖 X iajs-3014 97 31 = = SYM iajs-3014 97 32 1 1 NUM iajs-3014 97 33 , , PUNCT iajs-3014 97 34 2 2 NUM iajs-3014 97 35 … … SYM iajs-3014 97 36 𝑛 𝑛 NOUN iajs-3014 97 37 } } PUNCT iajs-3014 97 38 βˆͺ βˆͺ X iajs-3014 97 39 { { PUNCT iajs-3014 97 40 𝑣2𝑛+𝑖𝑣𝑖 𝑣2𝑛+𝑖𝑣𝑖 NUM iajs-3014 97 41 , , PUNCT iajs-3014 97 42 𝑣2𝑛+𝑖𝑣𝑛+𝑖 𝑣2𝑛+𝑖𝑣𝑛+𝑖 PROPN iajs-3014 97 43 ∢ ∢ NOUN iajs-3014 97 44 𝑖 𝑖 NOUN iajs-3014 97 45 = = SYM iajs-3014 97 46 1 1 NUM iajs-3014 97 47 , , PUNCT iajs-3014 97 48 2 2 NUM iajs-3014 97 49 … … SYM iajs-3014 97 50 𝑛 𝑛 NOUN iajs-3014 97 51 } } PUNCT iajs-3014 97 52 βˆͺ βˆͺ X iajs-3014 97 53 { { PUNCT iajs-3014 97 54 𝑣2𝑛+𝑖𝑣𝑖+1 𝑣2𝑛+𝑖𝑣𝑖+1 PROPN iajs-3014 97 55 , , PUNCT iajs-3014 97 56 𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 PROPN iajs-3014 97 57 ∢ ∢ NOUN iajs-3014 97 58 𝑖 𝑖 X iajs-3014 97 59 = = SYM iajs-3014 97 60 1 1 NUM iajs-3014 97 61 , , PUNCT iajs-3014 97 62 2 2 NUM iajs-3014 97 63 … … PUNCT iajs-3014 97 64 𝑛 𝑛 PRON iajs-3014 97 65 βˆ’ βˆ’ NOUN iajs-3014 97 66 1 1 NUM iajs-3014 97 67 } } PUNCT iajs-3014 97 68 . . PUNCT iajs-3014 98 1 for for ADP iajs-3014 98 2 𝑛 𝑛 PROPN iajs-3014 98 3 β‰₯ β‰₯ NUM iajs-3014 98 4 3 3 NUM iajs-3014 98 5 , , PUNCT iajs-3014 98 6 𝑛 𝑛 DET iajs-3014 98 7 β‰’ β‰’ NUM iajs-3014 98 8 0 0 NUM iajs-3014 98 9 ( ( PUNCT iajs-3014 98 10 mod mod NOUN iajs-3014 98 11 5 5 NUM iajs-3014 98 12 ) ) PUNCT iajs-3014 98 13 we we PRON iajs-3014 98 14 define define VERB iajs-3014 98 15 the the DET iajs-3014 98 16 labeling labeling NOUN iajs-3014 98 17 of of ADP iajs-3014 98 18 ( ( PUNCT iajs-3014 98 19 𝐢𝑛 𝐢𝑛 INTJ iajs-3014 98 20 Γ— Γ— NOUN iajs-3014 98 21 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 98 22 graph graph NOUN iajs-3014 98 23 as as ADP iajs-3014 98 24 : : PUNCT iajs-3014 98 25 πœ‡4 πœ‡4 NOUN iajs-3014 98 26 : : PUNCT iajs-3014 98 27 𝐸((𝐢𝑛 𝐸((𝐢𝑛 NOUN iajs-3014 98 28 Γ— Γ— NOUN iajs-3014 98 29 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 98 30 ) ) PUNCT iajs-3014 98 31 β†’ β†’ SYM iajs-3014 98 32 { { PUNCT iajs-3014 98 33 1 1 NUM iajs-3014 98 34 , , PUNCT iajs-3014 98 35 2 2 NUM iajs-3014 98 36 , , PUNCT iajs-3014 98 37 … … PUNCT iajs-3014 98 38 , , PUNCT iajs-3014 98 39 8𝑛 8𝑛 NOUN iajs-3014 98 40 } } PUNCT iajs-3014 98 41 . . PUNCT iajs-3014 99 1 such such ADJ iajs-3014 99 2 that that PRON iajs-3014 99 3 : : PUNCT iajs-3014 99 4 for for ADP iajs-3014 99 5 𝑖 𝑖 PRON iajs-3014 99 6 = = SYM iajs-3014 99 7 1 1 NUM iajs-3014 99 8 , , PUNCT iajs-3014 99 9 2 2 NUM iajs-3014 99 10 , , PUNCT iajs-3014 99 11 … … PUNCT iajs-3014 99 12 , , PUNCT iajs-3014 99 13 𝑛 𝑛 ADJ iajs-3014 99 14 , , PUNCT iajs-3014 99 15 πœ‡4(𝑣3𝑛+1𝑣𝑛+𝑖 πœ‡4(𝑣3𝑛+1𝑣𝑛+𝑖 ADV iajs-3014 99 16 ) ) PUNCT iajs-3014 99 17 = = SYM iajs-3014 99 18 𝑖 𝑖 SYM iajs-3014 99 19 πœ‡4(𝑣𝑛+𝑖𝑣𝑖 πœ‡4(𝑣𝑛+𝑖𝑣𝑖 PROPN iajs-3014 99 20 ) ) PUNCT iajs-3014 99 21 = = PUNCT iajs-3014 100 1 4𝑛 4𝑛 NOUN iajs-3014 101 1 + + CCONJ iajs-3014 101 2 3𝑖 3𝑖 NUM iajs-3014 101 3 βˆ’ βˆ’ PROPN iajs-3014 101 4 1 1 NUM iajs-3014 101 5 , , PUNCT iajs-3014 101 6 πœ‡4(𝑣2𝑛+𝑖𝑣𝑖 πœ‡4(𝑣2𝑛+𝑖𝑣𝑖 ADV iajs-3014 101 7 ) ) PUNCT iajs-3014 101 8 = = SYM iajs-3014 101 9 4𝑛 4𝑛 PROPN iajs-3014 101 10 + + CCONJ iajs-3014 101 11 3𝑖 3𝑖 NUM iajs-3014 101 12 , , PUNCT iajs-3014 101 13 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖 X iajs-3014 101 14 ) ) PUNCT iajs-3014 101 15 = = SYM iajs-3014 101 16 2𝑛 2𝑛 PROPN iajs-3014 102 1 + + CCONJ iajs-3014 102 2 2𝑖 2𝑖 NUM iajs-3014 102 3 βˆ’ βˆ’ NOUN iajs-3014 103 1 1 1 X iajs-3014 103 2 . . PUNCT iajs-3014 104 1 and and CCONJ iajs-3014 104 2 for for ADP iajs-3014 104 3 𝑖 𝑖 PRON iajs-3014 104 4 = = SYM iajs-3014 104 5 1 1 NUM iajs-3014 104 6 , , PUNCT iajs-3014 104 7 2 2 NUM iajs-3014 104 8 , , PUNCT iajs-3014 104 9 … … PUNCT iajs-3014 104 10 𝑛 𝑛 PRON iajs-3014 104 11 βˆ’ βˆ’ PROPN iajs-3014 104 12 1 1 NUM iajs-3014 104 13 , , PUNCT iajs-3014 104 14 πœ‡4(𝑣𝑖𝑣𝑖+1 πœ‡4(𝑣𝑖𝑣𝑖+1 NOUN iajs-3014 104 15 ) ) PUNCT iajs-3014 104 16 = = SYM iajs-3014 104 17 7𝑛 7𝑛 NOUN iajs-3014 104 18 + + SYM iajs-3014 104 19 𝑖 𝑖 NUM iajs-3014 104 20 , , PUNCT iajs-3014 104 21 πœ‡4(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 πœ‡4(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 X iajs-3014 104 22 ) ) PUNCT iajs-3014 104 23 = = SYM iajs-3014 104 24 𝑛 𝑛 PROPN iajs-3014 104 25 + + SYM iajs-3014 104 26 𝑖 𝑖 SYM iajs-3014 104 27 , , PUNCT iajs-3014 104 28 πœ‡4(𝑣2𝑛+𝑖𝑣𝑖+1 πœ‡4(𝑣2𝑛+𝑖𝑣𝑖+1 NOUN iajs-3014 104 29 ) ) PUNCT iajs-3014 104 30 = = SYM iajs-3014 104 31 4𝑛 4𝑛 NOUN iajs-3014 105 1 + + CCONJ iajs-3014 105 2 3𝑖 3𝑖 NUM iajs-3014 105 3 + + CCONJ iajs-3014 105 4 1 1 NUM iajs-3014 105 5 , , PUNCT iajs-3014 105 6 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 PROPN iajs-3014 105 7 ) ) PUNCT iajs-3014 105 8 = = SYM iajs-3014 106 1 2𝑛 2𝑛 PROPN iajs-3014 106 2 + + CCONJ iajs-3014 106 3 2𝑖. 2𝑖. NUM iajs-3014 106 4 finally finally ADV iajs-3014 106 5 , , PUNCT iajs-3014 106 6 πœ‡4(𝑣2𝑛𝑣𝑛+1 πœ‡4(𝑣2𝑛𝑣𝑛+1 PROPN iajs-3014 106 7 ) ) PUNCT iajs-3014 106 8 = = SYM iajs-3014 106 9 2𝑛 2𝑛 NUM iajs-3014 106 10 , , PUNCT iajs-3014 106 11 πœ‡4(𝑣𝑛𝑣1 πœ‡4(𝑣𝑛𝑣1 NOUN iajs-3014 106 12 ) ) PUNCT iajs-3014 106 13 = = SYM iajs-3014 106 14 8𝑛 8𝑛 NOUN iajs-3014 106 15 , , PUNCT iajs-3014 106 16 πœ‡4(𝑣3𝑛𝑣1 πœ‡4(𝑣3𝑛𝑣1 NOUN iajs-3014 106 17 ) ) PUNCT iajs-3014 106 18 = = PUNCT iajs-3014 107 1 4𝑛 4𝑛 NOUN iajs-3014 108 1 + + CCONJ iajs-3014 108 2 1 1 NUM iajs-3014 108 3 , , PUNCT iajs-3014 108 4 πœ‡4(𝑣3𝑛𝑣𝑛+1 πœ‡4(𝑣3𝑛𝑣𝑛+1 PROPN iajs-3014 108 5 ) ) PUNCT iajs-3014 108 6 = = SYM iajs-3014 109 1 4𝑛. 4𝑛. NUM iajs-3014 109 2 for for ADP iajs-3014 109 3 the the DET iajs-3014 109 4 vertex vertex NOUN iajs-3014 109 5 - - PUNCT iajs-3014 109 6 weights weight NOUN iajs-3014 109 7 we we PRON iajs-3014 109 8 get get VERB iajs-3014 109 9 : : PUNCT iajs-3014 109 10 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 109 11 ( ( PUNCT iajs-3014 109 12 𝑣1 𝑣1 PROPN iajs-3014 109 13 ) ) PUNCT iajs-3014 109 14 = = SYM iajs-3014 109 15 πœ‡4(𝑣1 πœ‡4(𝑣1 AUX iajs-3014 109 16 𝑣𝑛 𝑣𝑛 NOUN iajs-3014 109 17 ) ) PUNCT iajs-3014 109 18 + + CCONJ iajs-3014 109 19 πœ‡4(𝑣1𝑣2 πœ‡4(𝑣1𝑣2 NOUN iajs-3014 109 20 ) ) PUNCT iajs-3014 109 21 + + NUM iajs-3014 109 22 πœ‡4(𝑣1𝑣𝑛+1 πœ‡4(𝑣1𝑣𝑛+1 ADJ iajs-3014 109 23 ) ) PUNCT iajs-3014 109 24 + + NUM iajs-3014 109 25 πœ‡4(𝑣1𝑣3𝑛 πœ‡4(𝑣1𝑣3𝑛 NUM iajs-3014 109 26 ) ) PUNCT iajs-3014 110 1 + + CCONJ iajs-3014 110 2 πœ‡4(𝑣1𝑣2𝑛+1 πœ‡4(𝑣1𝑣2𝑛+1 X iajs-3014 110 3 ) ) PUNCT iajs-3014 110 4 = = PUNCT iajs-3014 110 5 ( ( PUNCT iajs-3014 110 6 8𝑛 8𝑛 NOUN iajs-3014 110 7 ) ) PUNCT iajs-3014 111 1 + + CCONJ iajs-3014 111 2 ( ( PUNCT iajs-3014 111 3 7𝑛 7𝑛 NUM iajs-3014 111 4 + + NOUN iajs-3014 111 5 1 1 NUM iajs-3014 111 6 ) ) PUNCT iajs-3014 111 7 + + CCONJ iajs-3014 111 8 ( ( PUNCT iajs-3014 111 9 4𝑛 4𝑛 NOUN iajs-3014 111 10 + + CCONJ iajs-3014 111 11 2 2 X iajs-3014 111 12 ) ) PUNCT iajs-3014 111 13 + + CCONJ iajs-3014 111 14 ( ( PUNCT iajs-3014 111 15 4𝑛 4𝑛 NOUN iajs-3014 111 16 + + CCONJ iajs-3014 111 17 1 1 X iajs-3014 111 18 ) ) PUNCT iajs-3014 111 19 + + CCONJ iajs-3014 111 20 ( ( PUNCT iajs-3014 111 21 4𝑛 4𝑛 NOUN iajs-3014 111 22 + + CCONJ iajs-3014 111 23 3 3 X iajs-3014 111 24 ) ) PUNCT iajs-3014 111 25 = = NOUN iajs-3014 111 26 27𝑛 27𝑛 X iajs-3014 112 1 + + CCONJ iajs-3014 112 2 7 7 NUM iajs-3014 112 3 , , PUNCT iajs-3014 112 4 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 112 5 ( ( PUNCT iajs-3014 112 6 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 112 7 ) ) PUNCT iajs-3014 112 8 = = SYM iajs-3014 113 1 πœ‡4(𝑣𝑖 πœ‡4(𝑣𝑖 PROPN iajs-3014 113 2 𝑣𝑖+1 𝑣𝑖+1 NUM iajs-3014 113 3 ) ) PUNCT iajs-3014 113 4 + + NUM iajs-3014 113 5 πœ‡4(π‘£π‘–π‘£π‘–βˆ’1 πœ‡4(π‘£π‘–π‘£π‘–βˆ’1 X iajs-3014 113 6 ) ) PUNCT iajs-3014 113 7 + + X iajs-3014 113 8 πœ‡4(𝑣𝑖𝑣2𝑛+𝑖 πœ‡4(𝑣𝑖𝑣2𝑛+𝑖 PROPN iajs-3014 113 9 ) ) PUNCT iajs-3014 113 10 + + X iajs-3014 113 11 πœ‡4(𝑣𝑖𝑣2𝑛+π‘–βˆ’1 πœ‡4(𝑣𝑖𝑣2𝑛+π‘–βˆ’1 ADJ iajs-3014 113 12 ) ) PUNCT iajs-3014 113 13 + + PUNCT iajs-3014 113 14 πœ‡4(𝑣𝑖𝑣𝑛+𝑖 πœ‡4(𝑣𝑖𝑣𝑛+𝑖 PROPN iajs-3014 113 15 ) ) PUNCT iajs-3014 113 16 for for ADP iajs-3014 113 17 𝑖 𝑖 NOUN iajs-3014 113 18 = = SYM iajs-3014 113 19 2 2 NUM iajs-3014 113 20 , , PUNCT iajs-3014 113 21 3 3 NUM iajs-3014 113 22 , , PUNCT iajs-3014 113 23 … … PUNCT iajs-3014 113 24 , , PUNCT iajs-3014 113 25 𝑛 𝑛 DET iajs-3014 113 26 βˆ’ βˆ’ PROPN iajs-3014 113 27 1 1 NUM iajs-3014 113 28 , , PUNCT iajs-3014 113 29 = = PRON iajs-3014 113 30 ( ( PUNCT iajs-3014 113 31 7𝑛 7𝑛 NUM iajs-3014 113 32 + + SYM iajs-3014 113 33 𝑖 𝑖 NOUN iajs-3014 113 34 ) ) PUNCT iajs-3014 113 35 + + CCONJ iajs-3014 113 36 ( ( PUNCT iajs-3014 113 37 7𝑛 7𝑛 NUM iajs-3014 113 38 + + NOUN iajs-3014 113 39 𝑖 𝑖 NOUN iajs-3014 113 40 βˆ’ βˆ’ NOUN iajs-3014 113 41 1 1 NUM iajs-3014 113 42 ) ) PUNCT iajs-3014 113 43 + + CCONJ iajs-3014 113 44 ( ( PUNCT iajs-3014 113 45 4𝑛 4𝑛 NOUN iajs-3014 113 46 + + CCONJ iajs-3014 113 47 3𝑖 3𝑖 NUM iajs-3014 113 48 ) ) PUNCT iajs-3014 114 1 + + CCONJ iajs-3014 114 2 ( ( PUNCT iajs-3014 114 3 4𝑛 4𝑛 NOUN iajs-3014 114 4 + + CCONJ iajs-3014 114 5 3𝑖 3𝑖 NUM iajs-3014 114 6 βˆ’ βˆ’ NOUN iajs-3014 114 7 2 2 NUM iajs-3014 114 8 ) ) PUNCT iajs-3014 114 9 + + CCONJ iajs-3014 114 10 ( ( PUNCT iajs-3014 114 11 4𝑛 4𝑛 NOUN iajs-3014 114 12 + + CCONJ iajs-3014 114 13 3𝑖 3𝑖 NUM iajs-3014 114 14 βˆ’ βˆ’ NOUN iajs-3014 114 15 1 1 NUM iajs-3014 114 16 ) ) PUNCT iajs-3014 114 17 = = NOUN iajs-3014 114 18 26𝑛 26𝑛 NOUN iajs-3014 114 19 + + NUM iajs-3014 114 20 11𝑖 11𝑖 NUM iajs-3014 114 21 βˆ’ βˆ’ ADP iajs-3014 114 22 4 4 NUM iajs-3014 114 23 for for ADP iajs-3014 114 24 𝑖 𝑖 NOUN iajs-3014 114 25 = = SYM iajs-3014 114 26 2 2 NUM iajs-3014 114 27 , , PUNCT iajs-3014 114 28 3 3 NUM iajs-3014 114 29 , , PUNCT iajs-3014 114 30 … … PUNCT iajs-3014 114 31 , , PUNCT iajs-3014 114 32 𝑛 𝑛 PRON iajs-3014 114 33 βˆ’ βˆ’ PROPN iajs-3014 114 34 1 1 NUM iajs-3014 114 35 , , PUNCT iajs-3014 114 36 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 114 37 ( ( PUNCT iajs-3014 114 38 𝑣𝑛 𝑣𝑛 NOUN iajs-3014 114 39 ) ) PUNCT iajs-3014 114 40 = = SYM iajs-3014 114 41 πœ‡4(𝑣𝑛𝑣1 πœ‡4(𝑣𝑛𝑣1 NOUN iajs-3014 114 42 ) ) PUNCT iajs-3014 114 43 + + ADJ iajs-3014 114 44 πœ‡4(π‘£π‘›π‘£π‘›βˆ’1 πœ‡4(π‘£π‘›π‘£π‘›βˆ’1 NOUN iajs-3014 114 45 ) ) PUNCT iajs-3014 114 46 + + CCONJ iajs-3014 114 47 πœ‡4(𝑣𝑛𝑣3𝑛 πœ‡4(𝑣𝑛𝑣3𝑛 NOUN iajs-3014 114 48 ) ) PUNCT iajs-3014 114 49 + + SYM iajs-3014 114 50 πœ‡4(𝑣𝑛𝑣2𝑛 πœ‡4(𝑣𝑛𝑣2𝑛 NOUN iajs-3014 114 51 ) ) PUNCT iajs-3014 114 52 + + NUM iajs-3014 114 53 πœ‡4(𝑣𝑛𝑣3π‘›βˆ’1 πœ‡4(𝑣𝑛𝑣3π‘›βˆ’1 NOUN iajs-3014 114 54 ) ) PUNCT iajs-3014 114 55 = = PUNCT iajs-3014 114 56 ( ( PUNCT iajs-3014 114 57 8𝑛 8𝑛 NOUN iajs-3014 114 58 ) ) PUNCT iajs-3014 115 1 + + CCONJ iajs-3014 115 2 ( ( PUNCT iajs-3014 115 3 8𝑛 8𝑛 NOUN iajs-3014 115 4 βˆ’ βˆ’ PROPN iajs-3014 115 5 1 1 X iajs-3014 115 6 ) ) PUNCT iajs-3014 115 7 + + CCONJ iajs-3014 115 8 ( ( PUNCT iajs-3014 115 9 7𝑛 7𝑛 NOUN iajs-3014 115 10 ) ) PUNCT iajs-3014 115 11 + + CCONJ iajs-3014 115 12 ( ( PUNCT iajs-3014 115 13 7𝑛 7𝑛 NUM iajs-3014 115 14 βˆ’ βˆ’ NOUN iajs-3014 115 15 1 1 NUM iajs-3014 115 16 ) ) PUNCT iajs-3014 115 17 + + CCONJ iajs-3014 115 18 ( ( PUNCT iajs-3014 115 19 7𝑛 7𝑛 NUM iajs-3014 115 20 βˆ’ βˆ’ NOUN iajs-3014 115 21 2 2 NUM iajs-3014 115 22 ) ) PUNCT iajs-3014 115 23 = = NOUN iajs-3014 115 24 37𝑛 37𝑛 NUM iajs-3014 115 25 βˆ’ βˆ’ PROPN iajs-3014 115 26 4 4 NUM iajs-3014 115 27 , , PUNCT iajs-3014 115 28 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 115 29 ( ( PUNCT iajs-3014 115 30 𝑣𝑛+1 𝑣𝑛+1 NOUN iajs-3014 115 31 ) ) PUNCT iajs-3014 115 32 = = SYM iajs-3014 115 33 πœ‡4(𝑣𝑛+1𝑣1 πœ‡4(𝑣𝑛+1𝑣1 PROPN iajs-3014 115 34 ) ) PUNCT iajs-3014 115 35 + + NUM iajs-3014 115 36 πœ‡4(𝑣𝑛+1𝑣3𝑛 πœ‡4(𝑣𝑛+1𝑣3𝑛 ADJ iajs-3014 115 37 ) ) PUNCT iajs-3014 115 38 + + NUM iajs-3014 115 39 πœ‡4(𝑣𝑛+1𝑣2𝑛+1 πœ‡4(𝑣𝑛+1𝑣2𝑛+1 NUM iajs-3014 115 40 ) ) PUNCT iajs-3014 115 41 + + PUNCT iajs-3014 115 42 πœ‡4(𝑣𝑛+1𝑣𝑛+2 πœ‡4(𝑣𝑛+1𝑣𝑛+2 ADJ iajs-3014 115 43 ) ) PUNCT iajs-3014 115 44 + + NOUN iajs-3014 115 45 πœ‡4(𝑣𝑛+1𝑣2𝑛 πœ‡4(𝑣𝑛+1𝑣2𝑛 NOUN iajs-3014 115 46 ) ) PUNCT iajs-3014 115 47 + + CCONJ iajs-3014 115 48 πœ‡4(𝑣𝑛+1𝑣3𝑛+1 πœ‡4(𝑣𝑛+1𝑣3𝑛+1 PROPN iajs-3014 115 49 ) ) PUNCT iajs-3014 115 50 = = NOUN iajs-3014 115 51 ( ( PUNCT iajs-3014 115 52 4𝑛 4𝑛 NOUN iajs-3014 115 53 + + CCONJ iajs-3014 115 54 2 2 X iajs-3014 115 55 ) ) PUNCT iajs-3014 115 56 + + CCONJ iajs-3014 115 57 ( ( PUNCT iajs-3014 115 58 4𝑛 4𝑛 NUM iajs-3014 115 59 ) ) PUNCT iajs-3014 116 1 + + CCONJ iajs-3014 116 2 ( ( PUNCT iajs-3014 116 3 2𝑛 2𝑛 NOUN iajs-3014 116 4 + + PROPN iajs-3014 116 5 1 1 NUM iajs-3014 116 6 ) ) PUNCT iajs-3014 116 7 + + CCONJ iajs-3014 116 8 ( ( PUNCT iajs-3014 116 9 𝑛 𝑛 PROPN iajs-3014 116 10 + + NOUN iajs-3014 116 11 1 1 NUM iajs-3014 116 12 ) ) PUNCT iajs-3014 116 13 + + CCONJ iajs-3014 116 14 ( ( PUNCT iajs-3014 116 15 2𝑛 2𝑛 NUM iajs-3014 116 16 ) ) PUNCT iajs-3014 117 1 + + CCONJ iajs-3014 117 2 1 1 NUM iajs-3014 117 3 = = NOUN iajs-3014 117 4 13𝑛 13𝑛 NUM iajs-3014 117 5 + + SYM iajs-3014 117 6 5 5 NUM iajs-3014 117 7 , , PUNCT iajs-3014 117 8 ihjpas ihjpa VERB iajs-3014 117 9 . . PUNCT iajs-3014 118 1 36(1)2023 36(1)2023 NUM iajs-3014 118 2 289 289 NUM iajs-3014 118 3 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 118 4 ( ( PUNCT iajs-3014 118 5 𝑣𝑛+𝑖 𝑣𝑛+𝑖 X iajs-3014 118 6 ) ) PUNCT iajs-3014 118 7 = = PUNCT iajs-3014 118 8 πœ‡4(𝑣𝑛+𝑖 πœ‡4(𝑣𝑛+𝑖 PROPN iajs-3014 118 9 𝑣𝑖 𝑣𝑖 ADP iajs-3014 118 10 ) ) PUNCT iajs-3014 118 11 + + CCONJ iajs-3014 118 12 πœ‡4(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 πœ‡4(𝑣𝑛+𝑖𝑣𝑛+𝑖+1 X iajs-3014 118 13 ) ) PUNCT iajs-3014 118 14 + + NUM iajs-3014 118 15 πœ‡4(𝑣𝑛+𝑖𝑣𝑛+π‘–βˆ’1 πœ‡4(𝑣𝑛+𝑖𝑣𝑛+π‘–βˆ’1 NUM iajs-3014 118 16 ) ) PUNCT iajs-3014 118 17 + + NUM iajs-3014 118 18 πœ‡4(𝑣𝑛+𝑖𝑣2𝑛+π‘–βˆ’1 πœ‡4(𝑣𝑛+𝑖𝑣2𝑛+π‘–βˆ’1 NOUN iajs-3014 118 19 ) ) PUNCT iajs-3014 118 20 + + NOUN iajs-3014 118 21 πœ‡4(𝑣𝑛+𝑖𝑣2𝑛+𝑖 πœ‡4(𝑣𝑛+𝑖𝑣2𝑛+𝑖 NOUN iajs-3014 118 22 ) ) PUNCT iajs-3014 118 23 + + CCONJ iajs-3014 118 24 πœ‡4(𝑣𝑛+𝑖𝑣3𝑛+1 πœ‡4(𝑣𝑛+𝑖𝑣3𝑛+1 NOUN iajs-3014 118 25 ) ) PUNCT iajs-3014 118 26 for for ADP iajs-3014 118 27 𝑖 𝑖 NOUN iajs-3014 118 28 = = SYM iajs-3014 118 29 2 2 NUM iajs-3014 118 30 , , PUNCT iajs-3014 118 31 3 3 NUM iajs-3014 118 32 , , PUNCT iajs-3014 118 33 … … PUNCT iajs-3014 118 34 , , PUNCT iajs-3014 118 35 𝑛 𝑛 DET iajs-3014 118 36 βˆ’ βˆ’ PROPN iajs-3014 118 37 1 1 NUM iajs-3014 118 38 = = SYM iajs-3014 118 39 ( ( PUNCT iajs-3014 118 40 4𝑛 4𝑛 NOUN iajs-3014 118 41 + + CCONJ iajs-3014 118 42 3𝑖 3𝑖 NUM iajs-3014 118 43 βˆ’ βˆ’ NOUN iajs-3014 118 44 1 1 NUM iajs-3014 118 45 ) ) PUNCT iajs-3014 118 46 + + CCONJ iajs-3014 118 47 ( ( PUNCT iajs-3014 118 48 𝑛 𝑛 PROPN iajs-3014 118 49 + + NUM iajs-3014 118 50 𝑖 𝑖 X iajs-3014 118 51 ) ) PUNCT iajs-3014 118 52 + + CCONJ iajs-3014 118 53 ( ( PUNCT iajs-3014 118 54 𝑛 𝑛 PROPN iajs-3014 118 55 + + NUM iajs-3014 118 56 𝑖 𝑖 SYM iajs-3014 118 57 βˆ’ βˆ’ NOUN iajs-3014 118 58 1 1 NUM iajs-3014 118 59 ) ) PUNCT iajs-3014 118 60 + + CCONJ iajs-3014 118 61 ( ( PUNCT iajs-3014 118 62 2𝑛 2𝑛 NOUN iajs-3014 118 63 + + CCONJ iajs-3014 118 64 2𝑖 2𝑖 NUM iajs-3014 118 65 βˆ’ βˆ’ NOUN iajs-3014 118 66 2 2 X iajs-3014 118 67 ) ) PUNCT iajs-3014 118 68 + + CCONJ iajs-3014 118 69 ( ( PUNCT iajs-3014 118 70 2𝑛 2𝑛 NOUN iajs-3014 118 71 + + CCONJ iajs-3014 118 72 2𝑖 2𝑖 NUM iajs-3014 118 73 βˆ’ βˆ’ NOUN iajs-3014 118 74 1 1 X iajs-3014 118 75 ) ) PUNCT iajs-3014 118 76 + + CCONJ iajs-3014 118 77 𝑖 𝑖 X iajs-3014 118 78 for for ADP iajs-3014 118 79 𝑖 𝑖 PRON iajs-3014 118 80 = = SYM iajs-3014 118 81 2 2 NUM iajs-3014 118 82 , , PUNCT iajs-3014 118 83 3 3 NUM iajs-3014 118 84 , , PUNCT iajs-3014 118 85 … … PUNCT iajs-3014 118 86 , , PUNCT iajs-3014 118 87 𝑛 𝑛 DET iajs-3014 118 88 βˆ’ βˆ’ PROPN iajs-3014 118 89 1 1 NUM iajs-3014 118 90 , , PUNCT iajs-3014 118 91 = = PRON iajs-3014 118 92 10𝑛 10𝑛 NOUN iajs-3014 118 93 + + NUM iajs-3014 118 94 10𝑖 10𝑖 NOUN iajs-3014 118 95 βˆ’ βˆ’ ADP iajs-3014 118 96 5 5 NUM iajs-3014 118 97 for for ADP iajs-3014 118 98 𝑖 𝑖 NOUN iajs-3014 118 99 = = SYM iajs-3014 118 100 2 2 NUM iajs-3014 118 101 , , PUNCT iajs-3014 118 102 3 3 NUM iajs-3014 118 103 , , PUNCT iajs-3014 118 104 … … PUNCT iajs-3014 118 105 𝑛 𝑛 PRON iajs-3014 118 106 βˆ’ βˆ’ PROPN iajs-3014 118 107 1 1 NUM iajs-3014 118 108 , , PUNCT iajs-3014 118 109 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 118 110 ( ( PUNCT iajs-3014 118 111 𝑣2𝑛 𝑣2𝑛 PROPN iajs-3014 118 112 ) ) PUNCT iajs-3014 118 113 = = SYM iajs-3014 118 114 πœ‡4(𝑣2𝑛𝑣𝑛+1 πœ‡4(𝑣2𝑛𝑣𝑛+1 NOUN iajs-3014 118 115 ) ) PUNCT iajs-3014 118 116 + + CCONJ iajs-3014 118 117 πœ‡4(𝑣2𝑛𝑣2π‘›βˆ’1 πœ‡4(𝑣2𝑛𝑣2π‘›βˆ’1 NUM iajs-3014 118 118 ) ) PUNCT iajs-3014 118 119 + + CCONJ iajs-3014 118 120 πœ‡4(𝑣2𝑛𝑣3𝑛+1 πœ‡4(𝑣2𝑛𝑣3𝑛+1 NUM iajs-3014 118 121 ) ) PUNCT iajs-3014 118 122 + + CCONJ iajs-3014 118 123 πœ‡4(𝑣2𝑛𝑣3𝑛 πœ‡4(𝑣2𝑛𝑣3𝑛 VERB iajs-3014 118 124 ) ) PUNCT iajs-3014 118 125 + + NOUN iajs-3014 118 126 πœ‡4(𝑣2𝑛𝑣𝑛 πœ‡4(𝑣2𝑛𝑣𝑛 NOUN iajs-3014 118 127 ) ) PUNCT iajs-3014 119 1 + + CCONJ iajs-3014 119 2 πœ‡4(𝑣2𝑛𝑣3π‘›βˆ’1 πœ‡4(𝑣2𝑛𝑣3π‘›βˆ’1 NUM iajs-3014 119 3 ) ) PUNCT iajs-3014 119 4 = = SYM iajs-3014 119 5 ( ( PUNCT iajs-3014 119 6 2𝑛 2𝑛 NUM iajs-3014 119 7 ) ) PUNCT iajs-3014 120 1 + + CCONJ iajs-3014 120 2 ( ( PUNCT iajs-3014 120 3 2𝑛 2𝑛 NUM iajs-3014 120 4 βˆ’ βˆ’ PROPN iajs-3014 120 5 1 1 NUM iajs-3014 120 6 ) ) PUNCT iajs-3014 120 7 + + CCONJ iajs-3014 120 8 ( ( PUNCT iajs-3014 120 9 𝑛 𝑛 X iajs-3014 120 10 ) ) PUNCT iajs-3014 120 11 + + CCONJ iajs-3014 120 12 ( ( PUNCT iajs-3014 120 13 4𝑛 4𝑛 NOUN iajs-3014 120 14 βˆ’ βˆ’ NOUN iajs-3014 120 15 1 1 NUM iajs-3014 120 16 ) ) PUNCT iajs-3014 120 17 + + CCONJ iajs-3014 120 18 ( ( PUNCT iajs-3014 120 19 7𝑛 7𝑛 NUM iajs-3014 120 20 βˆ’ βˆ’ NOUN iajs-3014 120 21 1 1 NUM iajs-3014 120 22 ) ) PUNCT iajs-3014 120 23 + + CCONJ iajs-3014 120 24 ( ( PUNCT iajs-3014 120 25 4𝑛 4𝑛 NOUN iajs-3014 120 26 βˆ’ βˆ’ NOUN iajs-3014 121 1 2 2 X iajs-3014 121 2 ) ) PUNCT iajs-3014 121 3 = = NOUN iajs-3014 121 4 20𝑛 20𝑛 NUM iajs-3014 121 5 βˆ’ βˆ’ PROPN iajs-3014 121 6 5 5 NUM iajs-3014 121 7 , , PUNCT iajs-3014 121 8 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 121 9 ( ( PUNCT iajs-3014 121 10 𝑣2𝑛+𝑖 𝑣2𝑛+𝑖 PROPN iajs-3014 121 11 ) ) PUNCT iajs-3014 121 12 = = SYM iajs-3014 121 13 πœ‡4(𝑣2𝑛+𝑖 πœ‡4(𝑣2𝑛+𝑖 PROPN iajs-3014 121 14 𝑣𝑖 𝑣𝑖 PROPN iajs-3014 121 15 ) ) PUNCT iajs-3014 121 16 + + NUM iajs-3014 121 17 πœ‡4(𝑣2𝑛+𝑖𝑣𝑖+1 πœ‡4(𝑣2𝑛+𝑖𝑣𝑖+1 NOUN iajs-3014 121 18 ) ) PUNCT iajs-3014 121 19 + + NUM iajs-3014 121 20 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖 X iajs-3014 121 21 ) ) PUNCT iajs-3014 122 1 + + CCONJ iajs-3014 122 2 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 πœ‡4(𝑣2𝑛+𝑖𝑣𝑛+𝑖+1 PROPN iajs-3014 122 3 ) ) PUNCT iajs-3014 122 4 for for ADP iajs-3014 122 5 𝑖 𝑖 NOUN iajs-3014 122 6 = = SYM iajs-3014 122 7 1 1 NUM iajs-3014 122 8 , , PUNCT iajs-3014 122 9 2 2 NUM iajs-3014 122 10 , , PUNCT iajs-3014 122 11 … … PUNCT iajs-3014 122 12 , , PUNCT iajs-3014 122 13 𝑛 𝑛 DET iajs-3014 122 14 βˆ’ βˆ’ PROPN iajs-3014 122 15 1 1 NUM iajs-3014 122 16 , , PUNCT iajs-3014 122 17 = = PRON iajs-3014 122 18 ( ( PUNCT iajs-3014 122 19 4𝑛 4𝑛 NOUN iajs-3014 122 20 + + CCONJ iajs-3014 122 21 3𝑖 3𝑖 NUM iajs-3014 122 22 ) ) PUNCT iajs-3014 123 1 + + CCONJ iajs-3014 123 2 ( ( PUNCT iajs-3014 123 3 4𝑛 4𝑛 NOUN iajs-3014 123 4 + + CCONJ iajs-3014 123 5 3𝑖 3𝑖 NUM iajs-3014 123 6 + + CCONJ iajs-3014 123 7 1 1 NUM iajs-3014 123 8 ) ) PUNCT iajs-3014 123 9 + + CCONJ iajs-3014 123 10 ( ( PUNCT iajs-3014 123 11 2𝑛 2𝑛 NOUN iajs-3014 123 12 + + CCONJ iajs-3014 123 13 2𝑖 2𝑖 NUM iajs-3014 123 14 βˆ’ βˆ’ NOUN iajs-3014 123 15 1 1 X iajs-3014 123 16 ) ) PUNCT iajs-3014 123 17 + + CCONJ iajs-3014 123 18 ( ( PUNCT iajs-3014 123 19 2𝑛 2𝑛 PROPN iajs-3014 123 20 + + CCONJ iajs-3014 123 21 2𝑖 2𝑖 NUM iajs-3014 123 22 ) ) PUNCT iajs-3014 123 23 for for ADP iajs-3014 123 24 𝑖 𝑖 NOUN iajs-3014 123 25 = = SYM iajs-3014 123 26 1 1 NUM iajs-3014 123 27 , , PUNCT iajs-3014 123 28 2 2 NUM iajs-3014 123 29 , , PUNCT iajs-3014 123 30 … … PUNCT iajs-3014 123 31 , , PUNCT iajs-3014 124 1 𝑛 𝑛 DET iajs-3014 124 2 βˆ’ βˆ’ PROPN iajs-3014 124 3 1 1 NUM iajs-3014 124 4 , , PUNCT iajs-3014 124 5 = = SYM iajs-3014 124 6 12𝑛 12𝑛 X iajs-3014 124 7 + + CCONJ iajs-3014 124 8 10𝑖 10𝑖 NOUN iajs-3014 124 9 for for ADP iajs-3014 124 10 𝑖 𝑖 PRON iajs-3014 124 11 = = SYM iajs-3014 124 12 1 1 NUM iajs-3014 124 13 , , PUNCT iajs-3014 124 14 2 2 NUM iajs-3014 124 15 , , PUNCT iajs-3014 124 16 … … PUNCT iajs-3014 124 17 , , PUNCT iajs-3014 124 18 𝑛 𝑛 DET iajs-3014 124 19 βˆ’ βˆ’ PROPN iajs-3014 124 20 1 1 NUM iajs-3014 124 21 , , PUNCT iajs-3014 124 22 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 124 23 ( ( PUNCT iajs-3014 124 24 𝑣3𝑛 𝑣3𝑛 NUM iajs-3014 124 25 ) ) PUNCT iajs-3014 124 26 = = SYM iajs-3014 124 27 πœ‡4(𝑣3𝑛𝑣1 πœ‡4(𝑣3𝑛𝑣1 NOUN iajs-3014 124 28 ) ) PUNCT iajs-3014 124 29 + + CCONJ iajs-3014 124 30 πœ‡4(𝑣3𝑛𝑣𝑛 πœ‡4(𝑣3𝑛𝑣𝑛 X iajs-3014 124 31 ) ) PUNCT iajs-3014 124 32 + + SYM iajs-3014 124 33 πœ‡4(𝑣3𝑛𝑣2𝑛 πœ‡4(𝑣3𝑛𝑣2𝑛 X iajs-3014 124 34 ) ) PUNCT iajs-3014 124 35 + + CCONJ iajs-3014 124 36 πœ‡4(𝑣3𝑛𝑣𝑛+1 πœ‡4(𝑣3𝑛𝑣𝑛+1 PROPN iajs-3014 124 37 ) ) PUNCT iajs-3014 124 38 = = PUNCT iajs-3014 124 39 ( ( PUNCT iajs-3014 124 40 4𝑛 4𝑛 NOUN iajs-3014 124 41 + + CCONJ iajs-3014 124 42 1 1 X iajs-3014 124 43 ) ) PUNCT iajs-3014 124 44 + + CCONJ iajs-3014 124 45 ( ( PUNCT iajs-3014 124 46 7𝑛 7𝑛 NOUN iajs-3014 124 47 ) ) PUNCT iajs-3014 124 48 + + CCONJ iajs-3014 124 49 ( ( PUNCT iajs-3014 124 50 4𝑛 4𝑛 NOUN iajs-3014 124 51 βˆ’ βˆ’ NOUN iajs-3014 124 52 1 1 NUM iajs-3014 124 53 ) ) PUNCT iajs-3014 124 54 + + CCONJ iajs-3014 124 55 ( ( PUNCT iajs-3014 124 56 4𝑛 4𝑛 NUM iajs-3014 124 57 ) ) PUNCT iajs-3014 124 58 = = PUNCT iajs-3014 125 1 19𝑛. 19𝑛. NUM iajs-3014 125 2 finally finally ADV iajs-3014 125 3 : : PUNCT iajs-3014 125 4 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 125 5 ( ( PUNCT iajs-3014 125 6 𝑣3𝑛+1 𝑣3𝑛+1 ADJ iajs-3014 125 7 ) ) PUNCT iajs-3014 125 8 = = SYM iajs-3014 125 9 βˆ‘ βˆ‘ PUNCT iajs-3014 125 10 πœ‡3(𝑣3𝑛+1 πœ‡3(𝑣3𝑛+1 PROPN iajs-3014 125 11 𝑣𝑛+𝑖 𝑣𝑛+𝑖 PROPN iajs-3014 125 12 ) ) PUNCT iajs-3014 125 13 𝑛 𝑛 PROPN iajs-3014 125 14 𝑖=1 𝑖=1 PUNCT iajs-3014 125 15 = = SYM iajs-3014 125 16 βˆ‘(𝑖 βˆ‘(𝑖 PROPN iajs-3014 125 17 ) ) PUNCT iajs-3014 126 1 𝑛 𝑛 PROPN iajs-3014 126 2 𝑖=1 𝑖=1 PUNCT iajs-3014 126 3 = = PUNCT iajs-3014 126 4 𝑛(𝑛 𝑛(𝑛 PROPN iajs-3014 126 5 + + CCONJ iajs-3014 126 6 1 1 NUM iajs-3014 126 7 ) ) PUNCT iajs-3014 126 8 2 2 NUM iajs-3014 126 9 . . PUNCT iajs-3014 127 1 so so ADV iajs-3014 127 2 , , PUNCT iajs-3014 127 3 the the DET iajs-3014 127 4 weights weight NOUN iajs-3014 127 5 of of ADP iajs-3014 127 6 all all DET iajs-3014 127 7 vertices vertex NOUN iajs-3014 127 8 are be AUX iajs-3014 127 9 distinct distinct ADJ iajs-3014 127 10 and and CCONJ iajs-3014 127 11 the the DET iajs-3014 127 12 following follow VERB iajs-3014 127 13 notes note NOUN iajs-3014 127 14 are be AUX iajs-3014 127 15 observed observe VERB iajs-3014 127 16 : : PUNCT iajs-3014 127 17 1π‘€π‘‘πœ‡4 1π‘€π‘‘πœ‡4 NUM iajs-3014 127 18 ( ( PUNCT iajs-3014 127 19 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 127 20 ) ) PUNCT iajs-3014 127 21 < < X iajs-3014 127 22 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 127 23 ( ( PUNCT iajs-3014 127 24 𝑣𝑖+1 𝑣𝑖+1 NUM iajs-3014 127 25 ) ) PUNCT iajs-3014 127 26 for for ADP iajs-3014 127 27 𝑖 𝑖 PRON iajs-3014 127 28 = = SYM iajs-3014 127 29 1,2 1,2 NUM iajs-3014 127 30 , , PUNCT iajs-3014 127 31 … … PUNCT iajs-3014 127 32 , , PUNCT iajs-3014 127 33 , , PUNCT iajs-3014 128 1 𝑛 𝑛 DET iajs-3014 128 2 βˆ’ βˆ’ NOUN iajs-3014 128 3 1 1 NUM iajs-3014 128 4 . . X iajs-3014 129 1 2the 2the NUM iajs-3014 129 2 weight weight NOUN iajs-3014 129 3 of of ADP iajs-3014 129 4 vertex vertex NOUN iajs-3014 129 5 ( ( PUNCT iajs-3014 129 6 𝑣𝑛 𝑣𝑛 NOUN iajs-3014 129 7 ) ) PUNCT iajs-3014 129 8 is be AUX iajs-3014 129 9 greater great ADJ iajs-3014 129 10 than than ADP iajs-3014 129 11 the the DET iajs-3014 129 12 weight weight NOUN iajs-3014 129 13 of of ADP iajs-3014 129 14 any any DET iajs-3014 129 15 other other ADJ iajs-3014 129 16 vertex vertex NOUN iajs-3014 129 17 in in ADP iajs-3014 129 18 the the DET iajs-3014 129 19 ( ( PUNCT iajs-3014 129 20 𝐢𝑛 𝐢𝑛 NOUN iajs-3014 129 21 Γ— Γ— NOUN iajs-3014 129 22 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 129 23 graph graph NOUN iajs-3014 129 24 . . PUNCT iajs-3014 130 1 thus thus ADV iajs-3014 130 2 , , PUNCT iajs-3014 130 3 the the DET iajs-3014 130 4 strong strong ADJ iajs-3014 130 5 face face NOUN iajs-3014 130 6 prism prism NOUN iajs-3014 130 7 graph graph NOUN iajs-3014 130 8 ( ( PUNCT iajs-3014 130 9 𝐢𝑛 𝐢𝑛 INTJ iajs-3014 130 10 Γ— Γ— NOUN iajs-3014 130 11 𝑃2)βˆ— 𝑃2)βˆ— NOUN iajs-3014 130 12 is be AUX iajs-3014 130 13 vertex vertex NOUN iajs-3014 130 14 antimagic antimagic ADJ iajs-3014 130 15 edge edge NOUN iajs-3014 130 16 labeling labeling NOUN iajs-3014 130 17 , , PUNCT iajs-3014 130 18 for for ADP iajs-3014 130 19 every every DET iajs-3014 130 20 𝑛 𝑛 PROPN iajs-3014 130 21 , , PUNCT iajs-3014 130 22 𝑛 𝑛 DET iajs-3014 130 23 β‰₯ β‰₯ NOUN iajs-3014 130 24 3 3 NUM iajs-3014 130 25 , , PUNCT iajs-3014 130 26 𝑛 𝑛 PRON iajs-3014 130 27 β‰’ β‰’ NUM iajs-3014 130 28 0 0 NUM iajs-3014 130 29 ( ( PUNCT iajs-3014 130 30 mod mod PROPN iajs-3014 130 31 5 5 NUM iajs-3014 130 32 ) ) PUNCT iajs-3014 130 33 . . PUNCT iajs-3014 131 1 however however ADV iajs-3014 131 2 when when SCONJ iajs-3014 131 3 𝑛 𝑛 PROPN iajs-3014 131 4 = = SYM iajs-3014 131 5 5 5 NUM iajs-3014 131 6 , , PUNCT iajs-3014 131 7 we we PRON iajs-3014 131 8 can can AUX iajs-3014 131 9 easily easily ADV iajs-3014 131 10 show show VERB iajs-3014 131 11 that that SCONJ iajs-3014 131 12 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 131 13 ( ( PUNCT iajs-3014 131 14 𝑣6 𝑣6 PROPN iajs-3014 131 15 ) ) PUNCT iajs-3014 131 16 = = PUNCT iajs-3014 131 17 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 131 18 ( ( PUNCT iajs-3014 131 19 𝑣11 𝑣11 X iajs-3014 131 20 ) ) PUNCT iajs-3014 131 21 and and CCONJ iajs-3014 131 22 similarly similarly ADV iajs-3014 131 23 when when SCONJ iajs-3014 131 24 𝑛 𝑛 PROPN iajs-3014 131 25 = = SYM iajs-3014 131 26 10 10 NUM iajs-3014 131 27 , , PUNCT iajs-3014 131 28 then then ADV iajs-3014 131 29 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 131 30 ( ( PUNCT iajs-3014 131 31 𝑣9 𝑣9 ADV iajs-3014 131 32 ) ) PUNCT iajs-3014 131 33 = = PUNCT iajs-3014 132 1 π‘€π‘‘πœ‡4 π‘€π‘‘πœ‡4 PROPN iajs-3014 132 2 ( ( PUNCT iajs-3014 132 3 𝑣11 𝑣11 X iajs-3014 132 4 ) ) PUNCT iajs-3014 132 5 and and CCONJ iajs-3014 132 6 so so ADV iajs-3014 132 7 on on ADV iajs-3014 132 8 . . PUNCT iajs-3014 133 1 theorem theorem VERB iajs-3014 133 2 5 5 NUM iajs-3014 133 3 the the DET iajs-3014 133 4 strong strong ADJ iajs-3014 133 5 face face NOUN iajs-3014 133 6 friendship friendship NOUN iajs-3014 133 7 graph graph NOUN iajs-3014 133 8 ( ( PUNCT iajs-3014 133 9 𝑇𝑛)βˆ— 𝑇𝑛)βˆ— PROPN iajs-3014 133 10 is be AUX iajs-3014 133 11 vael vael ADJ iajs-3014 133 12 for for ADP iajs-3014 133 13 every every DET iajs-3014 133 14 𝑛 𝑛 PRON iajs-3014 133 15 β‰₯ β‰₯ NOUN iajs-3014 133 16 3 3 NUM iajs-3014 133 17 . . PUNCT iajs-3014 134 1 proof proof NOUN iajs-3014 134 2 : : PUNCT iajs-3014 134 3 we we PRON iajs-3014 134 4 define define VERB iajs-3014 134 5 the the DET iajs-3014 134 6 vertex vertex NOUN iajs-3014 134 7 and and CCONJ iajs-3014 134 8 edge edge NOUN iajs-3014 134 9 sets set NOUN iajs-3014 134 10 of of ADP iajs-3014 134 11 ( ( PUNCT iajs-3014 134 12 𝑇𝑛)βˆ— 𝑇𝑛)βˆ— X iajs-3014 134 13 graph graph NOUN iajs-3014 134 14 as as SCONJ iajs-3014 134 15 follows follow VERB iajs-3014 134 16 ; ; PUNCT iajs-3014 134 17 𝑉 𝑉 PROPN iajs-3014 134 18 ( ( PUNCT iajs-3014 134 19 ( ( PUNCT iajs-3014 134 20 𝑇𝑛)βˆ— 𝑇𝑛)βˆ— X iajs-3014 134 21 ) ) PUNCT iajs-3014 134 22 = = SYM iajs-3014 134 23 { { PUNCT iajs-3014 134 24 𝑣 𝑣 NOUN iajs-3014 134 25 , , PUNCT iajs-3014 134 26 𝑣𝑖 𝑣𝑖 ADP iajs-3014 134 27 : : PUNCT iajs-3014 134 28 𝑖 𝑖 SYM iajs-3014 134 29 = = SYM iajs-3014 134 30 1 1 NUM iajs-3014 134 31 , , PUNCT iajs-3014 134 32 2 2 NUM iajs-3014 134 33 , , PUNCT iajs-3014 134 34 … … NUM iajs-3014 134 35 2𝑛 2𝑛 NUM iajs-3014 134 36 } } PUNCT iajs-3014 134 37 βˆͺ βˆͺ ADJ iajs-3014 134 38 { { PUNCT iajs-3014 134 39 𝑒𝑖 𝑒𝑖 NOUN iajs-3014 134 40 : : PUNCT iajs-3014 134 41 𝑖 𝑖 SYM iajs-3014 134 42 = = SYM iajs-3014 134 43 1 1 NUM iajs-3014 134 44 , , PUNCT iajs-3014 134 45 2 2 NUM iajs-3014 134 46 , , PUNCT iajs-3014 134 47 … … PUNCT iajs-3014 134 48 𝑛 𝑛 X iajs-3014 134 49 } } PUNCT iajs-3014 134 50 , , PUNCT iajs-3014 134 51 𝐸((𝑇𝑛)βˆ— 𝐸((𝑇𝑛)βˆ— NOUN iajs-3014 134 52 ) ) PUNCT iajs-3014 135 1 = = SYM iajs-3014 135 2 { { PUNCT iajs-3014 135 3 𝑣𝑣𝑖 𝑣𝑣𝑖 NOUN iajs-3014 135 4 ∢ ∢ NOUN iajs-3014 135 5 𝑖 𝑖 NOUN iajs-3014 135 6 = = SYM iajs-3014 135 7 1 1 NUM iajs-3014 135 8 , , PUNCT iajs-3014 135 9 2 2 NUM iajs-3014 135 10 , , PUNCT iajs-3014 135 11 … … NUM iajs-3014 135 12 2𝑛 2𝑛 NUM iajs-3014 135 13 } } PUNCT iajs-3014 135 14 βˆͺ βˆͺ ADJ iajs-3014 135 15 { { PUNCT iajs-3014 135 16 𝑣𝑒𝑖 𝑣𝑒𝑖 ADJ iajs-3014 135 17 ∢ ∢ NOUN iajs-3014 135 18 𝑖 𝑖 X iajs-3014 135 19 = = SYM iajs-3014 135 20 1 1 NUM iajs-3014 135 21 , , PUNCT iajs-3014 135 22 2 2 NUM iajs-3014 135 23 , , PUNCT iajs-3014 135 24 … … PUNCT iajs-3014 135 25 𝑛 𝑛 X iajs-3014 135 26 } } PUNCT iajs-3014 135 27 βˆͺ βˆͺ X iajs-3014 135 28 { { PUNCT iajs-3014 135 29 𝑣𝑖 𝑣𝑖 ADP iajs-3014 135 30 𝑣𝑖+1 𝑣𝑖+1 NUM iajs-3014 135 31 ∢ ∢ NOUN iajs-3014 135 32 𝑖 𝑖 X iajs-3014 135 33 = = SYM iajs-3014 135 34 1 1 NUM iajs-3014 135 35 , , PUNCT iajs-3014 135 36 3 3 NUM iajs-3014 135 37 , , PUNCT iajs-3014 135 38 … … PUNCT iajs-3014 135 39 2𝑛 2𝑛 NOUN iajs-3014 135 40 βˆ’ βˆ’ NOUN iajs-3014 135 41 1 1 NUM iajs-3014 135 42 } } PUNCT iajs-3014 135 43 βˆͺ βˆͺ ADJ iajs-3014 135 44 { { PUNCT iajs-3014 135 45 𝑒𝑖𝑣2π‘–βˆ’1 𝑒𝑖𝑣2π‘–βˆ’1 NUM iajs-3014 135 46 , , PUNCT iajs-3014 135 47 𝑒𝑖 𝑒𝑖 VERB iajs-3014 135 48 𝑣2𝑖 𝑣2𝑖 NOUN iajs-3014 135 49 ∢ ∢ NOUN iajs-3014 135 50 𝑖 𝑖 X iajs-3014 135 51 = = SYM iajs-3014 135 52 1 1 NUM iajs-3014 135 53 , , PUNCT iajs-3014 135 54 2 2 NUM iajs-3014 135 55 , , PUNCT iajs-3014 135 56 … … PUNCT iajs-3014 135 57 , , PUNCT iajs-3014 135 58 𝑛 𝑛 NOUN iajs-3014 135 59 } } PUNCT iajs-3014 135 60 . . PUNCT iajs-3014 136 1 for for ADP iajs-3014 136 2 every every DET iajs-3014 136 3 𝑛 𝑛 DET iajs-3014 136 4 β‰₯ β‰₯ NOUN iajs-3014 136 5 3 3 NUM iajs-3014 136 6 , , PUNCT iajs-3014 136 7 we we PRON iajs-3014 136 8 define define VERB iajs-3014 136 9 the the DET iajs-3014 136 10 labeling labeling NOUN iajs-3014 136 11 of of ADP iajs-3014 136 12 ( ( PUNCT iajs-3014 136 13 𝑇𝑛)βˆ— 𝑇𝑛)βˆ— X iajs-3014 136 14 graph graph NOUN iajs-3014 136 15 as as ADP iajs-3014 136 16 : : PUNCT iajs-3014 136 17 πœ‡5((𝑇𝑛)βˆ— πœ‡5((𝑇𝑛)βˆ— PROPN iajs-3014 136 18 ) ) PUNCT iajs-3014 136 19 β†’ β†’ SYM iajs-3014 136 20 { { PUNCT iajs-3014 136 21 1 1 NUM iajs-3014 136 22 , , PUNCT iajs-3014 136 23 2 2 NUM iajs-3014 136 24 , , PUNCT iajs-3014 136 25 … … PUNCT iajs-3014 136 26 , , PUNCT iajs-3014 136 27 6𝑛 6𝑛 NOUN iajs-3014 136 28 } } PUNCT iajs-3014 136 29 . . PUNCT iajs-3014 137 1 such such ADJ iajs-3014 137 2 that that SCONJ iajs-3014 137 3 : : PUNCT iajs-3014 137 4 1 1 NUM iajs-3014 137 5 πœ‡5(𝑣𝑣𝑖 πœ‡5(𝑣𝑣𝑖 NOUN iajs-3014 137 6 ) ) PUNCT iajs-3014 137 7 = = PRON iajs-3014 137 8 { { PUNCT iajs-3014 137 9 3𝑖 3𝑖 NUM iajs-3014 137 10 βˆ’ βˆ’ PROPN iajs-3014 137 11 2 2 NUM iajs-3014 137 12 for for ADP iajs-3014 137 13 𝑖 𝑖 NOUN iajs-3014 137 14 = = SYM iajs-3014 137 15 1 1 NUM iajs-3014 137 16 , , PUNCT iajs-3014 137 17 3 3 NUM iajs-3014 137 18 , , PUNCT iajs-3014 137 19 … … PUNCT iajs-3014 137 20 , , PUNCT iajs-3014 137 21 2𝑛 2𝑛 PROPN iajs-3014 137 22 βˆ’ βˆ’ PROPN iajs-3014 137 23 1 1 NUM iajs-3014 137 24 , , PUNCT iajs-3014 137 25 3𝑖 3𝑖 NUM iajs-3014 137 26 βˆ’ βˆ’ PROPN iajs-3014 137 27 3 3 NUM iajs-3014 137 28 for for ADP iajs-3014 137 29 𝑖 𝑖 NOUN iajs-3014 137 30 = = SYM iajs-3014 137 31 2 2 NUM iajs-3014 137 32 , , PUNCT iajs-3014 137 33 4 4 NUM iajs-3014 137 34 , , PUNCT iajs-3014 137 35 … … PUNCT iajs-3014 137 36 , , PUNCT iajs-3014 137 37 2𝑛 2𝑛 NUM iajs-3014 137 38 , , PUNCT iajs-3014 137 39 2 2 NUM iajs-3014 137 40 πœ‡5(𝑣𝑖𝑣𝑖+1 πœ‡5(𝑣𝑖𝑣𝑖+1 NOUN iajs-3014 137 41 ) ) PUNCT iajs-3014 137 42 = = SYM iajs-3014 137 43 3𝑖 3𝑖 NUM iajs-3014 137 44 βˆ’ βˆ’ NOUN iajs-3014 137 45 1 1 NUM iajs-3014 137 46 for for ADP iajs-3014 137 47 𝑖 𝑖 NOUN iajs-3014 137 48 = = SYM iajs-3014 137 49 1 1 NUM iajs-3014 137 50 , , PUNCT iajs-3014 137 51 3 3 NUM iajs-3014 137 52 , , PUNCT iajs-3014 137 53 … … PUNCT iajs-3014 137 54 , , PUNCT iajs-3014 137 55 2𝑛 2𝑛 PROPN iajs-3014 137 56 βˆ’ βˆ’ PROPN iajs-3014 137 57 1 1 NUM iajs-3014 137 58 , , PUNCT iajs-3014 137 59 3 3 NUM iajs-3014 137 60 πœ‡5(𝑣𝑒𝑖 πœ‡5(𝑣𝑒𝑖 NOUN iajs-3014 137 61 ) ) PUNCT iajs-3014 137 62 = = SYM iajs-3014 137 63 6𝑖 6𝑖 NOUN iajs-3014 137 64 βˆ’ βˆ’ NOUN iajs-3014 137 65 2 2 NUM iajs-3014 137 66 for for ADP iajs-3014 137 67 𝑖 𝑖 NOUN iajs-3014 137 68 = = SYM iajs-3014 137 69 1 1 NUM iajs-3014 137 70 , , PUNCT iajs-3014 137 71 2 2 NUM iajs-3014 137 72 , , PUNCT iajs-3014 137 73 … … PUNCT iajs-3014 137 74 , , PUNCT iajs-3014 137 75 𝑛 𝑛 NOUN iajs-3014 137 76 , , PUNCT iajs-3014 137 77 4 4 NUM iajs-3014 137 78 πœ‡5(𝑒𝑖𝑣2π‘–βˆ’1 πœ‡5(𝑒𝑖𝑣2π‘–βˆ’1 NOUN iajs-3014 137 79 ) ) PUNCT iajs-3014 137 80 = = SYM iajs-3014 137 81 6𝑖 6𝑖 NOUN iajs-3014 137 82 βˆ’ βˆ’ NOUN iajs-3014 137 83 1 1 NUM iajs-3014 137 84 for for ADP iajs-3014 137 85 𝑖 𝑖 NOUN iajs-3014 137 86 = = SYM iajs-3014 137 87 1 1 NUM iajs-3014 137 88 , , PUNCT iajs-3014 137 89 2 2 NUM iajs-3014 137 90 , , PUNCT iajs-3014 137 91 … … PUNCT iajs-3014 137 92 , , PUNCT iajs-3014 137 93 𝑛 𝑛 ADJ iajs-3014 137 94 , , PUNCT iajs-3014 137 95 5 5 NUM iajs-3014 137 96 πœ‡5(𝑒𝑖 πœ‡5(𝑒𝑖 PROPN iajs-3014 137 97 𝑣2𝑖 𝑣2𝑖 NOUN iajs-3014 137 98 ) ) PUNCT iajs-3014 137 99 = = PUNCT iajs-3014 137 100 6𝑖 6𝑖 NOUN iajs-3014 137 101 for for ADP iajs-3014 137 102 𝑖 𝑖 NOUN iajs-3014 137 103 = = SYM iajs-3014 137 104 1 1 NUM iajs-3014 137 105 , , PUNCT iajs-3014 137 106 2 2 NUM iajs-3014 137 107 , , PUNCT iajs-3014 137 108 … … PUNCT iajs-3014 137 109 , , PUNCT iajs-3014 137 110 𝑛. 𝑛. NOUN iajs-3014 137 111 for for ADP iajs-3014 137 112 the the DET iajs-3014 137 113 vertex vertex NOUN iajs-3014 137 114 - - PUNCT iajs-3014 137 115 weights weight NOUN iajs-3014 137 116 we we PRON iajs-3014 137 117 get get VERB iajs-3014 137 118 : : PUNCT iajs-3014 137 119 ihjpas ihjpas PROPN iajs-3014 137 120 . . PUNCT iajs-3014 138 1 36(1)2023 36(1)2023 NUM iajs-3014 138 2 290 290 NUM iajs-3014 138 3 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 NOUN iajs-3014 138 4 ( ( PUNCT iajs-3014 138 5 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 138 6 ) ) PUNCT iajs-3014 138 7 = = NOUN iajs-3014 138 8 { { PUNCT iajs-3014 138 9 πœ‡5(𝑣𝑖𝑣 πœ‡5(𝑣𝑖𝑣 NOUN iajs-3014 138 10 ) ) PUNCT iajs-3014 138 11 + + X iajs-3014 138 12 πœ‡5(𝑣𝑖𝑣𝑖+1 πœ‡5(𝑣𝑖𝑣𝑖+1 X iajs-3014 138 13 ) ) PUNCT iajs-3014 139 1 + + CCONJ iajs-3014 139 2 πœ‡5 πœ‡5 PROPN iajs-3014 139 3 ( ( PUNCT iajs-3014 139 4 𝑣𝑖 𝑣𝑖 PROPN iajs-3014 139 5 𝑒𝑖+1 𝑒𝑖+1 PROPN iajs-3014 139 6 2 2 NUM iajs-3014 139 7 ) ) PUNCT iajs-3014 139 8 for for ADP iajs-3014 139 9 𝑖 𝑖 NOUN iajs-3014 139 10 = = SYM iajs-3014 139 11 1 1 NUM iajs-3014 139 12 , , PUNCT iajs-3014 139 13 3 3 NUM iajs-3014 139 14 , , PUNCT iajs-3014 139 15 … … PUNCT iajs-3014 139 16 , , PUNCT iajs-3014 139 17 2𝑛 2𝑛 PROPN iajs-3014 139 18 βˆ’ βˆ’ PROPN iajs-3014 139 19 1 1 NUM iajs-3014 139 20 , , PUNCT iajs-3014 139 21 πœ‡5(𝑣𝑖 πœ‡5(𝑣𝑖 NUM iajs-3014 139 22 𝑣 𝑣 X iajs-3014 139 23 ) ) PUNCT iajs-3014 139 24 + + NUM iajs-3014 139 25 πœ‡5(π‘£π‘–π‘£π‘–βˆ’1 πœ‡5(π‘£π‘–π‘£π‘–βˆ’1 X iajs-3014 139 26 ) ) PUNCT iajs-3014 139 27 + + CCONJ iajs-3014 139 28 πœ‡5 πœ‡5 PROPN iajs-3014 139 29 ( ( PUNCT iajs-3014 139 30 𝑣𝑖 𝑣𝑖 NOUN iajs-3014 139 31 𝑒𝑖 𝑒𝑖 PRON iajs-3014 139 32 2 2 NUM iajs-3014 139 33 ) ) PUNCT iajs-3014 139 34 for for ADP iajs-3014 139 35 𝑖 𝑖 NOUN iajs-3014 139 36 = = SYM iajs-3014 139 37 2 2 NUM iajs-3014 139 38 , , PUNCT iajs-3014 139 39 4 4 NUM iajs-3014 139 40 , , PUNCT iajs-3014 139 41 … … PUNCT iajs-3014 139 42 , , PUNCT iajs-3014 139 43 2𝑛 2𝑛 NUM iajs-3014 139 44 , , PUNCT iajs-3014 139 45 = = PRON iajs-3014 139 46 { { PUNCT iajs-3014 139 47 ( ( PUNCT iajs-3014 139 48 3𝑖 3𝑖 NUM iajs-3014 139 49 βˆ’ βˆ’ PROPN iajs-3014 139 50 2 2 NUM iajs-3014 139 51 ) ) PUNCT iajs-3014 139 52 + + CCONJ iajs-3014 139 53 ( ( PUNCT iajs-3014 139 54 3𝑖 3𝑖 NUM iajs-3014 139 55 βˆ’ βˆ’ PROPN iajs-3014 139 56 1 1 NUM iajs-3014 139 57 ) ) PUNCT iajs-3014 139 58 + + CCONJ iajs-3014 139 59 ( ( PUNCT iajs-3014 139 60 3𝑖 3𝑖 NUM iajs-3014 139 61 + + CCONJ iajs-3014 139 62 2 2 NUM iajs-3014 139 63 ) ) PUNCT iajs-3014 139 64 for for ADP iajs-3014 139 65 𝑖 𝑖 NOUN iajs-3014 139 66 = = SYM iajs-3014 139 67 1 1 NUM iajs-3014 139 68 , , PUNCT iajs-3014 139 69 3 3 NUM iajs-3014 139 70 , , PUNCT iajs-3014 139 71 … … PUNCT iajs-3014 139 72 , , PUNCT iajs-3014 139 73 2𝑛 2𝑛 PROPN iajs-3014 139 74 βˆ’ βˆ’ PROPN iajs-3014 139 75 1 1 NUM iajs-3014 139 76 , , PUNCT iajs-3014 139 77 ( ( PUNCT iajs-3014 139 78 3𝑖 3𝑖 NUM iajs-3014 139 79 βˆ’ βˆ’ NOUN iajs-3014 139 80 3 3 NUM iajs-3014 139 81 ) ) PUNCT iajs-3014 139 82 + + CCONJ iajs-3014 139 83 ( ( PUNCT iajs-3014 139 84 3𝑖 3𝑖 NUM iajs-3014 139 85 βˆ’ βˆ’ NOUN iajs-3014 139 86 4 4 NUM iajs-3014 139 87 ) ) PUNCT iajs-3014 139 88 + + CCONJ iajs-3014 139 89 3𝑖 3𝑖 NUM iajs-3014 139 90 for for ADP iajs-3014 139 91 𝑖 𝑖 NOUN iajs-3014 139 92 = = SYM iajs-3014 139 93 2 2 NUM iajs-3014 139 94 , , PUNCT iajs-3014 139 95 4 4 NUM iajs-3014 139 96 , , PUNCT iajs-3014 139 97 … … PUNCT iajs-3014 139 98 , , PUNCT iajs-3014 139 99 2𝑛 2𝑛 NUM iajs-3014 139 100 , , PUNCT iajs-3014 139 101 = = PRON iajs-3014 139 102 { { PUNCT iajs-3014 139 103 9𝑖 9𝑖 NOUN iajs-3014 139 104 βˆ’ βˆ’ NOUN iajs-3014 139 105 1 1 NUM iajs-3014 139 106 for for ADP iajs-3014 139 107 i i PRON iajs-3014 139 108 = = NOUN iajs-3014 139 109 1 1 NUM iajs-3014 139 110 , , PUNCT iajs-3014 139 111 3 3 NUM iajs-3014 139 112 , , PUNCT iajs-3014 139 113 … … PUNCT iajs-3014 139 114 , , PUNCT iajs-3014 139 115 2n 2n NUM iajs-3014 139 116 βˆ’ βˆ’ NOUN iajs-3014 139 117 1 1 NUM iajs-3014 139 118 , , PUNCT iajs-3014 139 119 9𝑖 9𝑖 NOUN iajs-3014 139 120 βˆ’ βˆ’ NOUN iajs-3014 139 121 7 7 NUM iajs-3014 139 122 for for ADP iajs-3014 139 123 i i PRON iajs-3014 139 124 = = SYM iajs-3014 139 125 2 2 NUM iajs-3014 139 126 , , PUNCT iajs-3014 139 127 4 4 NUM iajs-3014 139 128 , , PUNCT iajs-3014 139 129 … … PUNCT iajs-3014 139 130 , , PUNCT iajs-3014 139 131 2n 2n NUM iajs-3014 139 132 , , PUNCT iajs-3014 139 133 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 NOUN iajs-3014 139 134 ( ( PUNCT iajs-3014 139 135 𝑒𝑖 𝑒𝑖 NOUN iajs-3014 139 136 ) ) PUNCT iajs-3014 139 137 = = SYM iajs-3014 139 138 πœ‡5(𝑒𝑖𝑣2π‘–βˆ’1 πœ‡5(𝑒𝑖𝑣2π‘–βˆ’1 NOUN iajs-3014 139 139 ) ) PUNCT iajs-3014 139 140 + + NUM iajs-3014 139 141 πœ‡5(𝑒𝑖𝑣2𝑖 πœ‡5(𝑒𝑖𝑣2𝑖 NOUN iajs-3014 139 142 ) ) PUNCT iajs-3014 139 143 + + NUM iajs-3014 139 144 πœ‡5(𝑒𝑖𝑣 πœ‡5(𝑒𝑖𝑣 PROPN iajs-3014 139 145 ) ) PUNCT iajs-3014 139 146 for for ADP iajs-3014 139 147 𝑖 𝑖 NOUN iajs-3014 139 148 = = SYM iajs-3014 139 149 1 1 NUM iajs-3014 139 150 , , PUNCT iajs-3014 139 151 2 2 NUM iajs-3014 139 152 , , PUNCT iajs-3014 139 153 … … PUNCT iajs-3014 139 154 , , PUNCT iajs-3014 139 155 𝑛 𝑛 NOUN iajs-3014 139 156 , , PUNCT iajs-3014 139 157 = = SYM iajs-3014 139 158 ( ( PUNCT iajs-3014 139 159 6𝑖 6𝑖 NOUN iajs-3014 139 160 βˆ’ βˆ’ NOUN iajs-3014 139 161 1 1 NUM iajs-3014 139 162 ) ) PUNCT iajs-3014 139 163 + + CCONJ iajs-3014 139 164 ( ( PUNCT iajs-3014 139 165 6𝑖 6𝑖 NOUN iajs-3014 139 166 ) ) PUNCT iajs-3014 139 167 + + CCONJ iajs-3014 139 168 ( ( PUNCT iajs-3014 139 169 6𝑖 6𝑖 NOUN iajs-3014 139 170 βˆ’ βˆ’ NOUN iajs-3014 139 171 2 2 NUM iajs-3014 139 172 ) ) PUNCT iajs-3014 139 173 = = NOUN iajs-3014 139 174 18𝑖 18𝑖 NUM iajs-3014 139 175 βˆ’ βˆ’ PROPN iajs-3014 139 176 3 3 NUM iajs-3014 139 177 for for ADP iajs-3014 139 178 𝑖 𝑖 PRON iajs-3014 139 179 = = SYM iajs-3014 139 180 1,2 1,2 NUM iajs-3014 139 181 , , PUNCT iajs-3014 139 182 … … PUNCT iajs-3014 139 183 , , PUNCT iajs-3014 139 184 𝑛. 𝑛. NOUN iajs-3014 139 185 finally finally ADV iajs-3014 139 186 : : PUNCT iajs-3014 139 187 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 NOUN iajs-3014 139 188 ( ( PUNCT iajs-3014 139 189 𝑣 𝑣 X iajs-3014 139 190 ) ) PUNCT iajs-3014 139 191 = = SYM iajs-3014 139 192 βˆ‘ βˆ‘ PUNCT iajs-3014 139 193 πœ‡5(𝑣𝑣𝑖 πœ‡5(𝑣𝑣𝑖 NUM iajs-3014 139 194 ) ) PUNCT iajs-3014 139 195 2𝑛 2𝑛 PROPN iajs-3014 139 196 𝑖=1 𝑖=1 PUNCT iajs-3014 140 1 + + CCONJ iajs-3014 140 2 βˆ‘ βˆ‘ PUNCT iajs-3014 140 3 πœ‡5(𝑣𝑒𝑖 πœ‡5(𝑣𝑒𝑖 X iajs-3014 140 4 ) ) PUNCT iajs-3014 140 5 𝑛 𝑛 PRON iajs-3014 140 6 𝑖=1 𝑖=1 PUNCT iajs-3014 140 7 = = PUNCT iajs-3014 140 8 ( ( PUNCT iajs-3014 140 9 3𝑛2 3𝑛2 NUM iajs-3014 140 10 βˆ’ βˆ’ PROPN iajs-3014 140 11 2𝑛 2𝑛 NUM iajs-3014 140 12 ) ) PUNCT iajs-3014 141 1 + + CCONJ iajs-3014 141 2 ( ( PUNCT iajs-3014 141 3 3𝑛2 3𝑛2 NUM iajs-3014 141 4 ) ) PUNCT iajs-3014 141 5 + + CCONJ iajs-3014 141 6 ( ( PUNCT iajs-3014 141 7 3𝑛2 3𝑛2 NUM iajs-3014 141 8 + + SYM iajs-3014 141 9 𝑛 𝑛 X iajs-3014 141 10 ) ) PUNCT iajs-3014 141 11 = = SYM iajs-3014 141 12 9𝑛2 9𝑛2 NUM iajs-3014 141 13 βˆ’ βˆ’ NOUN iajs-3014 141 14 𝑛. 𝑛. NOUN iajs-3014 141 15 which which PRON iajs-3014 141 16 implies imply VERB iajs-3014 141 17 that that SCONJ iajs-3014 141 18 : : PUNCT iajs-3014 141 19 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 NOUN iajs-3014 141 20 ( ( PUNCT iajs-3014 141 21 𝑣1 𝑣1 PROPN iajs-3014 141 22 ) ) PUNCT iajs-3014 141 23 < < X iajs-3014 141 24 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 25 ( ( PUNCT iajs-3014 141 26 𝑣2 𝑣2 PROPN iajs-3014 141 27 ) ) PUNCT iajs-3014 141 28 < < X iajs-3014 141 29 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 30 ( ( PUNCT iajs-3014 141 31 𝑒1 𝑒1 PROPN iajs-3014 141 32 ) ) PUNCT iajs-3014 141 33 < < X iajs-3014 141 34 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 35 ( ( PUNCT iajs-3014 141 36 𝑣3 𝑣3 ADJ iajs-3014 141 37 ) ) PUNCT iajs-3014 141 38 < < X iajs-3014 141 39 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 40 ( ( PUNCT iajs-3014 141 41 𝑣4 𝑣4 PROPN iajs-3014 141 42 ) ) PUNCT iajs-3014 141 43 < < X iajs-3014 141 44 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 45 ( ( PUNCT iajs-3014 141 46 𝑒2 𝑒2 PROPN iajs-3014 141 47 ) ) PUNCT iajs-3014 141 48 < < X iajs-3014 141 49 β‹― β‹― X iajs-3014 141 50 < < X iajs-3014 141 51 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 52 ( ( PUNCT iajs-3014 141 53 𝑣2π‘›βˆ’1 𝑣2π‘›βˆ’1 NOUN iajs-3014 141 54 ) ) PUNCT iajs-3014 141 55 < < X iajs-3014 141 56 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 57 ( ( PUNCT iajs-3014 141 58 𝑣2𝑛 𝑣2𝑛 PROPN iajs-3014 141 59 ) ) PUNCT iajs-3014 141 60 < < X iajs-3014 141 61 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 62 ( ( PUNCT iajs-3014 141 63 𝑒𝑛 𝑒𝑛 NOUN iajs-3014 141 64 ) ) PUNCT iajs-3014 141 65 < < X iajs-3014 141 66 π‘€π‘‘πœ‡5 π‘€π‘‘πœ‡5 PROPN iajs-3014 141 67 ( ( PUNCT iajs-3014 141 68 𝑣 𝑣 NOUN iajs-3014 141 69 ) ) PUNCT iajs-3014 141 70 . . PUNCT iajs-3014 142 1 based base VERB iajs-3014 142 2 on on ADP iajs-3014 142 3 vertex vertex NOUN iajs-3014 142 4 weights weight NOUN iajs-3014 142 5 , , PUNCT iajs-3014 142 6 the the DET iajs-3014 142 7 vertices vertex NOUN iajs-3014 142 8 weights weight NOUN iajs-3014 142 9 are be AUX iajs-3014 142 10 all all ADV iajs-3014 142 11 distinct distinct ADJ iajs-3014 142 12 . . PUNCT iajs-3014 143 1 thus thus ADV iajs-3014 143 2 , , PUNCT iajs-3014 143 3 the the DET iajs-3014 143 4 strong strong ADJ iajs-3014 143 5 face face NOUN iajs-3014 143 6 friendship friendship NOUN iajs-3014 143 7 graph graph NOUN iajs-3014 143 8 ( ( PUNCT iajs-3014 143 9 𝑇𝑛)βˆ— 𝑇𝑛)βˆ— PROPN iajs-3014 143 10 is be AUX iajs-3014 143 11 vertex vertex NOUN iajs-3014 143 12 antimagic antimagic ADJ iajs-3014 143 13 edge edge NOUN iajs-3014 143 14 labeling labeling NOUN iajs-3014 143 15 , , PUNCT iajs-3014 143 16 for for ADP iajs-3014 143 17 every every DET iajs-3014 143 18 𝑛 𝑛 PRON iajs-3014 143 19 β‰₯ β‰₯ NOUN iajs-3014 143 20 3 3 NUM iajs-3014 143 21 . . NOUN iajs-3014 143 22 3 3 NUM iajs-3014 143 23 . . X iajs-3014 143 24 conclusion conclusion NOUN iajs-3014 143 25 in in ADP iajs-3014 143 26 this this DET iajs-3014 143 27 paper paper NOUN iajs-3014 143 28 , , PUNCT iajs-3014 143 29 we we PRON iajs-3014 143 30 proved prove VERB iajs-3014 143 31 that that SCONJ iajs-3014 143 32 some some DET iajs-3014 143 33 families family NOUN iajs-3014 143 34 of of ADP iajs-3014 143 35 graphs graph NOUN iajs-3014 143 36 related relate VERB iajs-3014 143 37 to to ADP iajs-3014 143 38 strong strong ADJ iajs-3014 143 39 face face NOUN iajs-3014 143 40 graphs graph NOUN iajs-3014 143 41 , , PUNCT iajs-3014 143 42 admit admit VERB iajs-3014 143 43 vertex vertex NOUN iajs-3014 143 44 antimagic antimagic ADJ iajs-3014 143 45 edge edge NOUN iajs-3014 143 46 labeling labeling NOUN iajs-3014 143 47 . . PUNCT iajs-3014 144 1 our our PRON iajs-3014 144 2 future future ADJ iajs-3014 144 3 work work NOUN iajs-3014 144 4 will will AUX iajs-3014 144 5 involve involve VERB iajs-3014 144 6 finding find VERB iajs-3014 144 7 another another DET iajs-3014 144 8 family family NOUN iajs-3014 144 9 of of ADP iajs-3014 144 10 strong strong ADJ iajs-3014 144 11 face face NOUN iajs-3014 144 12 graphs graph NOUN iajs-3014 144 13 that that PRON iajs-3014 144 14 admits admit VERB iajs-3014 144 15 antimagic antimagic ADJ iajs-3014 144 16 labeling labeling NOUN iajs-3014 144 17 . . PUNCT iajs-3014 145 1 references reference NOUN iajs-3014 145 2 1.hartsfield 1.hartsfield NUM iajs-3014 145 3 , , PUNCT iajs-3014 145 4 n. n. NOUN iajs-3014 145 5 ; ; PUNCT iajs-3014 145 6 ringel ringel NOUN iajs-3014 145 7 , , PUNCT iajs-3014 145 8 g. g. NOUN iajs-3014 145 9 pearls pearl NOUN iajs-3014 145 10 in in ADP iajs-3014 145 11 graph graph NOUN iajs-3014 145 12 theory theory NOUN iajs-3014 145 13 . . PUNCT iajs-3014 146 1 academic academic ADJ iajs-3014 146 2 press press PROPN iajs-3014 146 3 , , PUNCT iajs-3014 146 4 boston boston PROPN iajs-3014 146 5 san san PROPN iajs-3014 146 6 diego diego PROPN iajs-3014 146 7 new new PROPN iajs-3014 146 8 york york PROPN iajs-3014 146 9 -london.1990 -london.1990 PROPN iajs-3014 146 10 . . PROPN iajs-3014 146 11 2.miller 2.miller NUM iajs-3014 146 12 , , PUNCT iajs-3014 146 13 m. m. NOUN iajs-3014 146 14 ; ; PUNCT iajs-3014 146 15 phanalasy phanalasy PROPN iajs-3014 146 16 , , PUNCT iajs-3014 146 17 o. o. PROPN iajs-3014 146 18 ; ; PUNCT iajs-3014 146 19 ryan ryan PROPN iajs-3014 146 20 , , PUNCT iajs-3014 146 21 j. j. PROPN iajs-3014 146 22 ; ; PUNCT iajs-3014 146 23 rylands ryland NOUN iajs-3014 146 24 , , PUNCT iajs-3014 146 25 l. l. 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PUNCT iajs-3014 153 1 on on ADP iajs-3014 153 2 face face VERB iajs-3014 153 3 antimagic antimagic ADJ iajs-3014 153 4 labeling labeling NOUN iajs-3014 153 5 of of ADP iajs-3014 153 6 strong strong ADJ iajs-3014 153 7 face face NOUN iajs-3014 153 8 plane plane NOUN iajs-3014 153 9 graphs graph NOUN iajs-3014 153 10 . . PUNCT iajs-3014 154 1 appl appl PROPN iajs-3014 154 2 . . PROPN iajs-3014 154 3 math math PROPN iajs-3014 154 4 . . PUNCT iajs-3014 155 1 sci sci PROPN iajs-3014 155 2 . . PROPN iajs-3014 155 3 2017 2017 NUM iajs-3014 155 4 , , PUNCT iajs-3014 155 5 11 11 NUM iajs-3014 155 6 , , PUNCT iajs-3014 155 7 77–91 77–91 NUM iajs-3014 155 8 . . PUNCT iajs-3014 156 1 5.vasuki 5.vasuki NUM iajs-3014 156 2 , , PUNCT iajs-3014 156 3 b. b. PROPN iajs-3014 156 4 ; ; PUNCT iajs-3014 156 5 shobana shobana PROPN iajs-3014 156 6 , , PUNCT iajs-3014 156 7 l. l. PROPN iajs-3014 156 8 ; ; PUNCT iajs-3014 156 9 ahmed ahmed PROPN iajs-3014 156 10 , , PUNCT iajs-3014 156 11 m. m. 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PROPN iajs-3014 160 1 a a DET iajs-3014 160 2 dynamic dynamic ADJ iajs-3014 160 3 survey survey NOUN iajs-3014 160 4 of of ADP iajs-3014 160 5 graph graph NOUN iajs-3014 160 6 labeling labeling NOUN iajs-3014 160 7 . . PUNCT iajs-3014 161 1 the the DET iajs-3014 161 2 electronic electronic ADJ iajs-3014 161 3 journal journal NOUN iajs-3014 161 4 of of ADP iajs-3014 161 5 combinatorics combinatoric NOUN iajs-3014 161 6 , , PUNCT iajs-3014 161 7 # # SYM iajs-3014 161 8 ds6 ds6 NOUN iajs-3014 161 9 . . PUNCT iajs-3014 161 10 2020 2020 NUM iajs-3014 161 11 . . PUNCT iajs-3014 162 1 7.bača 7.bača NUM iajs-3014 162 2 , , PUNCT iajs-3014 162 3 m. m. NOUN iajs-3014 162 4 ; ; PUNCT iajs-3014 162 5 miller miller PROPN iajs-3014 162 6 , , PUNCT iajs-3014 162 7 m. m. NOUN iajs-3014 162 8 ; ; PUNCT iajs-3014 162 9 ryan ryan PROPN iajs-3014 162 10 , , PUNCT iajs-3014 162 11 j. j. 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NOUN iajs-3014 163 12 ; ; PUNCT iajs-3014 163 13 semaničovΓ‘ semaničovΓ‘ NOUN iajs-3014 163 14 - - PUNCT iajs-3014 163 15 feňovčíkovΓ‘ feňovčíkovΓ‘ PROPN iajs-3014 163 16 , , PUNCT iajs-3014 163 17 a. a. NOUN iajs-3014 163 18 local local ADJ iajs-3014 163 19 antimagic antimagic ADJ iajs-3014 163 20 vertex vertex NOUN iajs-3014 163 21 coloring coloring NOUN iajs-3014 163 22 of of ADP iajs-3014 163 23 a a DET iajs-3014 163 24 graph graph NOUN iajs-3014 163 25 . . PUNCT iajs-3014 164 1 graphs graph NOUN iajs-3014 164 2 combin combin NOUN iajs-3014 164 3 . . PROPN iajs-3014 165 1 2017 2017 NUM iajs-3014 165 2 , , PUNCT iajs-3014 165 3 33 33 NUM iajs-3014 165 4 , , PUNCT iajs-3014 165 5 275–285 275–285 NUM iajs-3014 165 6 . . PUNCT iajs-3014 166 1 9.baca 9.baca NUM iajs-3014 166 2 , , PUNCT iajs-3014 166 3 m. m. NOUN iajs-3014 166 4 ; ; PUNCT iajs-3014 166 5 miller miller PROPN iajs-3014 166 6 , , PUNCT iajs-3014 166 7 m. m. 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