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1 Geometc-athmetc Idex ad Zageb Idces of Ceta Specal Molecula Gaphs efe X, e Gao School of Tousm ad Geogaphc Sceces, Yua Nomal Uesty Kumg , Cha School of Ifomato Scece ad Techology, Yua Nomal Uesty Kumg , Cha Abstact: I ths pape, we deteme the Geometc-athmetc dexad Zageb dcesof fa molecula gaph, wheel molecula gaph, gea fa molecula gaph, gea wheel molecula gaph, ad the -cooa molecula gaphs Keywods: Chemcal gaph theoy, Geometc-athmetc dex, Zageb dex, a molecula gaph, heel molecula gaph, Gea fa moleculagaph, Gea wheel moleculagaph, -cooa moleculagaph Coucl fo Ioate Reseach Joual: Joual of Adaces Chemsty Vol 0, No edtoacole@gmalcom wwwcaccom Pee Reew Reseach Publshg System 54 P a g e J u e 0, 0 4
2 INTRODUCTION ee dex,edge ee dex, Hype-wee dex, Geometc-athmetc dex ad Zageb dces ae toduced to eflect ceta stuctual featues of ogac molecules Seeal papes cotbuted to deteme the dstace-based dex of specal molecula gaphs (See Ya et al, [] ad [], Gao ad Sh [3] fo moe detal LetP ad C be path ad cycle wth etces The molecula gaph {} P s called a fa molecula gaph ad the molecula gaph {} C s called a wheel molecula gaph Molecula gaph I (G s called - cow molecula gaph of G whch splcg hag edges fo eey etex G By addg oe etex eey two adacet etces of the fa path P of fa molecula gaph, the esultg molecula gaph s a subdso molecula gaph called gea fa molecula gaph, deote as By addg oe etex eey two adacet etces of the wheel cycle C of wheel molecula gaph, The esultg molecula gaph s a subdso molecula gaph, called gea wheel molecula gaph, deoted as By cosdeg the degees of etces G, Vukcec ad utula [4]deeloped the Geometc-athmetc dex, shotly GA dex, whch s defed by GA( G ue ( G d( u d(, d( u d( wheed(u ad d( ae the degees of u ad, espectely ad The (fst ad secod Zageb dces hae bee toduced by Gutma ad Tastc[5] as the fom M ( G ( d (, V ( G M ( G d( u d( ue ( G O the othe had, fo a moleculagaph G, the modfed secod Zageb dex M ( G s defed as M ( G ue ( G d( u d( Seeal papes cotbuted o detemg the Zageb dces of specal molecula gaphs ca efe to [6-0] I ths pape, we peset the Geometc-athmetc dex of I (, I (, I ( ad I ( dces of I (, I (, I ( ad I ( ae deed Also, the Zageb GEOMETRIC-ARITHMETICINDEX Theoem GA( I( 4 ( ( ( ( (3 3 4 ( (3 ( 3 (3 ( ( Poof Let P ad the hagg etces of be,,, ( Let be a etex besde P, ad 55 P a g e J u e 0, 0 4
3 the hagg etces of be GA( I (, d( d( d( d(,, By the defto of Geometc-athmetc dex, we hae d( d( d( d( d( d( d( d( d( d( d d( ( 4 ( ( ( ( (3 ( 3 4 ( (3 ( 3 (3 (3 ( 5 3 Coollay GA( 4 ( ( 3 ( 3 4 Theoem GA( I( ( (3 3 (3 ( Poof Let C ad,,, be the hagg etces of ( Let be a etex besde C, ad,,, GA( I ( be the hagg etces of By the defto of Geometc-athmetc dex, we hae d( d( d( d( d( d( d( d( d( d( d( d( d( d( d d( ( ( (3 3 (3 ( Coollay GA( 3 3 Theoem 3 GA( I( 4 ( ( ( ( (3 3 4 ( ( ( 4( (3 ( 5 ( 3 PoofLet P ad, be the addg etex betwee ad Let,,, be the hagg etces of ( Let,,,,,, be the hagg etces of, ( - Let be a etex besde P, ad the hagg etces of be,,, By tue of the defto of Geometc-athmetc dex, we get 56 P a g e J u e 0, 0 4
4 GA( I ( d( d( d( d( d( d( d( d( d( d( d d( ( d( d(, d( d(, d( d(,, d(, d( d, d( d(, ( d(, 4 ( ( ( ( (3 ( 3 4 ( 3 ( 3 4 ( ( ( (3 ( ( 5 Coollay3 GA( 4 ( 3 3 ( ( ( (3 ( ( 5 ( 3 4( Theoem4 GA( I( ( ( (3 ( 5 3 Poof Let C ad be a etex besde C,, be the addg etex betwee ad Let,,, be the hagg etces of ad,,, be the hagg etces of ( Let,, ad,,,,,, be the hagg etces of, deduce ( I ew of the defto of Geometc-athmetc dex, we GA( I ( d( d( d( d( d( d( d( d( d( d( d d( ( d( d(, d( d(, d( d(,, d(, d( d, d( d(, ( d(, ( ( (3 ( 5 (3 ( 5 3 Coollay 4 GA( ZAGREB INDICES Usg the otatos defed aboe secto, ad combg wth the deftos of Zageb dces, we get the followg computatoal fomulas 57 P a g e J u e 0, 0 4
5 ( ( M I ( d ( ( ( d ( ( d ( ( ( (3 ( d ( ( ( ( ( M I ( d ( M ( 9 0 ( ( d ( ( d ( (3 ( d ( ( (9 9 M ( 9 M ( ( I ( d ( ( d ( ( d ( ( ( d (, ( ( (3 ( M ( 3 4 M ( ( I ( d ( ( d ( ( d ( ( ( d (, (3 ( ( d ( ( ( ( d (, ( ( d ( ( d (, ( (4 3 M ( 3 ( ( M I d( d( d( d( d( d( d( d( 58 P a g e J u e 0, 0 4
6 ( (( ( ( ( (3 (( (3 ( 3(3 (3 ( ( ( (3 3 ( M ( ( ( M I d( d( d( d( d( d( ( ( (3 (3 (3 (3 d( d( (3 ( M ( 3 9 M ( ( I d( d( d( d( d( d( d( d(, d(, d( d(, d(, ( (( ( ( ( (3 ( ( ( (3 (( ( ( (3 ( (( ( ( (3 ( ( ( (5 3 ( M ( M ( ( I d( d( d( d( d( d( d( d(, d(, d( d(, d(, ( ( (3 (3 (3 ( (3 ( ( (5 ( 9 3 M ( 3 59 P a g e J u e 0, 0 4
7 M ( ( I ( d( d( ( d( d( ( d( d( ( ( ( (3 ( 3 M ( ( (3 (3 (3 ( d( d( M ( ( I ( d( d( ( d( d( ( d( d( ( (3 (3 (3 3 ( d( d( M ( 3 9 M ( ( I ( d( d( ( d( d( ( d( d( ( d( d(, ( d(, d( ( d(, d(, ( ( ( ( (3 3 ( ( ( ( (3 ( M ( 3 3 M ( ( I ( d( d( ( d( d( ( d( d( ( d( d(, ( d(, d( ( d( d(,, ( (3 3 (3 ( M ( 3 60 P a g e J u e 0, 0 4
8 4 CONCLUSION I ths pape, we deteme the Geometc-athmetc dex ad Zageb dces of fa molecula gaph, wheel molecula gaph, gea fa molecula gaph, gea wheel molecula gaph, ad the -cooa molecula gaphs 5 Ackowledgemets st, we thak the eewes fo the costucte commets mpog the qualty of ths papeths wok was suppoted pat by the Natoal Natual Scece oudato of Cha (6607, ad the Key Scece ad Techology Reseach Poect of Educato Msty (00 e also would lke to thak the aoymous efeees fo podg us wth costucte commets ad suggestos REERENCES [] L Ya, Y L, Gao, J L, O the exteal hype-wee dex of gaphs, Joual of Chemcal ad Phamaceutcal Reseach, 04, 6(3: [] L Ya, Y L, Gao, J L, PI dex fo some specal gaphs, Joual of Chemcal ad Phamaceutcal Reseach, 03, 5(:60-64 [3] Gao, L Sh, ee dex of gea fa gaph ad gea wheel gaph, Asa Joual of Chemsty, 04, 6(: [4] D Vukcec, B utula, Topologcal dex based o the atos of geometcal ad athmetcal meas of ed-etexdegees of edges, J Math Chem,009, 4: [5] I Gutma, NTastc, Gaph theoy ad molecula obtals Total 4 -electo eegy of alteate hydocaboschem Phys Lett, 97, 7: [6] K Das, IGutma, B Zhou, New uppe bouds o Zageb dces, J Math Chem, 009, 46, 54-5 [7] P Ra, VLokesha, The Smaadache-Zageb dces o the thee gaph opeatos, It J MathComb,00, 3: -0 [8] P Ra, VLokesha, ICagul, O the Zageb dces of the le gaphs of the subdso gaphs Appl MathComput, 0, 8(3: [9] P Ra, VLokesha, MRaa, O Zageb dces of the subdso gaphs, It J Math Sc Eg Appl,00, 4: -8 [0] K Das, Atom-bod coectty dex of gaphs, Dscete Appl Math, 00, 58: P a g e J u e 0, 0 4
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