Some topological indices of graphs and some inequalities

Transcription

1 Iraia Joral of athatical Chistr, Vol, No, Fbrar 0, pp7 80 IJC So topological idics of graphs ad so iqalitis OHARRAB, AND B KHZRI OHADDA Dpartt of athatics, Prsia lf Uirsit, Bshhr7569, Ira Dpartt of athatics, Paa Noor Uirsit P O Bo: , Shiraz, Ira Rcid Jl, 0 ABSTRACT Lt b a graph I this papr, w std th cctric coctiit id, th w rsio of th scod Zagrb id ad th forth gotric arithtic id Th basic proprtis of ths ol graph dscriptors ad so iqalitis for th ar stablishd Kwords: Topological id, cctric coctiit, gotric arithtic, Zagrb id, Cach Schwarz iqalit INTRODUCTION A topological id for a graph is a rical qatit which is iariat dr atoorphiss of th graph Th siplst topological idics ar th br of rtics ad dgs of th graph Throghot this papr, all graphs ar assd to b sipl coctd with rtics ad dgs that ar dirctd I th last fw ars, th br of proposd olclar dscriptors is rapidl growig ] A spcial class of ths dscriptors copriss is calld topological idics Topological idics ar sall dfid ia th olclar graph raph thor is a athatical discipli blogig to discrt athatics For or iforatio o graph thor ad applicatio i chistr w rfr to 5] Sppos that V, is a graph with th rt st V ad th dg st, that V ad O of th rct olclar dscriptors dfid b graph dgr is th gotric arithtic idics of graphs ad its ariats Th gral forla for th gotric arithtic id is gi b gral Q Q / Q Q, whr for a rt, th br Q is ail:ogharab@gailco

2 7 OHARRAB AND B KHZRI OHADDA so qatit that, i a iq ar, ca b associatd with th rt 6] As t, for topological idics blogig to th -fail ha b cocid, ad as th first, scod, third ad th forth gotric arithtic id Th first gotric arithtic id of a graph is dfid as dd / d d, whr d is th dgr of th rt i th graph Th scod gotric arithtic id is calclatd b th forla /, whr for a dg, is th br of rtics that ar closr to th rt tha to th rt ad is dfid aalogosl Th third gotric arithtic id is dfid b th forla /, whr for a dg, is th br of dgs that ar closr to th rt tha to th rt ad is dfid aalogosl Lt a, b, b rtics ad b a dg of a graph Th distac btw a ad b that dotd b d a, b is th lgth of a shortst path coctig a ad b i th graph Th distac btw th rt ad th dg is dfid b d, i{ d,, d, } Th cctricit of a rt is dotd b ad is gi b a{ d, V} Th ai al of cctricit or all rtics of is calld th diatr of ad dotd b D Also, th ii al of cctricit aog th rtics of is calld th radis of ad dotd b r Th cctric coctiit id of th graph is dfid as V d Th forth gotric arithtic id of is dfid as / Th Zagrb grop idics of a graph ha b itrodcd or tha thirt ars ago b ta ad Triajstic 5] horbai 6] has dfid two w rsio of Zagrb idics as follows: ad It is as to s that for a graph, XAPLS Dirctl fro th dfiitio, w calclat th cctric coctiit, th forth gotric arithtic ad th w scod Zagrb idics of th rt coplt graph K, th coplt bipartit graph K r, t, th rt ccl graph C, th rt path P ad th rt star S Ths ar as follows:

3 So topological idics of graphs ad so iqalitis 75 K, K, K, K r, t rt, K r, t rt ad K If is, th for V, ad C, C, C If is odd, th for all V,, C ad C C If is, th P, / i i P i i ad / P 5 i i If i is odd th P, / i i P ad i / P i i i i Fiall, S, S ad S I this papr, th ai proprtis of, ad idics of graphs ar stablishd ad so bods for ths idics with rlatio btw th ar prstd AIN RSULTS AND DISCUSSION I this sctio, at first w calclat so bods for th forth gotric arithtic, th cctric coctiit ad w Zagrb idics of a graph, th prst so rlatios btw ths idics Th faos iqalit ab a b / ab for a positi ral brs a, b with qalit if ad ol if a b ad also th Cach-Schwarz iqalit ha b sd i th proof of th followig propositios Not that K if ad ol if for all, Propositio For a graph, i with qalit if ad ol if K, ii with qalit if ad ol if, k for so k, iii with qalit if ad ol if K Proof For proig this propositio, it is ogh to otic that for all V,, wh

4 76 OHARRAB AND B KHZRI OHADDA Propositio Sppos is a graph with dgs Th / with qalit if ad ol if S, whr S dots th rtics star Proof Sic for a dgs, or, w ca ass that If /, th / / Sppos f / B driatio, w ca cocld that f is icrasig o th closd itral /, ] So, for a /,], f f / This shows that / / ad / Now w pro that / if ad ol if S If S th for a dg, ad, so / If / ad is ot isoorphic to S, th thr ists a dg sch that or Sic or, th for /, / ad so / 6 / 5 I this cas, / 6 / 5 / Not It is as to s that if ' with = th ' / ' I this part, w prst so rlatioships btw th forth gotric arithtic id, th cctric coctiit id ad w Zagrb id of a graph Propositio Sppos is a graph th / Proof Sic for a rt, whr, w ca cocld that If f /, th o ca s th fctio f is icrasig o th closd itral, ] Thrfor, for all V, f ad so Propositio / with qalit if ad ol if K

5 So topological idics of graphs ad so iqalitis 77 Proof Sic for a rt, w ca cocld that for a dg, Thrfor, w ha: qalit if ad ol if for a dg proof ad th with w ha, which coplts or Propositio 5 Lt K Proof Sic b a graph th with qalit if ad ol if, ad b applig th Cach Schwarz Iqalit, Thrfor with qalit if ad ol Ths K If K th, ad so Propositio 6 Lt b a graph th with qalit if ad ol if K Proof B applig th hpothsis i th proof of Propositio, w ca s Thrfor, ' ' Now w clai that qalit holds if ad ol if if ad ol if, if ad ol if K Not w ca s that if for a dg,, th Thrfor, for ach, Propositio 7 If is a graph th Proof B dfiitio,

6 78 OHARRAB AND B KHZRI OHADDA ] ] ] ], ad so Propositio 8 Lt b a graph, th with qalit if ad ol if K Proof Tak t, so ] ] ] t t Thrfor ] t t ad so Th qalit holds if ad ol if K Propositio 9 If is a graph, th Proof Not that a positi ral brs b a, satisf th iqalit ab b a ad qalit holds if ad ol if b a W ca asil to s that, so with qalit if ad ol if, if ad ol if ad if ad ol if 5 that is a cotradictio So

7 So topological idics of graphs ad so iqalitis 79 Propositio 0 Sppos that is a graph, th th followig statts hold: i, ii ], iii ] a b Proof Th proof of this propositio b applig th iqalit ab for a positi ral brs a, b is siilar to th proof of propositi9 ad it is oittd Propositio Lt is a graph i If, i{, }, th ; ii If, i{, }, th Proof To pro this propositio, it is ogh to otic that if, i{, }, th ad if, i{, }, th ACKNOWLDNT This papr was spportd i part b th Rsarch Diisio of Prsia lf Uirsit RFRNCS R Todschii ad V Cosoi, olclar Dscriptors for Choiforatics, Wil VCH, Wihi, 009 H Hosoa, Topological id, A wl proposd qatit charactrizig th topological atr of strctral isors of satratd hdrocarbos, Bll Ch Soc Japa, 97 9 A raoac, O Ori, Faghai ad A R Ashrafi, Distac Proprt of Fllrs, Iraia J ath Ch, H Fath Tabar, J Nadjafi Arai, ogharrab ad A R Ashrafi, So Iqalitis for Szgd Lik Topological Idics of raphs, ATCH Co ath Copt Ch, Z Yarahadi ad S oradi, Scod ad third trals of catacodsd hagoal ssts with rspct to th PI id, Iraia J ath Ch, D Vkičić ad B Frtla, Topological id basd o th ratios of gotrical

8 80 OHARRAB AND B KHZRI OHADDA ad arithtical as of d rt dgrs of dgs J ath Ch, B Zho, I ta, B Frtla ad Z D, O two tps of gotric arithtic id, Ch Phs Ltt, K C Das, O gotric arithtic id of graphs, ATCH Co ath Copt Ch, Y Ya, B Zho ad N Triajstić, O gotric arithtic id, J ath Ch, ogharrab ad H Fath Tabar, So bods o id of graphs, ATCH Co ath Copt Ch, H Fath Tabar, B Frtla ad I ta, A w gotric arithtic id, J ath Ch, H Fath Tabar, Old ad w Zagrb idics of graphs, ATCH Co ath Copt Ch, H Fath Tabar, A Hazh ad S Hossi-Zadh, id of so graph opratios, FILOAT, 00 8 H FathTabar, A Azad ad L lahijad, So Topological Idics of Ttraric,-Adaata, Iraia J ath Ch, I ta ad N Triajstić, raph thor ad olclar orbitals, Total lctro rg of altrat hdrocarbos, Ch Phs Ltt, horbai, Prsoal coicatio, 0