Topological Indices of Hypercubes

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1 202, TextRoad Publcato ISSN Joural of Basc ad Appled Scetfc Research wwwtextroadcom Topologcal Idces of Hypercubes Sahad Daeshvar, okha Izbrak 2, Mozhga Masour Kalebar 3,2 Departmet of Idustral Egeerg, Easter Medterraea Uversty, North Cyprus (va Mers 0 Turkey) Departmet of Mathematcs, Tabrz Brach, Islamc Azad Uversty, Tabrz, Ira 3 Youg Researchers Club, Tabrz Brach, Islamc Azad Uversty, Tabrz, Ira ABSTRACT Weer dex of a graph s defed to be the sum of the dstaces of each par of vertces Schultz dex of a graph s sum of d(a+d ), where a ad d are the th etry of the adacecy ad dstace matrces of the graph ad d s degree of vertex for all, V ( ) Defto of Padmakar-Iva (PI) ad Szeged dces s based o the umber of vertces ad edges whch are earer to oe ed of a gve edge tha the other ed Topologcal dces of may molecules wth structures ear to cube has bee wdely studed chemstry I ths paper we gve explct formulas for four topologcal dces(weer, Schultz, PI ad Szeged) of hypercubes ad ther correspodg Eucldea graph, a weghted graph wth weght of a edge s equal to Eucldea dstace betwee ts edpots Also some bouds have bee deduced KEYWORDS: Topologcal dces, Hypercubes, Eucldea graphs INTRODUCTION raph theory s a useful tool may areas mathematcal chemstry Topologcal dces are graph theoretcal cocepts whch has a lot of applcatos chemstry [9] These dces are parameters over graphs of molecules whch has bee usually based o the cocept of dstace betwee vertces (or edges) of the graph [0] The frst dex of ths type has bee troduced 947 by HWeer [] Several other dces have bee defed later [0] A lot of work has bee doe o the dces ether from a chemcal pot of vew or through purely mathematcal approaches Some relatos has bee foud betwee dces [3], dces of some classes of graphs has bee detected[8], some complexty cocepts has bee studed o chemcal graphs [5] ad purely mathematcal work has bee doe o relatos of Weer dex ad matchg of a graph[4] Our work ams to fd out explct mathematcal formulas for four mportat dces of hypercubes I ths cotext ( V, E ) s a graph wth vertex set V ( ) { v,, v } ad edge set E e e m ( ) {,, } For smplcty we use V ad E for vertex ad edge set of respectvely The adacecy matrx of, wrtte A(), s the -by- matrx whch etry a, s the umber of edges wth edpots { v, v } [2] Also D(), dstace matrx of has d as ts th etry, where d s the dstace betwee vertces ad [] All over ths paper our graphs are smple ad coected so etres of adacecy matrx are 0 or I ext secto, we beg by the defto of four of these dces 2 Deftos The frst topologcal dex whch troduced by Weer was defed as follows due to the otatos whch are metoed last secto Defto 2 [9] The Weer dex of a graph s W ( ) d ( u, v ) 2 u, v V where d(u, v) s the dstace betwee vertces u ad v o the shortest path betwee them ad sum s take over all pars of vertces of Defto 22 [7] The Schultz dex of a graph s * Correspodg Author: Sahad Daeshvar, Departmet of Idustral Egeerg, Easter Medterraea Uversty, North Cyprus (va Mers 0) E-mal: sahaddaeshvar@emuedutr & sahaddaeshvar@yahoocom 50

2 Daeshvar et al, 202 MTI ( ) d ( a d ) where d s degree of th vertex ad a ad d are etres row ad colum of adacecy matrx A() ad dstace matrx D() of Defto 23 [4] Let e uv be a edge of wth edpots u ad v ad me ( u ) be the umber of vertces whch are closer to u tha v The the Szeged dex of a graph s S ( ) P m ( u ) m ( v ), where sum s take over all edges of Z e 2E e e The dstace from a vertex to a edge a graph s the mmum of the dstaces of the vertex from the ed-pots of the edge Wth respect to ths we have the followg defto Defto 24 [3] Let e uv be a edge of wth edpots u ad v ad e ( u ) be the umber of edges whch are closer to u tha v The, the PI dex of s PI ( ) P ( ( u ) ( v )), where sum s take over all edges of e 2E e e Structure of may of molecules for whch the dces has bee studed s smlar to a cube I addto Eucldea dstaces chemcal structures are as mportat as legths of paths ther graphs [6] Ths motvates us to look for dces of hypercubes To prepare mathematcal backgroud we eed the followg deftos Defto 25 [2] A -dmesoal cube or hypercube s the smple graph whose vertces are the - tuples wth etres {0, } ad whose edges are the pars of -tuples that dffer exactly oe posto Defto 26 [2] Eucldea graph s a weghted graph whch the weghts are equal to the Eucldea legths of the edges a specfed embeddg I ths cotext Eucldea graph of a graph s a ew graph, deoted by E, wth vertex set V ( E ) V ( ) ad edge set E ( E ) V ( ) V ( ) such that weght of each edge s equal to ts Eucldea legth (Correspodg to the above defto) For smplcty we use otato E stead of E I what follows, we fd the explct mathematcal formulas for Weer, Shultz, PI ad Szeged dces of hypercubes ad Eucldea graphs 3 Idces of hypercubes At frst, the Weer dex of hypercube s computed ad the result expressed as a explct mathematcal formula Theorem 3 Let be a -dmesoal cube The Weer dex of s W ( ) 4 Proof Note that s vertex trastve, so for each par u ad v of vertces of, sum of the dstaces of v from other vertces s equal to that of u Now for every,, f a vertex s at dstace from v (0,,0) the t must have exactly postos The umber of such vertces s So sum of dstaces of all vertces of from v s Now cosder that ( x ) x Takg 502

3 dervatos from both sdes of the equalty we have ( x ) x If we put x the 2 has 2 vertces, therefore W ( ) I the ext theorem the Szeged dex of hypercube Theorem 32 Szeged dex of -dmesoal cube s computed of order s Proof Let e uv be a arbtrary edge of ad me ( u ) the umber of the vertces of earer to u tha v Sce s vertex trastve the me ( u ) me ( v ) for every edge e uv Aga vertex trastvty allows us to suppose that u (0,, 0) ad v (,,0) wthout loss of geeralty If a vertex s at dstace from u, the ts dstace from v could ot be more tha ( ) Such a vertex should have coordates other tha the frst oe The umber of such vertces whch are at dstace from u ad at dstace from v s So Sz ( ) (2 ) A explct formula s cocluded for the PI dex of Theorem 33 PI ( ) ( )2 2 2 the followg theorem Proof Let e uv be a edge of ad e ( u ) be the umber of edges earer to u tha v Because of vertex ad edge trastvty of we ust eed to calculate e ( u ) for u (0,,0) ad v (,,0) If a edge s at dstace ( ) from u ad dstace from v the oe of ts edpots should have of ts coordates other tha the frst coordate ad the other edpot should have exactly oe extra a coordate other tha the former coordates So the umber of such edges s ( ) So PI ( ) 2 2 ( ) 0 2 ( ) The forth dex, Schultz dex, s foud for 2 ( )2 2 2 ( )2 ext theorem 503

4 Daeshvar et al, Theorem 34 MTI ( ) Proof We have MTI ( ) d ( a d ) Obvously d for every vertex of Also f two vertces are adacet the a ad d, the a d 2, otherwse a d d The 2 2 because each par of adacet vertces should have umber of par of adacet vertces s exactly oe dfferet coordate (whch s possble ways), ths coordate ca be 0 or (2 ways) ad each of the other coordates ca be ether 0 or ( 2 ways) Smlarly the umber of par of vertces dstace s 2 2 Therefore ) MTI ( ) 2( ( 2 ) At the ed of ths secto we ust meto relatos betwee some of these dces arse easly from ther formulas 2 Sz ( ) W ( ) ad PI ( ) 8 W ( ) 4 Idces of Eucldea graphs of -dmesoal cubes For fdg our desre dces for Euclda graph of -dmesoal cube we eed a lemma whch s metoed followg Lemma 3 If two vertces are at dstace from each other the ther Eucldea dstace s ad vce versa Proof Two vertces dstace have exactly dfferet coordates, so ther Eucldea dstace s Now, by usg the above lemma, we express the Weer, Shultz, PI ad Szeged dces of Euclda graph of -dmesoal cube Theorem 4 2 W ( E ) 2 Proof From the above lemma we ust eed to replace wth theorem 0 Usg Cauchy- Schwartz equalty we have 2 2 So W ( E ) 2 (I) Theorem 4 3 MTI ( E ) 2 W ( E ) 2 2 Proof From lemma 4 replace wth the proof of theorem 34 ad use theorem

5 Iequalty I, gves the followg boud ( ) 2 MTI ( E ) Theorem 44 PI ad Szeged dces of Eucldea graph of a -dmesoal cube are equal to PI ad Szeged dces of that -dmesoal cube respectvely Proof From lemma 4 t s obvous that for ay edge uv a arbtrary vertex w s earer to u f ad oly f t s earer to u E ad hece PI ad Szeged dces does ot dffer for ad ts Eucldea graph E 5 Cocluso We ca compute ad express completely the explct mathematcal formulas for four topologcal dces, Weer, Schultz, PI ad Szeged dces, for hypercubes ad Eucldea graph of -dmesoal hypercubes that these dces have applcatos mathematcal chemstry, chemometrcs ad aocomputatos REFERENCES [] Weer, H, 947, Structural determato of paraff bolg pots, J Am Chem Soc 69 pp 7-20 [2] Skea, S, 990, Implemetg Dscrete Mathematcs: Combatory ad raph Theory wth Mathematca, Addso-Wesley, Readg, MA [3] Dobry, A A, 999, Explct relato betwee Weer Idex ad the Schultz Idex of Catacadeced Bezeod raphs, Croatca Chemca Acta, 72, pp [4] Ya, Y, ad Yeh, Y, 2006, Coectos betwee Weer dex ad Matchg, Joural of Mathematcal Chemstry, 39(2), pp [5] Balaba, A T, Mlls, D, Kodal, V ad Basak, S C, 2006, Complexty of chemcal graphs terms of sze, brachg ad cyclcty, SAR ad SAR Evrometal Research, 7(4), pp [6] Ashraf, A, ad Hamadaa, M, 2005, Symmetry propertes of some Chemcal raphs, Croata Chemca Acta, 78(2), pp [7] Schultz, H P, 989, Topologcal orgac chemstry raph theory ad topologcal dces of alkaes, J Chem If Comput Sc, 29, pp [8] Dobry, A A, Etrger, R, utma, I, 200, Weer Idex of trees: theory ad applcatos, Acta Appl Math 66(3), pp [9] utma, I, ad Polasky, O E, 986, Mathematcal Cocepts Orgac Chemstry, Sprger-Verlag, Berl [0] Radc, M, Mhalc, Z, Nkolc, S, ad Trastc, N, 994, raphcal bod orders: Novel structural descrptors, J Chem If Comput Sc, 34, pp [] Buckly, F, ad Harary, H, 990, Dstace raphs, Peruses Books, New York [2] West, D B, 200, Itroducto to raph Theory, Secod edto, Pretce hall, Upper Saddle Rver [3] Khadkar, P V, ad Karmarkar, S, 2002, O The Estmato of PI Idex of Polyacees, Acta Chm Slov, 49, pp [4] Chepo, V, ad Klavzar, S, 997, The Weer Idex ad the Szeged Idex of Bezeod Systems Lear Tme, J Chem If Comput Sc, 37, pp