R Index of Some Graphs

Transcription

1 Aals of Pure ad Applied athematics Vol 6, No, 08, 6-67 IN: X P), olie) Published o Jauary 08 wwwresearchmathsciorg DOI: Aals of Departmet of athematicseethalakshmi Ramaswami College Tiruchirappalli, Tamil Nadu, Idia suma_shiasrc@yahoocom Receied 0 Noember 07; accepted December 07 Abstract A topological represetatio of a molecule is called molecular graph A molecular graph is a collectio of poits represetig the atoms i the molecule ad set of lies represetig the coalet bods These poits are amed ertices ad the lies are amed edges i molecular graph theory I this paper, the expressio of the ew R idex of path graph, star graph, wheel graph, gear graph, helm graph are deried Keywords: Degree of ertex, Neighbourhood, Topological idices A athematics ubject Classificatio 00): 05CXX, 94C5 Itroductio A topological idex is a umerical iariat that characterize the chemical properties of a molecule The Wieer idex WG) is a distace-based topological iariat much used i the study of the structure-property ad the structure-actiity relatioships of arious classes of biochemically iterestig compouds, which is itroduced by Harold Wieer i 947 for predictig boilig poits b p of alkaes based o the formula b p α W + βw) + γ,where α, β, γ are empirical costats, ad w) is called path umber It is defied as the half sum of the distaces betwee all pairs of ertices of G W G) d u, ), where du,) is the umber of edges i a shortest path that u, G coectig the ertices u & i G[,0] As of ow, iumerable olecular descriptors are beig proposed Recetly, degree based topological idices are also made a good correlatio with chemical properties of a molecule ome well-kow degree based topological idices are Radic idex, First ad ecod Zagreb idices, Reformulated first ad secod Zagreb idices, Atom-Bod Coectiity idex, Augmeted Zagreb idex, Harmoic idex, Geometric-arithmetic idexumcoectiity idex, etc[,,4-0,,4-7,9,0] The comparatie testig of these wellkow degrees based topological idices were gie i [] The cocept of R degree of a ertex ad R idex of a graph were itroduced by iileyma Ediz ad computed the R degree of a ertex ad R idex of some well-kow graphs i [8] Throughout this paper oly simple coected graphs was cosidered, ie coected graphs without self-loops ad parallel edges 6

2 Defiitios For a graph G, VG) ad EG) deote the set of all ertices ad edges respectiely The degree of the ertex is defied as the umber of edges icidet with ad deoted by d) The set of all ertices which are adjacet to is called the eighborhood of ad deg u ad deoted by N) For a ertex, the sum degree of is defied as ) u N ) for a ertex, the multiplicatio degree of is defied as deg u) degree of a ertex of a simple coected graph G is defied as u N ) first R idex of a simple coected graph G defied as R G) r ) ) G The R r ) + The The ecod R idex of a simple coected graph G defied as R G) r u) r ) u E idex of a simple coected graph G defied as R G) [ r u) + r ) ] u E Our otatio is stadard ad maily take from stadard books of graph theory [] R idex of some graphs Theorem Let P be the path graph with ertices ) the R P ) R P ) R P ) 6 4 P ) P be the Path graph with ertices ad - edges ie P ) V ad The Third R E For the Pedet ertices ad ad for the ertices ad For the rest of the, iteral ertices 4 Hece, r ) r ) 5, r ) r ) 8 After implificatio, R P ) ) ) 5 + 8) + 6 5) R P ) R P ) Theorem Let + 4 ) ) ) ) R ) R ) R ) be the tar graph with ertices ) the 64

3 ) E be the tar graph with ertices ad - edges ie ) For the Pedet ertices,, the cetral ertex ad Hece r ) r ), r ) After implificatio, R ) + 4 ) + ) ) R ) ) ) ) R ) ) + ) ) ) Theorem Let ) ) ) W be the Wheel graph with ertices 4) ) 0 4) + ) + ) ) 0 4) 7 + ) ) 5 + ) the V ad ad for W be the Wheel graph with ertices ad - edges is obtaied by coectig a sigle ertex to a ertices of a cycle of legth - ie V W ) ad W ) E For the ertices,, 5, + for the cetral ertex, ) ad 9 ) Hece r ) r ) ), r ) ) + After implificatio, ) ) ) Theorem 4 Let R H R H R H ) ) ) Wheel + ) ) 0 4) 7 + ) ) 5 + ) ) 0 4) + ) H be the Helm graph with - ertices 4 the + 4 ) + 64 ) ) 7 8) ) 4 ) ) ) 7 8) + 4 ) H be the Helm graph with - ertices ad - edges is obtaied from the W by addig a pedet edge at each ertex o the rim of ie V H ) ad H ) E W ad 65

4 For the ertices,,, i C, 8, + 6 ) ad for the cetral ertex, 4 ) ad 4 For the Pedet ertices 4 ad 4 Hece ) r ) 7 8 After implificatio, R H r, ) R H R H ) ) ) + 4 ) + 64 ) ) 7 8) ) 4 ) ) ) 7 8) + 4 ) r ) 4 + 4, r ) r ) 8 Theorem 5 Let G be the Helm graph with + ertices 4 the R G R G ) ) R G ) + ) + 5 ) 5 ) + + 9) ) ) ) 5 ) + ) + G be the Gear graph with + ertices ad edges is obtaied by isertig a ertex betwee each pair of adjacet ertices o the rim of a Wheel ie V G ) + ad E G ) W For the ertices,,, i C,, + 4 ) ad for the cetral ertex, ) ad For the isertig ertices 6 ad 9 Hece ) r ) 5 After implificatio, R G r, ) R G R G ) ) ) + ) + 5 ) ) 5 ) + + 9) ) ) ) 5 ) + ) r ) +, r ) r ) Coclusio I this paper, the expressio of the ew R idex of Path graphtar Graph, Wheel graph, Gear graph, Helm graph are deried REFERENCE BZhou ad NTriajstić, O a oel coectiity idex, J ath Chem, ) 5-70 BDurgiaekkalikeb, HRamae, O the Zagreb idices of semi total poit graphs of some graphs haded, Aals of Pure ad Applied athematics, ) 06)

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