Iteratioal Research Joural of Pure Algebra-6(7, 06, 34-347 Aailable olie through wwwrjpaifo ISSN 48 9037 MULTIPLICATIVE HYPER-ZAGREB INDICES AND COINDICES OF GRAPHS: COMPUTING THESE INDICES OF SOME NANOSTRUCTURES V R KULLI Departet of Matheatics, Gulbarga Uiersity, Gulbarga 58506, Idia (Receied O: -06-6; Reised & Accepted O: 5-07-6 ABSTRACT I this paper, we itroduce the first ad secod ultiplicatie hyper-zagreb idices of a graph The first ultiplicatie hyper-zagreb idex is defied as the product of squares of the su of the degrees of pairs of adjacet ertices The secod ultiplicatie hyper- Zagreb idex is defied as the product of squares of the product of the degrees of pairs of adjacet ertices Also we itroduce the first ad secod ultiplicatie hyper-zagreb coidices of a graph I this paper, the first ad secod ultiplicatie hyper-zagreb idices of cycles, coplete graphs, coplete bipartite graphs ad r-regular graphs are deteried Also we copute exact forulas of the ultiplicatie hyper-zagreb idices for G = TUSC 4 C 8 (S aotubes, G = VPHX [, ] aotues ad G = VPHY [, ] aotorus Keywords: Molecular graph, ultiplicatie hyper-zagreb idex, ultiplicatie hyper-zagreb coidex, aotubes, aotorus Matheatics Subject Classificatio: 05C05, 05C07 INTRODUCTION All graphs cosidered i this paper are fiite, coected, udirected without loops ad ultiple edges Ay udefied ter here ay be foud i Kulli [] Let G = (V(G, E(G be a graph with = V(G ertices ad = E(G edges The degree d G ( of a ertex is the uber of ertices adjacet to A olecular graph is a graph such that its ertices correspod to the atos ad the edges to the bods Cheical graph theory is a brach of atheatical cheistry which has a iportat effect o the deelopet of the cheical scieces I Cheical Sciece, the physico-cheical properties of cheical copouds are ofte odeled by eas of olecular graph based structure descriptors, which are referred to as topological idices The first ad secod ultiplicatie Zagreb idices of a graph G are defied as II ( G = dg ( u ad II ( G = dg ( u dg u V( G These two graph iariats are proposed by Todeshie et al i [] I [3], Eliasi, et al cosidered a ew ultiplicatie ersio of the first Zagreb idex as * II ( G = d G ( u + dg Recetly ay other ultiplicatie idices ad coidices of graphs were studied, for exaple, i [4, 5, 6, 7, 8, 9, 0,, ] I this paper, we iitiate a study of the ultiplicatie hyper-zagreb idices of graphs Correspodig Author: V R Kulli Departet of Matheatics, Gulbarga Uiersity, Gulbarga 58506, Idia Iteratioal Research Joural of Pure Algebra-Vol-6(7, July 06 34
V R Kulli / Multiplicatie Hyper-Zagreb Idices ad Coidices of Graphs: Coputig These Idices of / IRJPA- 6(7, July-06 FIRST MULTIPLICATIVE HYPER-ZAGREB INDEX We defie the first ultiplicatie hyper-zagreb idex of a graph Defiitio : The first ultiplicatie hyper-zagreb idex of a graph G is defied as HII ( G = d G ( u + dg e= Propositio : Let C be a cycle with 3 ertices The HII (C = 4 Proof: Let C be a cycle with 3 ertices Cosider HII ( C = dc ( u + d ( ( 4 C = + = u E( C Propositio 3: Let K be a coplete graph with ertices The HII ( K = ( ( Proof: Let K be a coplete graph with ertices The K has ( K ( ( K u E( K ( {( ( } HII K = d u + d = + = [( ] ( 06, RJPA All Rights Resered 343 ( edges Cosider Propositio 4: Let K, be a coplete bipartite graph with The HII (K, = (+ Proof: Let K, be a coplete bipartite graph with, + ertices ad edges Cosider HII ( K, = dk ( u d ( (, K +, = + u E( K, = ( + Corollary 5: Let K, be a star The HII (K, = (+ Theore A [, p3] Let G be a r-regular graph with ertices The G has Theore 6: Let G be a r-regular graph with ertices The HII (G = (r r Proof: Let G be a r-regular graph with ertices By Theore A, G has ( G ( G ( r ( HII G = d u + d = r + r = (r r 3 SECOND MULTIPLICATIVE HYPER-ZAGREB INDEX We defie the secod ultiplicatie hyper-zagreb idex of a graph r edges r edges Cosider Defiitio 7: The secod ultiplicatie hyper-zagreb idex of a graph G is defied as HII ( G = d G ( u dg e= Propositio 8: Let C be a cycle with 3 ertices The HII (C = 4 Proof: Let C be a cycle with 3 ertices Cosider HII ( C = dc ( u d ( ( 4 C = = u E( C Propositio 9: Let K be a coplete graph with ertices The HII (K = ( (
V R Kulli / Multiplicatie Hyper-Zagreb Idices ad Coidices of Graphs: Coputig These Idices of / IRJPA- 6(7, July-06 ( Proof: Let K be a coplete graph with ertices ad edges Cosider ( HII ( K = dk ( u d ( {( ( } K = u E( K = ( ( Propositio 0: Let K, be a coplete bipartite graph with The HII (K, = ( Proof: Let K, be a coplete bipartite graph with + ertices ad edges Cosider ( K ( (, K, ( HII K, = d u d = u E( K, = ( Corollary : Let K, be a star The KII (K, = Theore : Let G be a r-regular graph with ertices The HII (G = r r Proof: Let G be a r-regular graph with ertices By Theore A, G has r edges Cosider ( G ( G ( r ( HII G = d u d = r r = r r 4 FIRST AND SECOND MULTIPLICATIVE HYPER-ZAGREB COINDICES We defie the first ad secod ultiplicatie hyper-zagreb coidices of a graph Defiitio: The first ad secod ultiplicatie hyper-zagreb coidices of a graph G are defied as HII ( G = d G ( u + dg ( u E( G HII G dg u E( G u dg ( ( ( = 5 MULTIPLICATIVE HYPER ZAGREB INDICES OF TUSC 4 C 8 (S NANOTUBES Molecular graph TUSC 4 C 8 (S aotubes is a faily of aostructures that its structure cosists of cycles C 4 ad C 8 TUSC 4 C 8 (S aotubes is deoted by G = TUC 4 C 8 [, ] We copute the first ad secod ultiplicatie hyper-zagreb idices of G = TUC 4 C 8 [, ] aotubes Theore 3: Let G =TUC 4 C 8 [, ] be the TUSC 4 C 8 (S aotubes The HII (G = 4 4 5 8 4 4 6 HII (G = 4 4 5 8 6 4 4 Proof: Cosider G = TUC 4 C 8 [, ] (, {} aotubes We deote the uber of octagos C 8 i the first row of G by ad the uber of octagos C 8 i the first colu of G by I geeral case of the aotubes, there are 8 + 4 ertices/atos ad + 4 edges/bods, see Figure 3 - Figure- 06, RJPA All Rights Resered 344
V R Kulli / Multiplicatie Hyper-Zagreb Idices ad Coidices of Graphs: Coputig These Idices of / IRJPA- 6(7, July-06 We hae two partitios of the ertex set V(G as follows: V = { V(G/d G ( = }, V = + V 3 = { V(G/d G ( = 3}, V 3 = 8 By Figure, we hae three partitios of the edge set E(G as follows: * E = E = u E G / d u = d =, E = Now Now { ( G ( G ( } { ( G ( G ( } { ( G ( G ( } 4 4 4 E = E = u E G / d u =, d = 3, E = E = 4 5 6 5 6 E = E = u E G / d u = d = 3, E = E = 6 9 6 9 ( = ( + ( HII G d u d G G e= dg ( u dg dg ( u dg dg ( u dg = + + + u E4 u E5 u E6 4 ( 4 ( 5 ( 6 4 8 4 4 4 5 6 ( ( ( HII G = d u d G G e= dg ( u dg dg ( u dg dg ( u dg = * u E4 u E6 u E9 4 ( 4 ( 6 ( 9 4 8 4 4 4 6 9 6 MULTIPLICATIVE HYPER-ZAGREB INDICES OF V-PHENLENIC NANOTUES AND NANOTORUS Cheical Structures V-Pheyleic aotubes ad V-Pheyleic aotorus are widely used i Medical Sciece ad Pharaceutical field Thus we study ultiplicatie hyper-zagreb idices of these olecular structures fro a atheatical poit of iew I this sectio, we cosider the structures of V-Pheyleic aotubes VPHX[, ] ad V-Pheyleic aotorus VPHY[, ] (, {} ad copute their ultiplicatie hyper-zagreb idices Molecular graphs V-Pheyleic aotubes ad V-Pheyleic aotorus are two failies of aostructures that their structures cosist of cycles C 4, C 6 ad C 8 by differet copouds We deterie the first ad secod ultiplicatie hyper-zagreb idices of G = VPHX [, ] aotubes Theore 4: Let G = VPHX[, ] (, {} be the V-Pheyleic aotubes The HII (G = 5 8 8 0 6 HII (G = 6 8 9 8 0 Proof: Cosider G =VPHX[, ] (, {} aotubes We deote the uber of hexagos i the first row of G by ad the uber of hexagos i the first colu of G by I geeral case of this aotubes, there are 6 ertices/atos ad 9 edges/bods, see Figure Figure- 06, RJPA All Rights Resered 345
V R Kulli / Multiplicatie Hyper-Zagreb Idices ad Coidices of Graphs: Coputig These Idices of / IRJPA- 6(7, July-06 Fro the structure of G, we hae two partitios of the ertex set V(G as follows: V = V ( G : dg =, V = { } { ( G ( } V3 = V G : d = 3, V 3 = 6 Also fro the structure of G, we hae two partitios of the edge set E(G as follows: E = E * = e = u E G : d u =, d =, E = E * = 4 Now { ( G ( G ( } ( ( ( 5 6 5 6 { G G } E = E = e = u E G : d u = d = 3, E = E = 9 5 6 9 6 9 ( = ( + ( G G e= u E( G = dg ( u + d ( ( ( G dg u + d G u E u E HII G d u d 5 6 4 9 ( 5 ( 6 8 8 0 5 6 ( = ( ( G G e= u E( G HII G d u d dg ( u d ( ( ( G dg u d G = u E6 u E9 4 9 5 ( 6 ( 9 8 8 0 6 9 Now we copute the first ad secod ultiplicatie hyper-zagreb idices of G = VPHY[, ] aotorus Theore 5: Let G = VPHY[, ] (, {} be the V-Pheyleic aotorus The HII (G = 6 8 HII (G = 9 8 Proof: Cosider G = VPHY[, ] (, {} aotorus We deote the uber of hexagos i the first row of G by ad the uber of hexagos i the first colu of G by I geeral case of this aotorus, there are 6 ertices/atos ad 9 edges/bods, see Figure 3 Figure-3 Fro the structure G, there is oly oe partitio of the ertex set V(G as follows: V = V G : d = 3, V = 6 { ( G ( } 3 3 Also fro the structure of G, there is oly oe partitio of the edge set E(G as follows: E = E = u E G : d u = d = 3, E = E = 9 { ( G ( G ( } 6 9 6 9 06, RJPA All Rights Resered 346
V R Kulli / Multiplicatie Hyper-Zagreb Idices ad Coidices of Graphs: Coputig These Idices of / IRJPA- 6(7, July-06 Now HII( G = dg ( u + d ( G REFERENCES e= u E( G ( ( = dg u + d G u E6 6 8 6 ( 9 = = ( = ( ( HII G d u d G G e= u E( G = dg ( u d ( G * u E9 8 ( 9 = 9 = 9 VRKulli, College Graph Theory, Vishwa Iteratioal Publicatios, Gulbarga, Idia (0 RTodeshie ad V Cosoi, New local ertex iariats ad olecular descriptors based o fuctios of ertex degrees, MATCH Cou Math Coput Che 64(00 359-37 3 MEliasi, AIraaesh ad I Guta, Multiplicatie ersios of first Zagreb idex, MATCH Cou Math Coput Che 68(0 7-30 4 KCDas, A Yurttas, M Toga, AS Ceik ad N Cagul, The ultiplicatie Zagreb idices of graph operatios, Joural of Iequalities ad Applicatios, 03, 03:90, -4 5 VR Kulli, First ultiplicatie K Bahatti idex ad coidex of graphs, Aals of Pure ad Applied Matheatics, ( (06 79-8 6 VR Kulli, Secod ultiplicatie K Bahatti idex ad coidex of graphs, Joural of Coputer ad Matheatical Scieces, 7(5, (06 54-58 7 VR Kulli, Multiplicatie K hyper-bahatti idices ad coidices of graphs, Iteratioal Joural of Matheatical Archie, 7(6, (06 60-65 8 VR Kulli, o ultiplicatie K Bahatti idices ad ultiplicatie K hyper-bahatti idices of V-Pheyleic aotubes ad aotorus, Aals of Pure ad Applied Matheatics, (, (06, 45-50 9 J Liu ad Q Zhag, Sharp upper bouds for ultiplicatie Zagreb idices, MATCH Cou Math Coput Che 68(0 3-40 0 H Wag ad H Bao, A ote o ultiplicatie su Zagreb idex, South Asia J Math (6, (0 578-583 S Wag ad B Wei, Multiplicatie Zagreb idices of k-tree, Discrete Applied Math 80, (05 68-75 K Xu, KC Das ad K Tag, O the ultiplicatie Zagreb coidex of graphs, Opuscula Math 33(, 9-04 (03 Source of Support: Nil, Coflict of iterest: Noe Declared [Copy right 06, RJPA All Rights Resered This is a Ope Access article distributed uder the ters of the Iteratioal Research Joural of Pure Algebra (IRJPA, which perits urestricted use, distributio, ad reproductio i ay ediu, proided the origial work is properly cited] 06, RJPA All Rights Resered 347