Fourth Zagreb index of Circumcoronene series of Benzenoid

Transcription

1 Leonardo Electronc Journal of Practces and Technologes Issue 27 July-December 2015 Fourth Zagreb ndex of Crcumcoronene seres of Benzenod Mohammad Reza FARAHANI 1* and Rajesh M. R. KANNA 2 1 Department of Appled Mathematcs Iran Unversty of Scence and Technology (IUST) Narma Tehran Iran 2 Department of Mathematcs Maharan's Scence College for Women Mysore Inda E-mals: * mr_farahan@mathdep.ust.ac.r mrfarahan88@gmal.com Mr.rajeshanna@gmal.com * Correspondng author phone: Abstract A topologcal ndex of a graph s a numerc quantty related to a structure of a molecule whch s nvarant under graph automorphsm. Recently Ghorban and Hossenzadeh ntroduced Fourth Zagreb ndex of graphs. In ths paper we determne a closed formula of ths new topologcal ndex of the famous Benzenod famly named Crcumcoronene seres of Benzenod H. Keywords Molecular graph; Benzenod graph; Crcumcoronene seres of benzenod; Topologcal ndex; Eccentrcty ndex; Fourth Zagreb ndex Introducton A molecular graph s a smple graph such that ts vertces correspond to the atoms and the edges to the bonds. All graphs n ths paper are fnte and smple. For terms and concepts not defned here we refer the reader to any of several standard monographs such as e.g. [1-5]. Mathematcal chemstry s a branch of theoretcal chemstry for dscusson and predcton of the molecular structure usng mathematcal methods wthout necessarly referrng to quantum mechancs. Chemcal graph theory s a branch of mathematcal chemstry whch apples graph theory to mathematcal modellng of chemcal phenomena [1-5]. Ths theory had an mportant effect on the development of the chemcal scences. A graph nvarant s any functon on a graph that does not depend on a labellng of ts 155

2 Fourth Zagreb ndex of Crcumcoronene seres of Benzenod Mohammad R. FARAHANI Rajesh M.R. KANNA vertces. Such quanttes are also called topologcal ndces. One of the best nown and wdely used s the Zagreb topologcal ndex ntroduced n 1972 by Gutman and Trnajstć and s defned as the sum of the squares of the degrees of all vertces of G [2 5]. Let G be a connected graph wth vertex and edge sets V(G) and E(G) respectvely. For every vertex u V(G) the edge connectng u and v s denoted by uv and d G (u) (or d u ) denotes the degree of u n G (the number of frst neghbors of vertex u n G). The Zagreb ndces were orgnally defned as follows: M 1 (G)= v V(G) (d v ) 2 M 2 (G)= e=uv E(G) (d u d v ) here M 1 (G) and M 2 (G) denote the frst and the second Zagreb ndex respectvely. The frst Zagreb ndex can be also expressed as a sum over edges of G M 1 (G)= e=uv E(G) (d u +d v ) The readers nterested n more nformaton and some mathematcal propertes of Zagreb ndces for general graphs can be referred to [5-11]. as follows: In 2012 Ghorban and Hossenzadeh ntroduced a new verson of Zagreb ndex [12] M** 1 (G)= v V(G) ε(v) 2 In whch for a vertex v of V(G) ts eccentrcty ε(v) s the largest dstance between v and any other vertex u of G. ε(v)=max{d(uv) u V(G)} The maxmum eccentrcty over all vertces of G s called the dameter of G and denoted by D(G). Also the mnmum eccentrcty among vertces of G s called the radus and denoted by r(g). In other words D(G)=Max v V(G) {d(uv) u V(G)} R(G)=Mn v V(G) {Max{d(uv) u V(G)}} The eccentrc connectvty ndex ζ(g) of a graph G s defned as: ζ(g)= v V(G) d v ε(v) For further study and more detals see references [13-17]. The am of ths paper s to nvestgate a closed formula of new topologcal Zagreb ndex for Crcumcoronene seres of Benzenod H ( 1). In ths paper we call ths new Zagreb ndex by Fourth Zagreb ndex and denote by M 4 (G). 156

3 Leonardo Electronc Journal of Practces and Technologes Issue 27 July-December 2015 Materal and method In ths secton t s presented how to compute ths new topologcal ndex for Crcumcoronene seres of Benzenod H ( 1). The Crcumcoronene seres s part of Benzenod molecular graph famly (consst several copy of benzene C 6 ). The general representaton and some frst members of ths Benzenod graph famly are shown n Fgures 1 and 2 (see [14-20]). Fgure 1. Some frst members of Crcumcoronene seres of Benzenod H ( 1) [14-17] In ths artcle we use Cut method and Rng-cut method. Further studes on Cut Method and ts modfed verson Rng-cut method can be seen n [19 20]. Rng-cut method dvdes all vertces of G nto some parttons wth smlar mathematcal and topologcal propertes. For example readers can see rng-cuts of Crcumcoronene seres of Benzenod n Fgure 2. Fgure 2. The Rng-cuts for the general representaton of Crcumcoronene seres of Benzenod H ( 1) [14-17] 157

4 Fourth Zagreb ndex of Crcumcoronene seres of Benzenod Mohammad R. FARAHANI Rajesh M.R. KANNA Thus from the general representaton of Crcumcoronene seres of Benzenod H and ts rng-cuts n Fgure 2 one can see that [14-20]: I. The vertex/atom set of H s V(H )={ γ β Z & j Z & z Z 6 }. z j z l II. The edge/bond set of H s 1 E(H )={ β γ β γ + β γ and γ γ + 1 Z & j Z & z Z 6 }. z j z j z j z j 1 z j z j z z 1 III. The number of vertces s n = V(H ) =6 IV. The number of edges s e = E(H ) = = 1 =62. = = 1 + = 1 +6=92-3. = 1 V. =1..; j Z -1 & z Z 6 : ε( βz j) = d( βz j βz+ 3 j) + d( βz+ 3 j γz+ 3 j) =2(++1) ( ) + 1 VI. =1..; j Z & z Z 6 : εγ ( z j) = d( γ γ + ) + d( γ + γ + ) zj z 3j =2(+)-1. z 3j z 3j 4 1 2( ) VII. The radus number of H s equal to r(h )=2+1. VIII. The dameter number of H s equal to D(H )=4-1. IX. ζ(h )= where all vertces β γ are from I th rng cut R of H. such that =1 and j Z & z Z 6 z j z j (Z n s the cycle fnte group of order n). Results Now we can exhbt the closed formula of Fourth Zagreb ndex M 4 (H ) n followng theorem and ts proof. Theorem 1: Let G be the Crcumcoronene seres of Benzenod H ( 1). Then Fourth Zagreb ndex of H s equal to: M** 1 (H)=M 4 (H)= (1) Proof: Consder Crcumcoronene seres of Benzenod G=H ( 1) as shown n Fgure 2. Thus we refer to defnton of Fourth Zagreb ndex M 4 (H ) and compute ths ndex for G=H n general case ( Z) as follows: M** 1 (H )= v V(H) ε(v) 2 (2) 158

5 Leonardo Electronc Journal of Practces and Technologes M 4 (H )= 2 2 ε( γ ) z j + ε( βz j) V ( H ) β V ( H ) γz j z j Issue 27 July-December 2015 (3) = = [ ε( γ z j )] + [ ε ( β z j )] = 1 j= 1 z = 1 = 2 j= 1 z = [ ε( γz j)] + [ ε( βz j)] z = 1 = 1 j= 1 z = 1 = 2 j= 1 =6 (2+2-1) 2+ 6 (-1)(2+2-2) 2 = 1 = 2 =6 ( )+6 (-1)( ) = 1 = (16 16) 6 ( ) 6 ( 4 8 4) = = 1 = 1 = 1 = 1 2 ( 1) ( + 1)(2+ 1) ( 1) = ( 1) ( 1) ( ) ( ) =( )+( )+( ) -( ) Fnally Z the Fourth Zagreb ndex s equal to: M 4 (H )= (4) Obvously M 4 (H 1 =C 6 )=6(3) 2 =54 and the proof s complete. Concluson In ths paper we have computed closed formula of Fourth Zagreb ndex for famous Benzenod molecular graph "Crcumcoronene seres of Benzenod H ( 1)". References 1. Gutman I. Trnajstć N. Graph theory and molecular orbtals Total π electron energy of alternant hydrocarbons Chem Phys Lett p Randć M. On characterzaton of molecular branchng J. Am. Chem. Soc p

6 Fourth Zagreb ndex of Crcumcoronene seres of Benzenod Mohammad R. FARAHANI Rajesh M.R. KANNA 3. West D. B. Introducton to Graph Theory Upper Saddle Rver Prentce Hall Todeschn R. Consonn V. Handboo of Molecular Descrptors. Wley Wenhem Gutman I. Das K. C. The frst Zagreb ndex 30 years after MATCH Commun Math Comput Chem p Farahan M. R. Computng Zagreb ndex Zagreb Polynomal and some Connectvty ndex of Crcumcoronene seres Advances n Materals and Corroson p Nolc S. Kovacevc G. Mlcevc A. Trnajstć N. The Zagreb ndces 30 years after Croat Chem Acta p Braun J. Kerber A. Mernger M. Rucer C. Smlarty of molecular descrptors: the equvalence of Zagreb ndces and wal counts MATCH Commun Math Comput Chem p Khalfeh M. H. Yousef-Azar H. Ashraf A. R. The frst and second Zagreb ndces of graph operatons Dscr Appl Math p Zhou B. Gutman I. Relatons between Wener hyper Wener and Zagreb ndces Chem Phys Lett p Farahan M. R. Zagreb ndex Zagreb Polynomal of Crcumcoronene Seres of Benzenod Advances n Materals and Corroson p Ghorban M. Hossenzadeh M. A. A New Verson of Zagreb Indces Flomat (1) p Ashraf A. R. Ghorban M. Hemmas M. Eccentrc connectvty polynomal of an nfnte famly of Dendrmer Dgest Journal of Nanomaterals and Bostructures (3) p Alhan S. Iranmanesh M. A. Eccentrc connectvty polynomal of an nfnte famly of Dendrmer Dgest Journal of Nanomaterals and Bostructures (1) p Farahan M. R. Computng Eccentrcty Connectvty Polynomal of Crcumcoronene Seres of Benzenod H by Rng-Cut Method. Annals of West Unversty of Tmsoara- Mathematcs and Computer Scence (2) p Farahan M. R. Modfed Eccentrc Connectvty polynomal of Crcumcoronene Seres of Benzenod H Journal of Advances n Physcs (1) p Farahan M. R. Eccentrcty Verson of Atom-Bond Connectvty Index of Benzenod Famly ABC 5 (H ) World Appled Scences Journal (9) p

7 Leonardo Electronc Journal of Practces and Technologes Issue 27 July-December John P. E. Khadar P. V. Sngh J. A method of computng the PI ndex of Benzenod hydrocarbons usng orthogonal cuts J Math Chem (1) p Klavžar S. Gutman I. Mohar B. Labelng of Benzenod Systems whch Reflects the Vertex-Dstance Relatons J Chem Int Comput Sc p Klavžar S. A Brd s Eye Vew of The Cut Method and a Survey of Its Applcatons n Chemcal Graph Theory MATCH Commun Math Comput Chem p