On Topological Indices of Circumcoronene Series of Benzenoid

Transcription

1 Chemcal Methodologes 2(2018) Chemcal Methodologes journal homepage: Orgnal Research artcle On Topologcal Indces of Crcumcoronene Seres of Benzenod Yngyng Gao a, Mohammad Reza Farahan b *, Waqas Nazeer c a College of Pharmacy and Bologcal Engneerng, Chengdu Unversty, Chengdu, , Chna. b Department of Appled Mathematcs of Iran Unversty of Scence and Technology (IUST), Narma, Tehran 16844, Iran. c Dvson of Scence and Technology, Unversty of Educaton, Lahore 54000, Pastan A R T I C L E I N F O R M A T I O N Receved: 08 October 2017 Receved n revsed: 15 December 2017 Accepted: 01 January 2018 Avalable onlne: 05 January 2018 DOI: /chemm KEYWORDS Molecular Graph Crcumcoronene seres of Benzenod Topologcal ndex Multplcatve Zagreb Eccentrcty ndex A B S T R A C T Let G be a connected graph wth vertex and edge sets V (G) and E(G), respectvely. The frst Zagreb ndex M 1(G) was orgnally defned as the sum of the squares of the degrees of all vertces of G. Recently, we now a new verson of the frst Zagreb ndex as the Multplcatve Zagreb Eccentrcty ndex that ntroduced by Nlanjan De and ε(u) s the largest dstance between u and any other vertex v of G. In ths paper we compute ths new topologcal ndex of famous molecular graph Crcumcoronene Seres of Benzenod H. *Correspondng author: Emal: Mr_Farahan@Mathdep.ust.ac.r Department of Appled Mathematcs of Iran Unversty of Scence and Technology (IUST), Narma, Tehran 16844, Iran.

2 On Topologcal Indces of... P a g e 40 Graphcal Abstract Introducton Let G be a connected graph wth vertex and edge sets V (G) and E(G) and order n and sze m, respectvely. For every vertex uv (G), the edge connectng u and v s denoted by uv and the degree of any vertex s the number of frst neghbour of v and s denoted by d G(u) (or d u). Let the maxmum and mnmum degree of all the vertces of G are respectvely denoted by Δ and δ. The dstance of any two vertces u and v of s defned as the length of the shortest path connectng u and v and s denoted by d(u,v). 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 vv(g){d(u,v) uv(g)} (1) R(G)=Mn vv(g){max{d(u,v) uv(g)}} (2) 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 the chemcal phenomena [1-3]. Ths theory had an mportant effect on the development of the chemcal scences. A topologcal ndex of a graph s a number related to a graph whch s nvarant under graph automorphsms and s a numerc quantty from the structural graph of a molecule. One of the best nown and wdely used s the Zagreb topologcal ndex ntroduced n 1972 by I. Gutman and N. Trnajstć and s defned as the sum of the squares of the degrees of all vertces of G [4, 5]. The frst Zagreb ndex M 1(G) was orgnally defned as follows: M 1(G)= v V G (d v) 2 = e uv E G (d u+d v). (3)

3 Mohammad Reza Farahan et al. P a g e 41 where d v denotes the degree of vertex v n G. The multplcatve verson of ths frst Zagreb ndex was ntroduced by Todeschn et. al. [6, 7] and s defned as: ΠM 1(G)= uv E G d v 2 (4) The eccentrc verson of the frst Zagreb ndex was ntroduced by M. Ghorban et al. [8] and D. Vucevc et al. [9] n 2012 as follows: M** 1(G)= v V G ε(v) 2 (5) The eccentrcty of a vertex s the dstance between v and a vertex farthest from v and s denoted by ε(v). In other words, ε(v)=max{d(u,v) uv(g)}. The Eccentrc Connectvty ndex ζ(g) of a graph G s defned as [10]: v v G ζ(g) = d v ε(v), (6) where ε(u) s the eccentrcty of vertex u. Ths ndex ntroduced by Sharma et al. n 1997 [10-15]. Recently n 2012, Nlanjan De ntroduced a new verson of Frst Zagreb ndex [16] as the Multplcatve Zagreb Eccentrcty ndex and defned as follows: ΠE 1(G)= uv E G ε(v) 2, (7) where ε(u) s the largest dstance between u and any other vertex v of G. The readers nterested n more nformaton and some mathematcal propertes of Zagreb ndces for general graphs can be referred to [18-30]. The am of ths paper s to nvestgate a closed formula of ths new topologcal ndex Multplcatve Zagreb Eccentrcty ndex for famous Benzenod molecular graph Crcumcoronene seres of Benzenod H ( 1). Results and Dscusson The Crcumcoronene seres of Benzenod s a famous famly of Benzenod molecular graph, whch ths famly bult solely from benzene C 6 (or hexagons) on crcumference. For more detal of Benzenod molecular graphs, see the paper seres [29-38]. The general representatons and frst members of ths famly are shown n Fgures 1 and 2.

4 On Topologcal Indces of... P a g e 42 Fgure 1. Some frst members of Crcumcoronene Seres of Benzenod H ( 1) [29-33]. Theorem 1. [30] Let H be the Crcumcoronene Seres of Benzenod, 1. Then the Frst Zagreb ndex of H s equal to M 1(H )= Theorem 2. [15] Consder the Crcumcoronene seres of Benzenod H, 1. Then the Eccentrc Connectvty ndex of H s equal to ζ(h )= Theorem 3. [29]The frst eccentrc Zagreb ndex of the Crcumcoronene seres of Benzenod H, 1 s equal to M** 1(H )= Theorem 4. Let G be the Crcumcoronene seres of Benzenod H ( 1). Then the Multplcatve Zagreb Eccentrcty ndex of H s equal to: (8) ΠE 1 (H )= 1 Proof. Consder the Crcumcoronene seres of Benzenod H ( 1) as shown n Fgure2. And let the vertex/atom and edge/bond sets of H, 1 are equal to: V(H )={γ z,j,β z,j Z & jz & zz 6} and E(H )={ γ z,jβ z,j, γ z,j+1β z,j, γ -1 z,jβ z,j and γ z,jγ z,j+1 Z & jz & zz 6}. And the sze of vertex and edge sets of H are equal to and m = E(H ) =6 n = V(H ) = = = (9)

5 Mohammad Reza Farahan et al. P a g e 43 Fgure 2. The Crcumcoronene seres of Benzenod H ( 1) [29-33]. To compute the Multplcatve Zagreb Eccentrcty ndex of H, we used the Cut Method and the Rng-cut Method. The readers can consult [15, 29-33, 37, 38] for more nformaton about the Cut Method and t modfy verson Rng-cut Method. The Rng-cut Method dvdes all vertces of a graph G nto some parttons wth smlar mathematcal and topologcal propertes. We encourage readers see the rng-cuts of Crcumcoronene seres of Benzenod n Fgure 3, such that =1,, and jz & z, j,, zz 6 ; all vertces z j are from I th rng cut R of H. Now, from the general representaton of Crcumcoronene seres of Benzenod H n Fgure 2 and Fgure 3 and, we can see that =1..,; jz -1 & zz 6: ( ) d(, ) d(, ) z, j z, j z3, j z3, j z3, j ( ) 1 =2(++1) (10) ( z, j) d( z, j, z, j) d( z3, j, z3, j) =1..,; jz & zz 6: 4 1 2( ) =2(+)-1 (11) By above mansons, t s easy to see that the radus and dameter numbers of H are equal to R(H )=2+1 and D(H )=4-1, respectvely. Now by usng above menton results, we can compute the

6 On Topologcal Indces of... P a g e 44 Multplcatve Zagreb Eccentrcty ndex of Crcumcoronene seres of Benzenod H, Z as follows: = v V H ΠE 1(H )= ε(v) z, j z, j z, j z, j V H V H z, j z, j = j1 j1 z 1 1 z z, j, j z j 6 6 j = (12) Fgure 2. The Rng-cuts of Crcumcoronene seres of Benzenod H ( 1) [29-33]. Fnally, the Multplcatve Zagreb Eccentrcty ndex of H Z s equal to

7 Mohammad Reza Farahan et al. P a g e 45 ΠE 1(H )= (13) Concluson For a connected graph G, the Multplcatve Zagreb Eccentrcty ndex s equal to ΠE 1(G)= v V G ε(v) 2, where ε(v) denotes the eccentrcty of a vertex s the dstance between v and a vertex farthest from v. In ths paper, we obtaned a closed formula of ths new ndex the (Multplcatve Zagreb Eccentrcty ndex) of famous Benzenod molecular graph for the frst tme that we called Crcumcoronene seres of benzenod H ( 1). References [1] West D.B., Introducton to Graph Theory, Prentce Hall, Upper Saddle Rver, [2] Todeschn R., Consonn V. Handboo of Molecular Descrptors. Wley, Wenhem [3] Balaban A.T. (Ed.), From Chemcal Topology to Three-dmensonal Geometry, Plenum, New Yor, [4] Gutman I., Trnajstć N. Chem. Phys. Lett., 1972, 17:535 [5] Gutman I., Das K.C. MATCH Commun. Math. Comput. Chem., 2004, 50:83 [6] Todeschn R., Consonn V. MATCH Commun. Math. Comput. Chem., 2010, 64:359 [7] Todeschn R., Ballabo D., Consonn V., Novel molecular descrptors based on functons of the vertex degrees, n: I.Gutman, B. Furtula (Eds.), Novel Molecular Structure Descrptors - Theory and Applcatons I, Unv. Kragujevac, Kragujevac, 2010, [8] Ghorban M., Hossenzadeh M.A., Flomat, 2012, 26:93 [9] Vucevc D., Graovac A. Acta. Chm. Slov., 2010, 57;524 [10] Sharma V., Goswam R., Madan A.K. J. Chem. Inf. Comput. Sc., 1997, 37:273 [11] Sardana S., Madan A.K. MATCH Commun. Math. Comput. Chem., 2001, 43:85 [12] Gupta S., Sngh M., Madan A.K. J. Math. Anal. Appl., 2002, 266: 259 [13] Furtula B., Graovac A., Vučevć D. Dsc. Appl. Math., 2009, 157:2828 [14] Fschermann M., Homann A., Rautenbach D., Szeely L.A., Volmann L. Dscrete Appl. Math., 2002, 122:127 [15] Farahan M.R. Annals of West Unversty of Tmsoara-Mathematcs and Computer Scence., 2013, 51:29

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