Graphs serve many concrete real-world problems as mathematical models. A graph can be used to represent almost any physical situation involving discrete objects and relationships among them. It is used to model many problems of practical interest in chemical, biological, physical, social system and computer sciences and many branches of mathematics such as numerical analysis, probability, matrix theory, group theory and topology have their association with graph theory.

The graph theory was inborn in 1736 after the solution of famed Konigsberg bridge problem by Eulers. A Graph G is a pair (VG, EG), where VG is a non-empty set whose elements are called vertices of G and EG is a set of 2-element subsets of VG, whose elements are called edges of G. Mathematical chemistry is the branch of theoretical chemistry which deals with prediction and study of chemical structure with the assist of mathematical tools and without discussing quantum mechanics approach [1-2].

Chemical graph theory is the branch of Mathematical Chemistry which put on graph theory to mathematical modeling of chemical structure [3]. Chemical structures of molecules and molecular compound can be modeled by molecular graphs, by considering atoms as vertices and covalent bonds between the atoms as edges [4]. Topological indices are used in the development of Quantitative Structure-Activity and Structure-Property Relationship (QSAR/QSPR) in which the biological or other properties of molecules are correlated with a certain chemical compound [5-6]. Topology of a graph is illustrated by topological index which is a numerical object describes the physico-chemical properties of a molecular graph [7].

Topological index has a great importance in chemical graph theory. If G represents the class of all finite graphs then a topological index is a function R such that for any G, H G, Top(G) = Top (H) if G and H are isomorphic. There are two major types of topological indices, one is degree based and the other is distance based topological index. Distance based topological indices has more importance as compare to degree based topological indices.

Let G= (VG, EG) be a graph then the degree of a vertex is defined as all the number of edges that are incident to a vertex and denoted by deg(v). The eccentricity of avertex in a connected graph G is a largest distance from and defined as (v)=maxvuV(G) {d(v,u)}, where . Distance is defined as the shortest path attain by edges. The diameter is the largest eccentricity by all vertices of graph G and denoted by D (G).

Topological indices were firstly introduced by wiener (wiener index)in 1947 while studying the boiling points of the substance of paraffin [8]. The eccentric connectivity index(ECI), Connectivity eccentric index(CEI), Modified eccentric connectivity index(MECI), general sum connectivity index(GSCI), Zagreb Indices are distance based parameters introduced in recent years and consequently studied [9-13].

Degree based topological index was firstly introduced by Milan Randic in 1975 to reflect molecular branching [14]. Randic index is define as

### X(G) = ?_uv?E(G)?1/?(d_G (u)d_G (v) )

Atom-bond connectivity(ABC) index was introduced in 2009, Estrada et al [15].later on organic chemical compound were modeled by ABC index for thermodynamics properties [16].

ABC index is defined as ABC (G) =?_uv?E(G)??((d_G (u)+d_G (v)-2)/(d_G (u)d_G (v) ))

Recently, the fourth version of the atom-bond connectivity index (ABC4) is proposed by Ghorbani et al [17]. ABC4 index is defined as

ABC_4 (G) = ?_(lm?E(G))??((?_G (l)+ ?_G (m)-2)/(?_G (l) ?_G (m)))

Where ?_G(m) = ?_lm?E(G)?d_G (l) and? ??_(G ) (l)= ?_(lm?E(G))??d_G (m)?.

Vukicevic et al.[18] introduced first Geometric Arithmetic index (GA) of a connected graph in 2009 and defined as

GA (G)=?_(lm?E(G))?(2?(d_G (l)d_G (m)))/(d_G (l)?+d?_G (m))

The fifth version of GA index is GA5 introduced by Graovac et al. in 2011 and defined as

GA¬¬_5 (G) = ?_(lm?E(G))?(2?(?_G (l)?_G (m)))/(?_G ?(l)+??_G (m))

## Bibliography

[1] I.Gutman, O.E.Polansky, Mathematical Concepts in Organic Chemistry,

### Springer-Verlag, Berlin(1986).

[2] N.Trinajstic, I.Gutman, Mathematical Chemistry, Croat. Chem. Acta 75, 329 (2009).

[3] N.Trinajstic, Chemical Graph Theory. CRC Press, Bo ca Raton, FL. (1992).

[4] Danail Bonchev, D.H. Rouvray (eds.) (1991) Chemical Graph Theory: Introduction and Fundamentals ISBN 0-85626-454-7.

[5] L. Pogliani, Modeling Enthalpy and Hydration properties of inorganic compounds, Croatica Chemica Acta, vol. 3, pp. 803-817, 1997.

[6] L. B. Kier and L.H. Hall, An electrotopological state index for atom in molecules, pharmaceutical Research, vol. 7, no. 8, pp. 801-807, 1990.

[7] C. Hansch, L. Leo, exploring QSAR fundamentals and applicability in chemistry and biology, Amer. Chem. Soc., Washington DC, 1996.

[8] Weiner, H.J.: Structural determination of Paraffin boiling points. J. am. Chem. Soc 69, 17-20 (1947)

[9] A.R. Ashrafi, P. Nikzad, K. Austin, Connectivity index of the family of dendrimer nanostars,

Digest J. Nanomater. Biostruct. 4, 2009, 269-273.

[10] A.R. Ashrafi, M. Saheli, M. Ghorbani, The eccentric connectivity index of nanotubes and nanotori, 235. 4561-4566 (2011)

[11] M.M. Zobair, M.A. Malik, H. Shaker, N. Rehman, Eccentricity based topological invariants of triangulane dendrimers, Utilitas Mathematica 107(2018), pp. 193-206

[12]N. Prabhakara Rao, K. Lakshmi, Eccentric connectivity index of V-phenylenic nanotubes, vol. 6, no 1, 2010,p.81-87.

[13]I. Nadeem, H. Shaker, On eccentric connectivity index of TiO2 nanotubes, Acta Chim. Slov.2016,63, 363-368.

[14]M. Randic, J. Am. Chem. Soc. 97, 6609 (1975)

[15] E. Estrada, L. Torres, L. Rodriguez, I. Gutman, Indian J. Chem. 37A, 849 (1998)

[16] I. Gutman, J. Tosovic, S. Radenkovic, S. Markovic, Indian J. Chem. 51A, 690(2012)

[17] M. Ghorbani, M. A. Hosseinzadeh, Optoelectron. Adv. Mater.Rapid Commun.4,1419(2009)

[18] A. Khaksar, M. Ghorbani, H.R. Maimani, Optoelectron. Adv. Mat. 11, 1868 (2010)