International Journal of Computational Mathematics

Volume 2014, Article ID 436140, 8 pages

http://dx.doi.org/10.1155/2014/436140

## Modified Eccentric Connectivity of Generalized Thorn Graphs

^{1}Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata 700 040, India^{2}Department of Mathematics, National Institute of Technology, Durgapur 713 209, India^{3}Department of Mathematics, Aliah University, DN 20, Sector V, Salt Lake, Kolkata 700 091, India

Received 26 June 2014; Accepted 9 December 2014; Published 22 December 2014

Academic Editor: Sheung-Hung Poon

Copyright © 2014 Nilanjan De et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The thorn graph of a given graph is obtained by attaching pendent vertices to each vertex of . The pendent edges, called thorns of , can be treated as or , so that a thorn graph is generalized by replacing by and by and the respective generalizations are denoted by and . The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graph and the generalized thorn graphs and .

#### 1. Introduction

Let be a simple connected graph with vertex set and edge set , so that and . Let the vertices of be labeled as . For any vertex the number of neighbors of is defined as the degree of the vertex and is denoted by . Let denote the set of vertices which are the neighbors of the vertex , so that . Also let , that is, sum of degrees of the neighboring vertices of . The distance between the vertices and is equal to the length of the shortest path connecting and . Also for a given vertex , the eccentricity is the largest distance from to any other vertices of and the sum of eccentricities of all the vertices of is denoted by [1]. The eccentric connectivity index of a graph was proposed by Sharma et al. [2]. A lot of results related to chemical and mathematical study on eccentric connectivity index have taken place in the literature [3–5]. There are numerous modifications of eccentric connectivity index reported in the literature till date. These include edge versions of eccentric connectivity index [6], eccentric connectivity topochemical index [7], augmented eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and connective eccentricity index [10]. A modified version of eccentric connectivity index was proposed by Ashrafi and Ghorbani [11].

Similar to other topological polynomials, the corresponding polynomial, that is, the modified eccentric connectivity polynomial of a graph, is defined as so that the modified eccentric connectivity index is the first derivative of this polynomial for . Several studies on this modified eccentric connectivity index are also found in the literature. In [11], the modified eccentric connectivity polynomials for three infinite classes of fullerenes were computed. In [12], a numerical method for computing modified eccentric connectivity polynomial and modified eccentric connectivity index of one-pentagonal carbon nanocones was presented. In [13], some exact formulas for the modified eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction, and join of graphs were presented. Some upper and lower bounds for this modified eccentric connectivity index are recently obtained by the present authors [14].

The first and the second Zagreb indices of , denoted by and , respectively, are two of the oldest topological indices introduced in [15] by Gutman and Trinajstić and were defined as

Let be a -tuple of nonnegative integers. The thorn graph is the graph obtained from by attaching pendent vertices to the vertex of . In this paper, we assume . These pendent vertices are termed as thorns. The concept of thorn graphs was first introduced by Gutman [16]. A lot of studies on thorn graphs for different topological indices are made by several researchers in the recent past [17–24]. Very recently, De [25, 26] studied two eccentricity related topological indices, such as eccentric connectivity index and augmented eccentric connectivity indices, on thorn graphs.

The thorns of the thorn graph can be treated as or , so that the thorn graph can be generalized by replacing by and by and the generalizations are, respectively, denoted by and . In the following section, we present the explicit expressions of the modified eccentric connectivity index of thorn graph and its generalized forms and .

#### 2. Evaluation of Modified Eccentric Connectivity Index

The eccentric connectivity index [2] and connective eccentric index [10] of a graph are defined as

The modified eccentric connectivity index [11] is defined as

Total eccentricity index is defined as . Total eccentricity index of the generalized hierarchical product of graphs has been studied by De et al. recently [14].

Since the modified eccentric connectivity index is likely to have an application in drug discovery process, therefore, we evaluate the index in comparison to some other well known indices in this section.

The two graphs shown in Figure 1 have the same eccentricity connectivity index and connective eccentricity index, but they have different modified eccentric connectivity indices.