International Journal of Combinatorics

Volume 2014, Article ID 579257, 5 pages

http://dx.doi.org/10.1155/2014/579257

## On Some Bounds and Exact Formulae for Connective Eccentric Indices of Graphs under Some Graph Operations

^{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 14 July 2014; Accepted 14 November 2014; Published 24 December 2014

Academic Editor: Cai Heng Li

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 connective eccentric index of a graph is a topological index involving degrees and eccentricities of vertices of the graph. In this paper, we have studied the connective eccentric index for double graph and double cover. Also we give the connective eccentric index for some graph operations such as joins, symmetric difference, disjunction, and splice of graphs.

#### 1. Introduction

Let be a simple connected graph with vertex set and edge set . Let and be the number of vertices and edges of , respectively. We denote the degree of a vertex of by . For , the distance between and in is defined as the length of the shortest path between and in and is denoted by . For a given vertex of , the eccentricity is the largest distance from to any other vertices of . The sum of eccentricities of all the vertices of is denoted by [1]. If any vertex is adjacent to all the other vertices of then is called a well-connected vertex. Thus, if is a well-connected vertex, then . For example, all the vertices of a complete graph are well connected.

Recently, a number of topological indices involving vertex degree and eccentricity were subject to a lot of mathematical as well as chemical studies. A topological index of this type, introduced by Gupta et al. [2], was named as the connective eccentric index and was defined as Ghorbani [3] gave some bounds of connective eccentricity index and also computed this index for two infinite classes of dendrimers. De [4] reported some bounds for this index in terms of some graph invariants such as maximum and minimum degree, radius, diameter, first Zagreb index, and first Zagreb eccentricity index. In [5], Ghorbani and Malekjani computed the eccentric connectivity index and the connective eccentric index of an infinite family of fullerenes. In [6], Yu and Feng also derived some upper or lower bounds for the connective eccentric index and investigated the maximal and the minimal values of connective eccentricity index among all -vertex graphs with fixed number of pendent vertices.

In [3], Ghorbani showed that, for a vertex transitive graph , the connective eccentric index is given by where are the orbits of under its natural action on and , . In particular, if is a regular graph, then , where is the number of vertices of , which is a -regular graph, and is the radius of . Let , , , , denote the complete graph with vertices, the cycle on vertices, -dimensional hypercube, -sided prism, and the -sided antiprism, respectively. It can be easily verified that the explicit formulae for the connective eccentric index of , , , , are as follows.

Proposition 1. *Consider the following
*

*Proposition 2. Consider the following
*

*Proposition 3. Consider the following
*

*Proposition 4. Consider the following
*

*Proposition 5. Consider the following
*

*Several studies on different topological indices related to graph operations of different kinds are available in the literature [7–11].*

*In this paper, first we calculate connective eccentric index of double graph and double cover and hence the explicit formulae for the connective eccentric indices of join, symmetric difference, disjunction, and splice of graphs are obtained. For the definitions and different results on graph operations, such as join, symmetric difference, and disjunction, readers are referred to the book of Imrich and Klavžar [12].*

*2. Main Results*

*2. Main Results**In this section, first we define and then compute eccentric connectivity index of double graph and double cover graph.*

*2.1. Connective Eccentric Index of Double Graph and Double Cover*

*2.1. Connective Eccentric Index of Double Graph and Double Cover**Let us denote the double graph of a graph by , which is constructed from two copies of in the following manner [13, 14]. Let the vertex set of be , and the vertices of are given by the two sets and . Thus, for each vertex , there are two vertices and in . The double graph includes the initial edge set of each copy of , and, for any edge , two more edges and are added. The graph and its double graph is shown in Figure 1.*