Eccentric Connectivity Index and Polynomial of Some Graphs

Aims/ Objectives: Let G be a simple and connected graph with n vertices and m edges. The Eccentric connectivity index of G is defined as the summation of the product of degree and eccentricity of the vertices [1]. Eccentric connectivity polynomial is a topological polynomial of G which is related to its Eccentric connectivity index [2]. The objective of the study is to derive exact expressions of these indices for the double graph and extended double cover graph of a given graph. In addition to it, a lower bound of these invariants for the subdivision graph of the double graph and extended double cover graph of a given graph is also proposed.

Methodology: In this study, simple and connected graphs with n vertices and m edges are considered.
Results: In this article, exact expressions of Eccentric Connectivity index and Eccentric Connectivity Polynomial for the double graph of a given graph is presented. In addition to it, a lower bound of these invariants for the subdivision graph of the double graph of a given graph and a lower bound of these invariants for the extended double cover graph of a given graph is also proposed.

Conclusion: Eccentric Connectivity index and Eccentric Connectivity Polynomial of double graph and Extended double cover graphs can be expressed in terms of their parent graphs.