First and Second Zagreb Eccentricity Indices of Thorny Graphs

Abstract

The Zagreb eccentricity indices are the eccentricity reformulation of the Zagreb indices. Let H be a simple graph. The first Zagreb eccentricity index ( E 1 ( H ) ) is defined to be the summation of squares of the eccentricity of vertices, i.e., E 1 ( H ) = ∑ u ∈ V ( H ) Ɛ H 2 ( u ) . The second Zagreb eccentricity index ( E 2 ( H ) ) is the summation of product of the eccentricities of the adjacent vertices, i.e., E 2 ( H ) = ∑ u v ∈ E ( H ) Ɛ H ( u ) Ɛ H ( v ) . We obtain the thorny graph of a graph H by attaching thorns i.e., vertices of degree one to every vertex of H . In this paper, we will find closed formulation for the first Zagreb eccentricity index and second Zagreb eccentricity index of different well known classes of thorny graphs.

Keywords

• graphs
• vertices
• complete graph
• path
• star
• cycle