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 ( E1(H) ) is defined to be the summation of squares of the eccentricity of vertices, i.e., E1(H)=∑u∈V(H)εH2 (u) . The second Zagreb eccentricity index ( E2(H) ) is the summation of product of the eccentricities of the adjacent vertices, i.e., E2(H)=∑uv∈E(H)εH(u)εH(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.