The starlike trees with maximum and minimum second order connectivity index

Abstract

Let G be a simple graph and h≥0 be an integer. The higher order connectivity index R h (G) of G is defined as $$R_h(G)=\sum_{v_{i_1}v_{i_2}\cdots v_{i_{h+1}}} \frac{1}{\sqrt {d_{i_1}d_{i_2}\cdots d_{i_{h+1}}}},$$ where d i denotes the degree of the vertex v i and $v_{i_{1}}v_{i_{2}}\cdots v_{i_{h+1}}$ runs over all paths of length h in G. A starlike tree is a tree with unique vertex of degree greater than two. Rada and Araujo proved that the starlike trees which have equal connectivity index of order h for all h≥0 are isomorphic. By T(n) we denote the set of the starlike trees on n vertices. In this paper, we characterize the extremal starlike trees with maximum and minimum second order connectivity index in T(n).