On the extremal connectivity index of trees with k pendant vertices

Abstract

The connectivity index w _α(G) of a graph G is the sum of the weights (d(u)d(v)) ^αof all edges uv of G, where α is a nonzero real number and d(u) denotes the degree of the vertex u. Let J _n,k be the set of trees on n vertices with k pendant vertices. Liu, Lu and Tian [Discrete Appl. Math., 154:106-119,2006] determined the sharp upper and lower bounds of w ₁(T) for T ε _n, _k- In this paper, as a continuous of it, we characterize all the trees in .J _n, _k with the second largest and second smallest w ₁-values.