Bounds on the ABC spectral radius of a tree

Abstract

Let G be a simple connected graph with vertex set {1,2,..., n } and d i denote the degree of vertex i in G . The ABC matrix of G , recently introduced by Estrada, is the square matrix whose ij th entry is √(( d i +d j -2)/ d i d i ); if i and j are adjacent, and zero; otherwise. The entries in ABC matrix represent the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph. In this article, we provide bounds on ABC spectral radius of G in terms of the number of vertices in G . The trees with maximum and minimum ABC spectral radius are characterized. Also, in the class of trees on n vertices, we obtain the trees having first four values of ABC spectral radius and subsequently derive a better upper bound.