Bounds on the ABC spectral radius of a tree

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 ijth entry is root d(i)+d(j)-2/d(i)d(j); 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.