Bounds on the ABC spectral radius of a tree - Sasmita Barik; Sonu Rani (Electronic Journal of Graph Theory and Applications (EJGTA), 2020)
Let G be a simple connected graph with vertex set {1,2,...,n} and di 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 √((di+dj-2)/didi); 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.