Bounds on the ABC spectral radius of a tree

Sasmita Barik (School of Basic Sciences IIT Bhubaneswar, Bhubaneswar, 752050, India)
Sonu Rani (School of Basic Sciences IIT Bhubaneswar, Bhubaneswar, 752050, India)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 8, No 2 (2020): Electronic Journal of Graph Theory and Applications

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.