On extremality of ABC spectral radius of a tree

Abstract

The ABC matrix of a graph G, recently introduced by Estrada, is the square matrix of order |G| whose (i,j)-entry is equal to (di+dj−2)/(didj) if the i-th vertex and the j-th vertex of G are adjacent, and 0 otherwise, where di is the degree of the i-th vertex of G. The ABC spectral radius of G is the largest eigenvalue of the ABC matrix of G, which is denoted by ρ(Ω(G)). In this paper, we prove that for any tree T of order n≥3,2cos⁡πn+1≤ρ(Ω(T))≤n−2, with equality in the left (resp., right) inequality if and only if T is isomorphic to the path Pn (resp., the star K1,n−1).