Sharp bounds of the $A_α$-spectral radii of mixed trees

Abstract

A mixed tree is a tree in which both directed arcs and undirected edges may exist. Let T be a mixed tree with n vertices and m arcs, where an undirected edge is counted twice as arcs. Let A be the adjacency matrix of T . For α ∈ [0, 1], the matrix A α of T is defined to be αD + + (1 − α)A, where D + is the the diagonal out-degree matrix of T . The A α -spectral radius of T is the largest real eigenvalue of A α . We will give a sharp upper bound and a sharp lower bound of the A α -spectral radius of T .