Sharp bounds for the spectral radius of digraphs

Let G=(V,E)G=(V,E) be a digraph with n vertices and m   arcs without loops and multiarcs. The spectral radius ρ(G)ρ(G) of G   is the largest eigenvalue of its adjacency matrix. In this paper, the following sharp bounds on ρ(G)ρ(G) have been obtained.min{ti+tj+:(vi,vj)∈E}⩽ρ(G)⩽max{ti+tj+:(vi,vj)∈E}where G   is strongly connected and ti+ is the average 2-outdegree of vertex vivi. Moreover, each equality holds if and only if G is average 2-outdegree regular or average 2-outdegree semiregular.