Sharp bounds for the spectral radius of digraphs

Abstract

Let G = (V, E) be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius ρ (G) of G is the largest eigenvalue of its adjacency matrix. In this paper, the following sharp bounds on ρ (G) have been obtained.min {sqrt(ti+ tj+) : (vi, vj) ∈ E} ≤ ρ (G) ≤ max {sqrt(ti+ tj+) : (vi, vj) ∈ E}where G is strongly connected and ti+ is the average 2-outdegree of vertex vi. Moreover, each equality holds if and only if G is average 2-outdegree regular or average 2-outdegree semiregular. © 2008.