The A_α spectral radius and maximum outdegree of irregular digraphs

Let G be a digraph with adjacency matrix A(G). Let D(G) be the diagonal matrix with outdegrees of vertices of G. In this paper, we study the convex linear combinations of A(G) and D(G), defined as A_α(G)=αD(G)+(1−α)A(G),0≤α≤1. The largest modulus of the eigenvalues of A_α(G), is called the A_α spectral radius of G, denoted by λ_α(G). We establish some lower bounds on Δ⁺−λ_α(G) for strongly connected irregular digraphs with given maximum outdegree and some other parameters, where Δ⁺ is the maximum vertex outdegree of G.