Ordering digraphs with given maximum outdegree by their A_α spectral radius

Let G be a strongly connected digraph with n vertices and m arcs. For any real α∈[0,1], the A_α matrix of a digraph G is defined as A_α(G)=αD(G)+(1−α)A(G), where A(G) is the adjacency matrix of G and D(G) is the outdegrees diagonal matrix of G. The eigenvalue of A_α(G) with the largest modulus is called the A_α spectral radius of G, denoted by λ_α(G). In this paper, we first obtain an upper bound on λ_α(G) for [Formula presented]. Employing this upper bound, we prove that for two strongly connected digraphs G₁ and G₂ with n≥4 vertices and m arcs, and [Formula presented], if the maximum outdegree Δ⁺(G₁)≥2α(1−α)(m−n+1)+2α and Δ⁺(G₁)>Δ⁺(G₂), then λ_α(G₁)>λ_α(G₂). Moreover, we also give another upper bound on λ_α(G) for[Formula presented]. Employing this upper bound, we prove that for two strongly connected digraphs G₁ and G₂ with m arcs, and[Formula presented], if the maximum outdegree[Formula presented]and Δ⁺(G₁)>Δ⁺(G₂), then[Formula presented].