ON THE A_α SPECTRAL RADIUS OF STRONGLY CONNECTED DIGRAPHS

Let A(G) and D(G) be the adjacency matrix and the diagonal matrix with outdegrees of vertices of a digraph G, respectively. In 2017, Nikiforov proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. In 2019, Liu et al. extended the definition to digraphs. For any real α ∈ [0, 1], the matrix A_α (G) of a digraph G is defined as A_α (G) = αD(G) +(1 −α)A(G). The largest modulus of the eigen-values of A_α (G) is called the A_α spectral radius of G, denoted by λ_α (G) . This paper proves some extremal results about the A_α spectral radius λ_α (G) that generalize previous results about λ₀ (G) and (formula presented). We mainly characterize the extremal digraph with the maximum (or min-imum) A_α spectral radius among all ˜∞-digraphs and θ-digraphs on n vertices. Furthermore, 2 we determine the digraphs with the second and the third minimum A_α spectral radius among all strongly connected bicyclic digraphs. For 0 ≤ α ≤¹2, we also determine the digraphs with the second, the third and the fourth minimum A_α spectral radius among all strongly connected digraphs on n vertices. Finally, we characterize the digraph with the minimum A_α spectral radius among all strongly connected bipartite digraphs which contain a complete bipartite subdigraph.