On the $A_α$ spectral radius of strongly connected digraphs

Abstract

Let G be a digraph with adjacency matrix A(G). Let D(G) be the diagonal matrix with outdegrees of vertices of G. Nikiforov [16] proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. Liu et al.[14] extended the definition to digraphs. For any real α ∈ [0, 1], the matrix A α (G) of a digraph G is defined asThe largest modulus of the eigenvalues 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 spectral radius λ α (G) that generalize previous results about λ 0 (G) and λ 1 2 (G). In particular, we characterize the extremal digraph with the maximum (or minimum) A α spectral radius among all ∞-digraphs and θ-digraphs on n vertices. Furthermore, we determine the digraphs with the second and the third minimum A α spectral radius among all strongly connected bicyclic digraphs. For 0 ≤ α ≤ 1 2 , 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.