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 \cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. Liu et al. \cite{LWCL} extended the definition to digraphs. For any real $α\in[0,1]$, 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 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 $λ_{\frac{1}{2}}(G)$. In particular, we characterize the extremal digraph with the maximum (or minimum) $A_α$ spectral radius among all $\widetilde{\infty}$-digraphs and $\widetildeθ$-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 $0\leqα\leq\frac{1}{2}$, 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.