On the $A_{\alpha }$ 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 $\alpha \in [0,1]$, the matrix $A_{\alpha }(G)$ of a digraph $G$ is defined as $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G).$$ The largest modulus of the eigenvalues of $A_{\alpha }(G)$ is called the $A_{\alpha }$ spectral radius of $G$, denoted by ${\lambda }_{\alpha }(G)$. This paper proves some extremal results about the spectral radius ${\lambda }_{\alpha }(G)$ that generalize previous results about ${\lambda }_0(G)$ and ${\lambda }_{\frac{1}{2}}(G)$. In particular, we characterize the extremal digraph with the maximum (or minimum) $A_{\alpha }$ spectral radius among all $\tilde{\infty }$-digraphs and $\tilde{\theta }$-digraphs on $n$ vertices. Furthermore, we determine the digraphs with the second and the third minimum $A_{\alpha }$ spectral radius among all strongly connected bicyclic digraphs. For $0\le \alpha \le \frac{1}{2}$, we also determine the digraphs with the second, the third and the fourth minimum $A_{\alpha }$ spectral radius among all strongly connected digraphs on $n$ vertices. Finally, we characterize the digraph with the minimum $A_{\alpha }$ spectral radius among all strongly connected bipartite digraphs which contain a complete bipartite subdigraph.

20 pages,6 figures