New Bounds for the Spectral Radius and Low Energy of the $A_\alpha$-Matrix of Digraphs

Abstract

The $A_\alpha$-matrix of a digraph $D$ is defined as a linear convex combination $\alpha\operatorname{Deg}(D)+(1-\alpha)A(D)$ of the adjacency matrix $A(D)$ and the diagonal out-degree matrix $\operatorname{Deg}(D)$, where $\alpha\in[0,1]$. The low energy of $A_\alpha(D)$ is defined as the sum of the absolute values of the real parts of the eigenvalues of $A_\alpha(D)$. In this paper, we establish new upper bounds for the spectral radius of the $A_\alpha$-matrix and derive two Koolen--Moulton type upper bounds for its low energy, together with characterizations of the equality cases. Numerical comparisons further show that these bounds can be sharper than existing bounds for certain digraph families. Furthermore, when $\alpha=0$, our results recover several classical bounds, and in particular, the low-energy bounds generalizes the classical Koolen--Moulton bound.