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

TL;DR

This paper introduces new upper bounds for the spectral radius and low energy of the $A_\alpha$-matrix of digraphs, generalizing classical bounds and providing sharper estimates for certain digraph families.

Contribution

The paper derives novel bounds for the spectral radius and low energy of the $A_\alpha$-matrix, including characterizations of equality cases and numerical comparisons.

Findings

New upper bounds for spectral radius of $A_\alpha$-matrix.

Two Koolen--Moulton type bounds for low energy.

Bounds can be sharper than existing ones for specific 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.