On the Spread of Graph-Related Matrices

TL;DR

This paper investigates the spread of the $A_{\alpha}$-matrix of graphs, identifying the extremal graphs that maximize a specific eigenvalue difference, and confirms a related conjecture in spectral graph theory.

Contribution

It determines the unique extremal graph maximizing a weighted eigenvalue difference of the $A_{\alpha}$-matrix among connected graphs, confirming a conjecture and deriving a corollary for specific parameters.

Findings

Identified the extremal graph for the eigenvalue difference problem.

Confirmed a conjecture by Lin, Miao, and Guo.

Derived a corollary related to previous work with specific parameters.

Abstract

The spread of a real symmetric matrix is defined as the difference between its largest and smallest eigenvalue. The study of graph-related matrices has attracted considerable attention, leading to a substantial body of findings. In this paper, we investigate a general spread problem related to $A_{\alpha}$-matrix of graphs. The $A_{\alpha}$-matrix of a graph $G$, introduced by Nikiforov in 2017, is a convex combinations of its diagonal degree matrix $D(G)$ and adjacency matrix $A(G)$, defined as $A_{\alpha} (G) = \alpha D(G) + (1-\alpha) A(G)$. Let $\lambda_1^{(\alpha)} (G)$ and $\lambda_n^{(\alpha)} (G)$ denote the largest and smallest eigenvalues of $A_{\alpha} (G)$, respectively. We determined the unique graph that maximizes $\lambda^{(\alpha)}_1 (G) - \beta\cdot\lambda^{(\gamma)}_n (G)$ among all connected $n$-vertex graphs for sufficiently large $n$, where $0 \leq \alpha < 1$, $1/2\leq \gamma < 1$ and $0<\beta\gamma\leq 1$. As an application, we confirm a conjecture proposed by Lin, Miao, and Guo [Linear Algebra Appl. 606 (2020) 1--22]. In addition, one of main results in [SIAM J. Discrete Math. 38 (2024) 590--608] is a simple corollary of our result by choosing $\alpha = \gamma = 1/2$ and $\beta = 1$.