On the Spread of Graph-Related Matrices

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 ${\lambda }_1^{(\alpha )}(G)-\beta \cdot {\lambda }_n^{(\gamma )}(G)$ among all connected $n$-vertex graphs for sufficiently large $n$, where $0\le \alpha <1$, $1\mathrm{/}2\le \gamma <1$ and $0<\beta \gamma \le 1$. 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 $\alpha =\gamma =1\mathrm{/}2$ and $\beta =1$.