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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.14789 |
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Table of Contents:
- 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 $A_α$-matrix of graphs. The $A_α$-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 $A_α (G) = αD(G) + (1-α) A(G)$. Let $λ_1^{(α)} (G)$ and $λ_n^{(α)} (G)$ denote the largest and smallest eigenvalues of $A_α (G)$, respectively. We determined the unique graph that maximizes $λ^{(α)}_1 (G) - β\cdotλ^{(γ)}_n (G)$ among all connected $n$-vertex graphs for sufficiently large $n$, where $0 \leq α< 1$, $1/2\leq γ< 1$ and $0<βγ\leq 1$. 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 $α= γ= 1/2$ and $β= 1$.