On the spread of graph-related matrices

The spread of a real symmetric matrix is defined as the difference between its largest and smallest eigenvalues. 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 combination of its diagonal degree matrix D ( G ) and adjacency matrix A ( G ) , defined as A α ( G ) = α D ( 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 determine the unique graph that maximizes λ 1 ( α ) ( G ) − β ⋅ λ n ( γ ) ( G ) among all connected n -vertex graphs for sufficiently large n , where 0 ≤ α < 1 , 1 / 2 ≤ γ < 1 and 0 < β γ ≤ 1 . As an application, we confirm a conjecture proposed by Lin et al. (2020) [12] . In addition, one of main results in Liu (2024) [13] is a simple corollary of our result by choosing α = γ = 1 / 2 and β = 1 .