2024/12/19 by Liu, Lele, Fan, Yi-Zheng, Wang, Yi +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.14789
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 Aα-matrix of graphs. The Aα-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 Aα (G) = αD(G) + (1-α) A(G). Let λ1(α) (G) and λn(α) (G) denote the largest and smallest eigenvalues of Aα (G), respectively. We determined the unique graph that maximizes λ(α)1 (G) - β⋅λ(γ)n (G) among all connected n-vertex graphs for sufficiently large n, where 0 ≤ α< 1, 1/2≤ γ< 1 and 0<βγ≤ 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.