A note on the A(alpha)-spectral radius of graphs

作者：Lin, Huiqiu^*; Huang, Xing; Xue, Jie

来源：Linear Algebra and Its Applications, 2018, 557: 430-437.

DOI：10.1016/j.laa.2018.08.008

• A(alpha)-matrix
• A(alpha)-spectral radius
• Cut vertex
• Matching number

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real alpha is an element of [0, 1], Nikiforov (2017) [7] defined the matrix A(alpha) (G) as A(alpha)(G) = alpha D(G) + (1 - alpha)A(G). Let u and v be two vertices of a connected graph G. Suppose that u and v are connected by a path w(0) (= v)w(1) . . . w(s-1)ws (= u) where d(w(i)) = 2 for 1 <= i <= s - 1. Let G(p, s, q) (u, v) be the graph obtained by attaching the paths P-p to u and P-q to v. Let s = 0, 1. Nikiforov and Rojo (2018) [9] conjectured that rho(alpha) (G(p, s, q) (u , v)) < rho(alpha) (G(p, s, q) (u , v)) if p > q + 2. In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal A(alpha)-spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal A(alpha)-spectral radius with fixed order and matching number. These results generalize some known results.

• 出版日期2018-11-15
• 单位华东理工大学; 华东师范大学