ON THE Aσ-SPECTRAL RADII OF GRAPHS WITH SOME GIVEN PARAMETERS

作者：Li, Shuchao^*; Zhou, Zihan

来源：ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 2022, 52(3): 949-966.

DOI：10.1216/rmj.2022.52.949

• A(sigma)-index
• signless Laplacian spectral radius
• block graph
• independence number
• cut edge.

Given a graph G, the adjacency matrix and degree diagonal matrix of G are denoted by A(G) and D(G), respectively. In 2017, Nikiforov proposed the A (sigma) -matrix: A(sigma) (G) = sigma D(G) + (1 - sigma) A(G), where s. [0, 1]. The largest eigenvalue of this novel matrix is called the A sigma -index of G. Let B-n(a) be the class of n-vertex block graphs with independence number a and let G (n, k) be another class of n-vertex graphs with k cut edges. We show that the maximum A(sigma) -index, among all graphs G is an element of B-n(a) (resp. G. G (n, k)), is attained at a unique graph. It is surprising to see that in both cases, the extremal graphs are usually pineapple graphs. We use two methods to establish upper bounds on the A s -index of the corresponding extremal graphs. As a byproduct we obtain an upper bound for signless Laplacian spectral radius q1(G), when G is an element of B-n(a)