On the Aα-spectral radius of graphs with given size

For a simple graph G and alpha is an element of [0, 1], Nikiforov (2017) defined the A(alpha)(G)-matrix of G as A(alpha) = alpha D(G) + (1 - alpha)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal degree matrix of G, respectively. The A(alpha)-spectral radius of G, denoted by rho(alpha)(G), is the largest eigenvalue of A(alpha) (G). Wang and Guo (2022) proved that for an integer k >= 0 and a graph G of size m, if rho(1/2)(G) >= 1/2 (m - k + 1), then K-1,K-m-k subset of G. In this paper, we 2 extend their result to A alpha-spectral radius for alpha is an element of [1/2, 1). Moreover, for alpha is an element of [1/2, 1), we also determine the graph with maximum A(alpha)-spectral radius among graphs of given size and girth, which generalizes the result of Chen, Wang and Zhai (2022) for alpha = 1/2. (c) 2023 Elsevier B.V. All rights reserved.