The Aα spectral radius of graphs with given independence number n−4

Let G be a graph with adjacency matrix A(G) and degree diagonal matrix D(G). In 2017, Nikiforov (2017) defined the matrix A_α(G)=αD(G)+(1−α)A(G) for any real α∈[0,1]. The largest eigenvalue of A(G) is called the spectral radius of G, while the largest eigenvalue of A_α(G) is called the A_α spectral radius of G. Let G_n,i be the set of graphs of order n with independence number i. Recently, for all graphs in G_n,i having the minimum or the maximum of A, Q and A_α spectral radius where i∈{1,2,⌊n2⌋⌈n2⌉+1,n−3,n−2,n−1} there are some results given by Xu, Li and Sun et al., respectively. In 2022, Luo and Guo (2022) determined all graphs in G_n,n−4 having the minimum spectral radius. In this paper, we characterize the graphs in G_n,n−4 having the minimum and the maximum A_α spectral radius for α∈[12,1), respectively.