Graphs of fixed order and size with maximal Aα-index

For any real number alpha is an element of[0, 1], by the A(alpha)-matrixof a graph Gwe mean the matrix A(alpha)(G) = D-alpha(G) +(1 - alpha)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The largest eigenvalue of A(alpha)(G) is called the A(alpha)-index of G. In this paper, we settle the problem of characterizing graphs which attain the maximum Aa-index over G( n, n + k), the class of graphs with nvertices and n + kedges, for -1 <= k <= n - 3and 1/2= alpha < 1. The following result is obtained: for -1 <= k <= n - 3, when 12 <= alpha < 1, H-n,H-k is the unique graph in G( n, n + k) that maximizes the A(alpha)-index, except when ( n, k) =(4, -1), ( n, 2) or (7, 3) and alpha = 1/2, or ( n, k) =(5, 1) and alpha is an element of[ 1/2, 35-root 409/24]. When ( n, k, a) =(4, -1, 1/2), the optimal graphs are H(4,-1)and K-3 boolean OR K-1; when ( n, k, alpha) =( n, 2, 1/2), the optimal graphs are H(n,2)and G(n,2); when ( n, k, alpha) =(5, 1, 35-root 409/24), the optimal graphs are H(5,1)and K-4 boolean OR K-1; when ( n, k, a) =(7, 3, 1/2), the optimal graphs are H(7,3)and K-5. 2K(1); when ( n, k) =(5, 1) and 1/2 <= alpha < 35-root 409/24, K-4 boolean OR K(1)is the unique graph that maximizes the A(alpha)-index. Our work completes the corresponding work of Chang and Tam (2010) and Zhai et al. (2022) for the special case alpha = 1/2. As a by-product, we provide a new proof for the known result that for any positive integer mand any