Jixiang Meng

Let G be a graph with order n and size m. For any real number ? ? [0, 1], Nikiforov defined the matrix A?(G) = ?D(G) + (1 ? ?)A(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the vertex degrees. Let ?i (i = 1, 2,..., n) denote the eigenvalues of A?(G) and EA? (G) = Pni =1 |?i ? 2?m/n| denote the A?-energy of G. In this paper, we get some lower bounds of EA? (G) in terms of the order, the size and the first Zagreb index of G for ? ? [1/2, 1), and characterize the extremal graphs when attaining the bounds if G is regular. In addition, we give some lower and upper bounds of EA? (G) under the condition that ?1 + ?n ? 4?m/n.