Energy and resolvent energy of A_α(G)

Gutman defined graph energy and then different types of energy were introduced. The energy of a graph G, is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix.In this paper we study energy and resolvent energy of the convex linear combinations A_α (G) of a simple undirected graph G defined byA_α (G)≔αD(G)+(1-α)A(G) for any real 0≤α≤1 . We show that the solvent energy of A_α (G) is increasing in α, as well as the energy of A_α (G) is increasing in α if α> 1/2. we give a few additional bounds on energy of A_α (G) in terms of the degrees of the vertices of graph G. For regular graph G, we present lower and upper bounds on the energy of A_α (G). We compute energy and resolvent energy of path P_n and cycle C_n . Finally, we calculate the energy and resolvent energy of A_α (G) for complete graphs K_n, complete bipartite graphs K_(a,b), and stars K_(1,n-1) (S_n).