BOUNDS FOR THE α –ADJACENCY ENERGY OF A GRAPH

For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A_α(G) matrix is the convex combinations of D(G) and A(G), and is defined as A_α(G) = αD(G)+(1 − α)A(G), for 0 ≤ α ≤ 1. Let ρ1 ≥ ρ2 ≥ … ≥ ρn be the eigenvalues of Aα(G) (which we call α -adjacency eigenvalues of the graph G). The generalized adjacency energy also called α -adjacency energy of the graph G is defined as E_Aα (G) = (Formula presented.), where (Formula presented.) n is the average vertex degree, m is the size and n is the order of G. The α -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as E^A₀ (G) = E (G) and 2E ^A_1/2(G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained these bounds.