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(G) matrix is the convex combinations of D(G) and A(G), and is defined as A(G) = D(G)+(1)A(G), for 0 n be the eigenvalues of A(G) (which we call -adjacency eigenvalues of the graph G). The generalized adjacency energy also called -adjacency energy of the graph G is defined as EA (G) = is the average vertex degree, m is the size and n is the order of G. The -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as EA0 (G) = E (G) and 2E A 12 (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. [ABSTRACT FROM AUTHOR]