On α-adjacency energy of graphs and Zagreb index

Abstract

Let A(G) be the adjacency matrix and D(G) be the diagonal matrix of the vertex degrees of a simple connected graph G. Nikiforov defined the matrix of the convex combinations of D(G) and A(G) as for If are the eigenvalues of (which we call α-adjacency eigenvalues of G), the α-adjacency energy of G is defined as where n is the order and m is the size of G. We obtain upper and lower bounds for in terms of the order n, the size m and the Zagreb index Zg(G) associated to the structure of G. Further, we characterize the extremal graphs attaining these bounds.

Keywords

• adjacency matrix
• laplacian (signless laplacian) matrix
• degree regular graph
• α-adjacency matrix
• α-adjacency energy