Some New Bounds for a-Adjacency Energy of Graphs

Let G be a graph with the adjacency matrix A(G), and let D(G) be the diagonal matrix of the degrees of G. Nikiforov first defined the matrix A(a)(G) as A(a)(G) = a D(G) + (1 - a)A(G), 0 = a = 1, which shed new light on A(G) and Q(G) = D(G) + A(G), and yielded some surprises. The a-adjacency energy E-a(A) (G) of G is a new invariant that is calculated from the eigenvalues of A(a)(G). In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for E-a(A) (G) in terms of graph parameters (the order n, the edge size m, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between E-a(A) (G) and other energies such as the energy E(G). Some results can be applied to appropriately estimate the a-adjacency energy using some given graph parameters rather than by performing some tedious calculations.