Some New Bounds for α-Adjacency Energy of Graphs

Abstract

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 Aα(G) as Aα(G)=αD(G)+(1−α)A(G), 0≤α≤1, which shed new light on A(G) and Q(G)=D(G)+A(G), and yielded some surprises. The α−adjacency energy EAα(G) of G is a new invariant that is calculated from the eigenvalues of Aα(G). In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for EAα(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 EAα(G) and other energies such as the energy E(G). Some results can be applied to appropriately estimate the α-adjacency energy using some given graph parameters rather than by performing some tedious calculations.