Some New Bounds for α -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 α ( G ) as A α ( G ) = α D ( 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 E A α ( G ) of G is a new invariant that is calculated from the eigenvalues of A α ( G ) . In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for E A α ( 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 α ( 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.