Spectral Properties with the Difference between Topological Indices in Graphs

Let G be a graph of order n with vertices labeled asv 1 , v 2 , … , v n . Letd ibe the degree of the vertexv i , for i = 1,2 , … , n . The difference adjacency matrix of G is the square matrix of order n whosei , jentry is equal tod i + d j − 2 − 1/d id jif the verticesv iandv jof G are adjacent orv iv j ∈ E Gand zero otherwise. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix, that is, a modification of the classical adjacency matrix involving the degrees of the vertices. In this paper, some properties of its characteristic polynomial are studied. We also investigate the difference energy of a graph. In addition, we establish some upper and lower bounds for this new energy of graph.