Bounds and extremal graphs for Harary energy

Let G be a connected graph of order n and let RD(G) be the reciprocal distance matrix (also called Harary matrix) of the graph G. Let ρ1 ≥ ρ2 ≥⋯ ≥ ρn be the eigenvalues of the reciprocal distance matrix RD(G) of the connected graph G called the reciprocal distance eigenvalues of G. The Harary energy HE(G) of a connected graph G is defined as sum of the absolute values of the reciprocal distance eigenvalues of G, that is, HE(G) =∑i=1n|ρ i|. In this paper, we establish some new lower and upper bounds for HE(G), in terms of different graph parameters associated with the structure of the graph G. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of k largest adjacency eigenvalues of a connected graph.