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 ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n 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 = 1 n | 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. [ABSTRACT FROM AUTHOR]