Let $G$ $G$ be a connected graph of order $n$ $n$ and let $RD (G)$ $RD (G)$ be the reciprocal distance matrix (also called Harary matrix) of the graph $G$ $G$ . Let $ρ_{1} \geq ρ_{2} \geq \dots \geq ρ_{n}$ $ρ_{1} \geq ρ_{2} \geq \dots \geq ρ_{n}$ be the eigenvalues of the reciprocal distance matrix $RD (G)$ $RD (G)$ of the connected graph $G$ $G$ called the reciprocal distance eigenvalues of $G$ $G$ . The Harary energy $HE (G)$ $HE (G)$ of a connected graph $G$ $G$ is defined as sum of the absolute values of the reciprocal distance eigenvalues of $G$ $G$ , that is, $HE (G) = \sum_{i = 1}^{n} | ρ_{i} | .$ $HE (G) = \sum_{i = 1}^{n} | ρ_{i} | .$ In this paper, we establish some new lower and upper bounds for $HE (G),$ $HE (G),$ in terms of different graph parameters associated with the structure of the graph $G$ $G$ . We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of $k$ $k$ largest adjacency eigenvalues of a connected graph.

Communicated by Cunquan Zhang

Keywords:

AMSC: 05C50, 05C12, 15A18

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