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## New Bounds for the Harmonic Energy and Harmonic Estrada index of Graphs

Authors: Akbar Jahanbani
Keywords: Eigenvalue of graph, Energy, sum-connectivity index, Harmonic energy, Harmonic Estrada index.

### Abstract

Let $G$ be a finite simple undirected graph with $n$ vertices and $m$ edges. The Harmonic energy of a graph $G$, denoted by $\mathcal{H}E(G)$, is defined as the sum of the absolute values of all Harmonic eigenvalues of $G$. The Harmonic Estrada index of a graph $G$, denoted by $\mathcal{H}EE(G)$, is defined as $\mathcal{H}EE=\mathcal{H}EE(G)=\sum_{i=1}^{n}e^{\gamma_i},$ where $\gamma_1\geqslant \gamma_2\geqslant \dots\geqslant \gamma_n$ are the $\mathcal{H}$-$eigenvalues$ of $G$. In this paper we present some new bounds for $\mathcal{H}E(G)$ and $\mathcal{H}EE(G)$ in terms of number of vertices, number of edges and the sum-connectivity index.

Department of Mathematics,
Shahrood University of Technology,
Shahrood, Iran
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