**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

E-mail:

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