# Energy of strong reciprocal graphs

Document Type : Research Paper

Authors

1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

2 Science and Research Branch, Islamic Azad University

3 Faculty of Engineering Modern Technologies, Amol University of Special Modern Technologies, Amol, Iran

Abstract

The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute values of all eigenvalues of $G$. A graph $G$ is called reciprocal if $\frac{1}{\lambda}$ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $\lambda$ and $\frac{1}{\lambda}$ have the same multiplicities, for each eigenvalue $\lambda$, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631--633), it was conjectured that for every graph $G$ with maximum degree $\Delta(G)$ and minimum degree $\delta(G)$ whose adjacency matrix is non-singular, $\mathcal{E}(G) \geq \Delta(G) + \delta(G)$ and the equality holds if and only if $G$ is a complete graph. Here, we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if $G$ is a strong reciprocal graph, then $\mathcal{E}(G) \geq \Delta(G) + \delta(G) - \frac{1}{2}$. Recently, it has been proved that if $G$ is a reciprocal graph of order $n$ and its spectral radius, $\rho$, is at least $4\lambda_{min}$, where $\lambda_{min}$ is the smallest absolute value of eigenvalues of $G$, then $\mathcal{E}(G) \geq n+\frac{1}{2}$. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption.

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### History

• Receive Date: 04 July 2022
• Revise Date: 02 August 2022
• Accept Date: 05 August 2022
• Published Online: 01 September 2023