University of IsfahanTransactions on Combinatorics2251-865712320230901Energy of strong reciprocal graphs1651712681010.22108/toc.2022.134259.1999ENMaryamGhahremaniDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranAbolfazlTehranianScience and Research Branch, Islamic Azad UniversityHamidRasouliDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranMohammad AliHosseinzadehFaculty of Engineering Modern Technologies, Amol University of Special Modern Technologies, Amol, IranJournal Article20220704The 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.https://toc.ui.ac.ir/article_26810_877d7be13cf3ac469149b724447166ab.pdf