%0 Journal Article
%T Energy of strong reciprocal graphs
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Ghahremani, Maryam
%A Tehranian, Abolfazl
%A Rasouli, Hamid
%A Hosseinzadeh, Mohammad Ali
%D 2023
%\ 09/01/2023
%V 12
%N 3
%P 165-171
%! Energy of strong reciprocal graphs
%K Graph energy
%K Strong reciprocal graph
%K Non-singular graph
%R 10.22108/toc.2022.134259.1999
%X 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.
%U https://toc.ui.ac.ir/article_26810_877d7be13cf3ac469149b724447166ab.pdf