Energy of strong reciprocal graphs

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{۱}{\lambda} is an eigenvalue of G whenever \lambda is an eigenvalue of G. Further, if \lambda and \frac{۱}{\lambda} have the same multiplicities, for each eigenvalue \lambda, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. ۸۳ (۲۰۲۰) ۶۳۱--۶۳۳), 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{۱}{۲}. Recently, it has been proved that if G is a reciprocal graph of order n and its spectral radius, \rho, is at least ۴\lambda_{min}, where \lambda_{min} is the smallest absolute value of eigenvalues of G, then \mathcal{E}(G) \geq n+\frac{۱}{۲}. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption.

Keywords:

Graph energy , Strong reciprocal graph , Non-singular graph