ENERGY OF STRONG RECIPROCAL GRAPHS

The energy of a graph G, denoted by epsilon(G), is defined as the sum of absolute values of all eigenvalues of G. A graph G is called reciprocal if 1/lambda is an eigenvalue of G whenever lambda is an eigenvalue of G. Further, if lambda and 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, epsilon(G) >= 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 epsilon(G) >= Delta(G) + delta(G) - 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 epsilon(G) >= n + 1/2. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption.