Energy of strong reciprocal graphs

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. A graph G is called reciprocal if 1 is an eigenvalue of G whenever λ is an eigenvalue λ of G. Further, if λ and 1 have the same multiplicities, for each eigenvalue λ, 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 ∆(G) and minimum degree δ(G) whose adjacency matrix is non-singular, E(G) ≥ ∆(G) + δ(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 E(G) ≥ ∆(G) + δ(G) − 1 . Recently, it has been proved that if G is a reciprocal graph of order n and its spectral radius, ρ, is at least 4λmin, where λmin is the smallest absolute value of eigenvalues of G, then E(G) ≥ n + 1 . In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption.