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 λof 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.