The \emph{energy} of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. I. Milovanović, E. Milovanović et al. {\rm [extit{Discr. Math. Algorithms Appl.} extbf{11}(1) (2019) \#1950001]} established several inequalities relating the graph energy to the determinant of the adjacency matrix. We now sharpen the proofs and formulations of some key results from that work.