  Let G be a simple graph on n vertices with vertex set V(G). The energy of G,
denoted by, $\mathcal{E}(G)$ is the sum of all absolute values of the
eigenvalues of the adjacency matrix $A(G)$. It is the first eigenvalue-based
topological molecular index and is related to the molecular orbital energy
levels of ${\pi}$-electrons in conjugated hydrocarbons. Recently, the concept
of energy of a graph is extended to a self-loop graph. Let $S$ be a subset of
$V(G)$. The graph $G_S$ is obtained from the graph $G$ by attaching a self-loop
at each of the vertices of $G$ which are in the set $S$. The energy of the
self-loop graph $G_S$, denoted by $\mathcal{E}(G_S)$, is the sum of all
absolute eigenvalues of the matrix $A(G_S)$. Two non-isomorphic self-loop
graphs are equienergetic if their energies are equal. Akbari et al.
(2023)conjectured that there exist a subset $S$ of $V(G)$ such that
$\mathcal(G_S) > \mathcal{E}(G)$. In this paper, we confirm this conjecture.
Also, we construct pairs of equienergetic self-loop graphs of order 24n for all
n \ge 1.
