  Energy of a simple graph $G$, denoted by $\mathcal{E}(G)$, is the sum of the
absolute values of the eigenvalues of $G$. Two graphs with the same order and
energy are called equienergetic graphs. A graph $G$ with the property $G\cong
\overline{G}$ is called self-complementary graph, where $\overline{G}$ denotes
the complement of $G$. Two non-self-complementary equienergetic graphs $G_1$
and $G_2$ satisfying the property $G_1\cong \overline{G_2}$ are called
complementary equienergetic graphs. Recently, Ramane et al. [Graphs
equienergetic with their complements, MATCH Commun. Math. Comput. Chem. 82
(2019) 471-480] initiated the study of the complementary equienergetic regular
graphs and they asked to study the complementary equienergetic non-regular
graphs. In this paper, by developing some computer codes and by making use of
some software like Nauty, Maple and GraphTea, all the complementary
equienergetic graphs with at most 10 vertices as well as all the members of the
graph class $\Omega=\{G \ : \ \mathcal{E}(L(G)) = \mathcal{E}(\overline{L(G)})
\text{, the order of $G$ is at most 10}\}$ are determined, where $L(G)$ denotes
the line graph of $G$. In the cases where we could not find the closed forms of
the eigenvalues and energies of the obtained graphs, we verify the graph
energies using a high precision computing (2000 decimal places) of Maple. A
result about a pair of complementary equienergetic graphs is also given at the
end of this paper.
