  The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the
absolute values of all eigenvalues of $G$. Let $n$ be an even number and
$\mathbb{U}_{n}$ be the set of all conjugated unicyclic graphs of order $n$
with maximum degree at most $3$. Let $S_n^{\frac{n}{2}}$ be the radialene graph
obtained by attaching a pendant edge to each vertex of the cycle
$C_{\frac{n}{2}}$. In [Y. Cao et al., On the minimal energy of unicyclic
H\"{u}ckel molecular graphs possessing Kekul\'{e} structures, Discrete Appl.
Math. 157 (5) (2009), 913--919], Cao et al. showed that if $n\geq 8$,
$S_n^{\frac{n}{2}}\ncong G\in \mathbb{U}_{n}$ and the girth of $G$ is not
divisible by $4$, then $E(G)>E(S_n^{\frac{n}{2}})$. Let $A_n$ be the unicyclic
graph obtained by attaching a $4$-cycle to one of the two leaf vertices of the
path $P_{\frac{n}{2}-1}$ and a pendent edge to each other vertices of
$P_{\frac{n}{2}-1}$. In this paper, we prove that $A_n$ is the unique unicyclic
graph in $\mathbb{U}_{n}$ with minimal energy.
