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On the minimal energy of conjugated unicyclic graphs with maximum degree at most 3

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 Hückel molecular graphs possessing Kekulé 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.