The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of theabsolute 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 graphobtained 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 unicyclicH\"{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 notdivisible by $4$, then $E(G)>E(S_n^{\frac{n}{2}})$. Let $A_n$ be the unicyclicgraph obtained by attaching a $4$-cycle to one of the two leaf vertices of thepath $P_{\frac{n}{2}-1}$ 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 unicyclicgraph in $\mathbb{U}_{n}$ with minimal energy.
展开▼
机译：由$ E（G）$表示的图$ G $的能量定义为$ G $所有特征值的绝对值之和。令$ n $为偶数，$ \ mathbb {U} _ {n} $为阶数为$ n $且最大度为$ 3 $的所有共轭单环图的集合。令$ S_n ^ {\ frac {n} {2}} $为通过将垂线附加到循环$ C _ {\ frac {n} {2}} $的每个顶点而获得的径向烯图。在[Y. Cao等，关于具有Kekul'{e}结构的单环H \“ {u} ckel分子图的最小能量，Discrete Appl.Math。157（5）（2009），913--919]，Cao等表示如果$ n \ geq 8 $，$ S_n ^ {\ frac {n} {2}} \ ncong G \ in \ mathbb {U} _ {n} $并且$ G $的周长不能被$ 4整除$，然后$ E（G）> E（S_n ^ {\ frac {n} {2}}）$。令$ A_n $是通过将$ 4 $循环附加到路径的两个叶顶点之一获得的单周期图$ P _ {\ frac {n} {2} -1} $和相对于$ P _ {\ frac {n} {2} -1} $的两个顶点的下垂边缘。在本文中，我们证明了$ A_n $是$ \ mathbb {U} _ {n} $中具有最小能量的唯一单圈线图。
展开▼
