关于最大度不超过3的共轭单圈图的最小能量
On the minimal energy of conjugated unicyclic graphs with maximum degree at most 3
摘要：
图$G$的能量，记为$E(G)$，定义为$G$的所有特征值的绝对值之和。设$n$是偶数，$\mathbb{U}_{n}$是所有阶为$n$且最大度至多为$3$的共轭单圈图的集合.设$S_n^{\frac{n}{2}}$是通过将悬垂边附加到圈$C_{\frac{n}{2}}$的每个顶点而获得的径向烯图。在[Y. Cao等人，关于具有Kekulé结构的单圈Hückel分子图的最小能量，离散应用数学157（5）（2009），913- 919]，Cao等人证明了如果$n\geq 8$，$S_n^{\frac{n}{2}}\ncong G\in \mathbb{U}_{n}$和$G$的围长不能被$4$整除，则$E(G)>E(S_n^{\frac{n}{2}})$。设$A_n$是单圈图，它是通过在路径$P_{\frac{n}{2}-1}$的两个叶顶点之一上附加一个$4$ -圈，并在路径$P_{\frac{n}{2}-1}$的每一个叶顶点上附加一条悬挂边而得到的.本文证明了$A_n$是$\mathbb{U}_{n}$中唯一具有最小能量的单圈图。
英文摘要：
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.
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