Minimal Energy on Unicyclic Graphs

DOI：10.3770/j.issn:2095-2651.2014.04.004

 作者 单位 计省进 山东理工大学理学院, 山东 淄博 255049 瞿勇科 洛阳师范学院数学院, 河南 洛阳 471022

对于一个简单图$G$,图$G$的能量$E(G)$定义为它的邻接矩阵的所有特征值的绝对值之和.令$\mathscr{U}_{n}$表示所有具有$n$个的连通单圈图的集合.令$\mathscr{U}^{r}_{n}=\{G\in\mathscr{U}_{n}|\,d(x)=r$, for any vertex $x\in V(C_{\ell})\}$, 其中 $r\geq 2$且$C_{\ell}$是$G$中唯一的圈. $\mathscr{U}^{r}_{n}$中每个单圈图称为$r$-圈-正则图.在$\mathscr{U}_{n}^{4}$中,我们完全刻画了具有唯一的极小能量图$C_{9}^{3}(2,2,2)\circ S_{n-8}$. 此外, 该图在 $\mathscr{U}_{n}^{r}$ (for $r=3,4$)中也是唯一的极小能量图.

For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $\mathscr{U}_{n}$ denote the set of all connected unicyclic graphs with order $n$, and $\mathscr{U}^{r}_{n}=\{G\in\mathscr{U}_{n}|\,d(x)=r$ for any vertex $x\in V(C_{\ell})\}$, where $r\geq 2$ and $C_{\ell}$ is the unique cycle in $G$. Every unicyclic graph in $\mathscr{U}^{r}_{n}$ is said to be a cycle-$r$-regular graph. In this paper, we completely characterize that $C_{9}^{3}(2,2,2)\circ S_{n-8}$ is the unique graph having minimal energy in $\mathscr{U}_{n}^{4}$. Moreover, the graph with minimal energy is uniquely determined in $\mathscr{U}_{n}^{r}$ for $r=3,4$.