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单圈图的极小能量
引用本文:计省进,瞿勇科.单圈图的极小能量[J].数学研究及应用,2014,34(4):414-422.
作者姓名:计省进  瞿勇科
作者单位:山东理工大学理学院, 山东 淄博 255049;洛阳师范学院数学院, 河南 洛阳 471022
基金项目:国家自然科学基金(Grant No.11326216), 山东理工大学基金(Grant No.413010).
摘    要: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 Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un| d(x) = r for any vertex x ∈ V(Cl)}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39(2, 2, 2) ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4.

关 键 词:最小能量  单圈图  R-正则图  邻接矩阵  绝对值  特征值  联合国  NIS
收稿时间:2013/7/10 0:00:00
修稿时间:2014/2/24 0:00:00

Minimal Energy on Unicyclic Graphs
Shengjin JI and Yongke QU.Minimal Energy on Unicyclic Graphs[J].Journal of Mathematical Research with Applications,2014,34(4):414-422.
Authors:Shengjin JI and Yongke QU
Affiliation:School of Science, Shandong University of Technology, Shandong 255049, P. R. China;Department of Mathematics, Luoyang Normal University, Henan 471022, P. R. China
Abstract: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 $r=3,4$.
Keywords:graph energy  unicyclic graph  matching  quasi-order  
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