| 扈生彪.单圈图的最大特征值的上界的改进[J].数学研究及应用,2009,29(5):945~950 |
| 单圈图的最大特征值的上界的改进 |
| Improved Upper Bounds for the Largest Eigenvalue of Unicyclic Graphs |
| 投稿时间：2007-06-04 修订日期：2008-07-07 |
| DOI：10.3770/j.issn:1000-341X.2009.05.022 |
| 中文关键词: 单圈图 邻接矩阵 最大特征值. |
| 英文关键词:unicyclic graph adjacency matrix largest eigenvalue. |
| 基金项目:国家自然科学基金(No.10861009). |
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| 摘要点击次数: 2975 |
| 全文下载次数: 2461 |
| 中文摘要: |
| 设$G(V,E)$是一个单圈图,记$C_{m}$是长为$m$的圈且$C_{m}\subset G$, 设$u_{i}\in V(C_{m})$.$G-E(C_{m})$是$m$个树,被记为$T_{i} (i=1,2,...,m)$.设 $e_{u_{i}}(i=1,2,...,m)$是$T_{i}$中$u_{i}$的离心距且$$e_{c}=max\{e_{u_{i}}:i=1,2,...,m\}$$设 $k=e_{c}$ 1. 对于$j=1,2,...,k-1$,设$$\delta_{ij}=max\{d |
| 英文摘要: |
| Let $G(V,E)$ be a unicyclic graph, $C_{m}$ be a cycle of length $m$ and $C_{m}\subset G$, and $u_{i}\in V(C_{m})$. The $G-E(C_{m})$ are $m$ trees, denoted by $T_{i}$, $i=1,2,\ldots,m$. For $i=1,2,\ldots,m$, let $e_{u_{i}}$ be the excentricity of $u_{i}$ in $T_{i}$ and $$e_{c}=\max\{e_{u_{i}}: i=1,2,\ldots,m\}.$$ Let $k=e_{c}$ 1. For $j=1,2,\ldots,k-1$, let $$\delta_{ij}=\max\{d_{v}:{\rm dist}(v,u_{i})=j, v\in T_{i}\},$$ $$\delta_{j}=\max\{\delta_{ij}:i=1,2,\ldots,m\},$$ $$\delta_{0}=\max\{d_{u_{i}}:u_{i}\in V(C_{m})\}.$$ Then $$\lambda_{1}(G)\leq \max\{\max\limits_{2\leq j\leq k-2}(\sqrt{\delta_{j-1}-1} \sqrt{\delta_{j}-1}),2 \sqrt{\delta_{0}-2},\sqrt{\delta_{0}-2} \sqrt{\delta_{1}-1}\}.$$ If $G\cong C_{n}$, then the equality holds, where $\lambda_{1}(G)$ is the largest eigenvalue of the adjacency matrix of $G$. |
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