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

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.

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 $G$.