The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups
The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups
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DOI：
10.37236/8054
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发表时间：
2018-08
期刊：
影响因子：
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通讯作者：
Xueyi Huang;Qiongxiang Huang;Sebastian M. Cioabùa
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文献类型：
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作者：
Xueyi Huang;Qiongxiang Huang;Sebastian M. Cioabùa
设$G$为传递作用于$[n]=\{1,2,\ldots,n\}$上的有限群，$\Gamma=\mathrm{Cay}(G,T)$为$G$的Cayley图。如果$T$在共轭下闭合，则图$\Gamma$称为法线图。本文根据$\Gamma$的某些子图的第二个特征值，得到了图$\Gamma$邻接矩阵的第二个（最大）特征值的上界。利用这一结果，我们开发了一种递归方法来确定$S_n$的某些Cayley图的第二特征值，并且我们用$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$确定了$S_n$的大多数连接的正规Cayley图（及其部分子图）的第二特征值，其中$\mathrm{supp}(\tau)$是$[n]$中不固定的点的集合$\tau$。
Let $G$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $\Gamma=\mathrm{Cay}(G,T)$ be a Cayley graph of $G$. The graph $\Gamma$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $\Gamma$ in terms of the second eigenvalues of certain subgraphs of $\Gamma$. Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$, and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $S_n$ with $\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$, where $\mathrm{supp}(\tau)$ is the set of points in $[n]$ non-fixed by $\tau$.