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arXiv 1410.8166math.CO
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具有生成块转置集的对称群上的Cayley图

Cayley graph on symmetric groups with generating block transposition sets

Annachiara Korchmaros

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摘要：

本文研究了Cayley图$\Cay,$，其中生成集由所有的块转置组成.研究这些特殊凯莱图的动机来自生物信息学的当前研究。证明了${\rm{Aut}}(\Cay)$是右平移群与$\textsf{N}\rtimes \textsf{D}_{n+1},$的乘积，其中$\textsf{N}$是按元素固定$S_n$的子群，$\textsf{D}_{n+1}$是$2(n+1)$阶二面体群.我们猜想$\textsf{N}$是平凡的。证明了顶点集为$S_n$的子图$Γ$是$2(n-2)$ -正则图，其自同构群为$\textsf{D}_{n+1}$ .此外，$Γ$具有与$n+1$一样多的大小为$2.$的最大团。此外，其顶点是这些团中的顶点的子图$Γ(V)$是$3$ -正则、汉密尔顿和顶点传递图。

英文摘要：

This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that ${\rm{Aut}}(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing $S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order $2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose automorphism group is $\textsf{D}_{n+1}$. Furthermore, $Γ$ has as many as $n+1$ maximum cliques of size $2.$ Also, its subgraph $Γ(V)$ whose vertices are those in these cliques is a $3$-regular, Hamiltonian, and vertex-transitive graph.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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