论文标题
$ \ k_ {2^n，2^n} $的Edge-affine $ 2 $ -2 $ -ARC-TRANSTIVE盖的特征
A characterisation of edge-affine $2$-arc-transitive covers of $\K_{2^n,2^n}$
论文作者
论文摘要
我们介绍了\ emph {$ n $ dimensional混合二面群}的概念，这是一个通用类的组类，我们为其提供了图形理论表征。特别是，如果$ h $是$ n $二的混合二二二二二二二进制组，那么我们构建了一个边缘转换的cayley图$γ$ $ h $的$ h $，使得clique Graph $σ$的$γ$是$ 2 $ -Arc-arc-arc-arc-arc-arc-arc-arc-arc-arc-transitive proffor $ \ k_ {2^n，2^n} $ a subgref（ \ emph {edge-affine} $ \ k_ {2^n，2^n} $上的操作。相反，我们证明，如果$σ$是$ 2 $ -ARC的正常封面，为$ \ k_ {2^n，2^n} $，则具有$ \ aut（σ）$的子组，诱导\ emph {edge-egraffine}在$ \ k_的$ \ k_ {2^n，2^n} $的$ \ k_ emph {edge-favfine} $ fargrape $ { $ n $二维混合二面体组。此外，我们对$ n $ dimensional混合二面群的家族进行了明确的结构。该家庭解决了李提出的问题，即“基本” $ 2 $ -2 $ -ARC传递图的普通电源封面。特别是，我们为每个$ n \ geq 2 $构造了$ 2 $ -ARC的传播封面$ 2 $ -2 $ - 'BASIC'Graph $ \ k_ {2^n，2^n} $的订单。
We introduce the notion of an \emph{$n$-dimensional mixed dihedral group}, a general class of groups for which we give a graph theoretic characterisation. In particular, if $H$ is an $n$-dimensional mixed dihedral group then the we construct an edge-transitive Cayley graph $Γ$ of $H$ such that the clique graph $Σ$ of $Γ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducing a particular \emph{edge-affine} action on $\K_{2^n,2^n}$. Conversely, we prove that if $Σ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducing an \emph{edge-affine} action on $\K_{2^n,2^n}$, then the line graph $Γ$ of $Σ$ is a Cayley graph of an $n$-dimensional mixed dihedral group. Furthermore, we give an explicit construction of a family of $n$-dimensional mixed dihedral groups. This family addresses a problem proposed by Li concerning normal covers of prime power order of the `basic' $2$-arc-transitive graphs. In particular, we construct, for each $n\geq 2$, a $2$-arc-transitive normal cover of $2$-power order of the `basic' graph $\K_{2^n,2^n}$.
