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arXiv · 2304.10633

Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite $2$-arc-transitive graph which is not a Cayley graph

Daniel R. Hawtin·Cheryl E. Praeger·Jin-Xin Zhou
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Abstract

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition such that the automorphism group of the Cayley graph $\Cay(H,(X\cup Y)\setminus\{1\})$ is equal to $H: A(H,X,Y)$, where $A(H,X,Y)$ is the setwise stabiliser in $\Aut(H)$ of $X\cup Y$. We use this criterion to resolve a questions of Li, Ma and Pan from 2009, by constructing a $2$-arc transitive normal cover of order $2^{53}$ of the complete bipartite graph $\K_{16,16}$ and prove that it is \emph{not} a Cayley graph.

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Daniel R. Hawtin, Cheryl E. Praeger, Jin-Xin Zhou. 2023-04-20. Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite $2$-arc-transitive graph which is not a Cayley graph. https://arxiv.org/abs/2304.10633

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