The vertex primitive and vertex bi-primitive \(s\)-arc regular graphs
The goal of the paper is to determine all vertex primitive and vertex bi-primitive \(3\)-arc regular graphs. This result completes the classification of vertex primitive \(s\)-arc regular graphs with \(s\geq 2\) and the classification of vertex bi-primitive \(s\)-arc regular graphs with \(s\geq 3\). Let \(\Gamma\) be an undirected connected graph. An \(s\)-arc of \(\Gamma\) is a \((s+1)\)-tuple \((\alpha_0,\ldots,\alpha_s)\) of vertices of \(\Gamma\) such that \(\alpha_{i-1}\) and \(\alpha_i\) are adjacent for \(1\leq i\leq s\), and \(\alpha_{i-1}\neq\alpha_{i+1}\) for \(1\leq i\leq s-1\). If \(\Gamma\) is regular of valence at least \(3\), and \(G\leq\mathrm{Aut}(\Gamma)\) is transitive (respectively regular) on the set of \(s\)-arcs, then \(\Gamma\) is called \((G,s)\)-arc transitive (resp. arc-regular). The reference to \(G\) is avoided when \(G=\mathrm{Aut}(\Gamma)\). A permutation group \(G\) on \(V\) is bi-primitive if \(G\) is transitive and imprimitive and \(V\) has only one nontrivial \(G\)-invariant partition, with exactly two blocks. The paper has three main theorems. In the first, the structure of the vertex stabilizers in \(s\)-arc regular graphs is given. The second states that if \(\Gamma\) is a connected \((G,3)\)-arc regular graph such that \(G\) is primitive on the vertex set, then either {\parindent=7mm \begin{itemize}\item[(i)]\(\Gamma\) is the Petersen graph and \(\mathrm{Aut}(\Gamma)=G\simeq PGL(2,5)\), or \item[(ii)]\(\Gamma\) is the Coxeter graph and \(\mathrm{Aut}(\Gamma)=G\simeq PGL(2,7)\). \end{itemize}} The third one gives the structure of connected \((G,3)\)-arc regular graphs such that \(G\) is bi-primitive on the vertex set. It is shown that there are the following four possibilities: {\parindent=7mm \begin{itemize}\item[(i)]\(\Gamma=K_{p^d,p^d}\), where \(p\) is a prime; \item[(ii)]\(\Gamma\) is the standard double cover of the Petersen graph or the Coxeter graph, and \(G\simeq PGL(2,5)\times{\mathbb Z}_2\) or \(PGL(2,7)\times\mathbb{Z}_2\), respectively; \item[(iii)]\(\Gamma\) is cubic, and \(G\simeq PGL(2,11)\) or \(PGL(2,13)\); \item[(iv)]\(\Gamma\) is of valence \(4\), and \(G\simeq P\Gamma L(2,27)\). \end{itemize}} Moreover, in cases (ii)-(iv), \(\mathrm{Aut}(\Gamma)=G\) and \(\Gamma\) is \(3\)-arc regular. In case (i) \(\Gamma\) is \(3\)-arc regular if and only if \(p^d=3\).
scientific article; zbMATH DE number 5716643
scientific article