Finite vertex-primitive edge-transitive metacirculants.
Let \(\Gamma=(V,E)\) be a connected, undirected, simple graph. Then \(\Gamma\) is called a metacirculant if its automorphism group \(\Aut(\Gamma)\) contains a metacyclic group \(R\) which is transitive on the vertex set \(V\) (that is, \(R\) is a cyclic extension of a cyclic normal subgroup). It is a long standing problem to characterize metacirculants for which \(\Aut(\Gamma)\) is transitive on the edge set \(E\). The main result of the present paper is to solve this problem in the case that \(\Aut(\Gamma)\) acts primitively on \(V\). More precisely, assume that \(G\) is a subgroup of \(\Aut(\Gamma)\) such that \(R\leq G\) and that \(G\) is both transitive on \(E\) and primitive on \(V\), then there are nine classes of metacirculants which can arise. In particular, \(\Gamma\) is a Cayley graph except in the following cases: (i) \(\Gamma\) is the line graph of the complete graph \(K_p\) (or the complement of this line graph) for some prime \(p\equiv 1\pmod 4\); (ii) \(R=\mathbb Z_{17}:\mathbb Z_8\) and \(\Aut(\Gamma)=\mathrm{P}\Gamma\)L
scientific article; zbMATH DE number 6371198
scientific article