Three local actions in 6‐valent arc‐transitive graphs

Abstract

AbstractIt is known that there are precisely three transitive permutation groups of degree 6 that admit an invariant partition with three parts of size 2 such that the kernel of the action on the parts has order 4; these groups are called , and . For each , we construct an infinite family of finite connected 6‐valent graphs and arc‐transitive groups such that the permutation group induced by the action of the vertex‐stabiliser on the neighbourhood of a vertex is permutation isomorphic to , and such that is exponential in . These three groups were the only transitive permutation groups of degree at most 7 for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic 2‐arc‐transitive graphs such that the dimension of the 1‐eigenspace over the field of order 2 of the adjacency matrix of the graph grows linearly with the order of the graph.