Finite locally-quasiprimitive graphs

A finite graph is said to be locally-quasiprimitive relative to a subgroup G of automorphisms if, for all vertices α, the stabiliser in G of α is quasiprimitive on the set of vertices adjacent to α. (A permutation group is said to be quasiprimitive if all of its non-trivial normal subgroups are transitive.) The graph theoretic condition of local quasiprimitivity is strictly weaker than the conditions of local primitivity and 2-arc transitivity which have been studied previously. It is shown that the family of locally-quasiprimitive graphs is closed under the formation of a certain kind of quotient graph, called a normal quotient, induced by a normal subgroup. Moreover, a locally-quasiprimitive graph is proved to be a multicover of each of its normal quotients. Thus finite locally-quasiprimitive graphs which are minimal in the sense that they have no non-trivial proper normal quotients form an important sub-family, since each finite locally-quasiprimitive graph has at least one such graph as a normal quotient. These minimal graphs in the family are called "basic" locally-quasiprimitive graphs, and their structure is analysed. The process of constructing locally-quasiprimitive graphs with a given locally-quasiprimitive graph Σ as a normal quotient is then considered. It turns out that this can be viewed as a problem of constructing covering graphs of certain multigraphs associated with Σ. Further, it is shown that, under certain conditions, a locally-quasiprimitive graph can be reconstructed from knowledge of two of its normal quotients. Finally a series of open problems is presented.