Analysing finite locally 𝑠-arc transitive graphs

Abstract

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G G and are either locally ( G , s ) (G,s) –arc transitive for s ≥ 2 s \geq 2 or G G –locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G G . Given a normal subgroup N N which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of N N preserves both local primitivity and local s s –arc transitivity and leads us to study graphs where G G acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for G G in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.