Analysing finite locally 𝑠-arc transitive graphs

Abstract

Abstract. We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G and are either locally (G, s)-arc transitive for s ≥ 2 or G-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G. Given a normal subgroup N which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of N preserves both local primitivity and local s-arc transitivity and leads us to study graphs where G acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for 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.

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“…Before proceeding, we need the following definition. (The normal quotient technique is a very important idea introduced in [13] which has proven useful in the investigation of many graphs [8,11].) Definition 4.2.…”

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“…(2) The triple (Γ , n, AutΓ ) lies in Table 1. [10] for the definition of the coset graph). However, it seems infeasible to determine all the possible values of p (and so the corresponding symmetric graphs Γ ) for general square-free integer n.…”

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“…The study of transitive graphs is a natural topic of algebraic graph theory. Basic references are and ; see also , and .…”