Abstract
An s-geodesic of a graph is a path of length s such that the first and last vertices are at distance s. We study finite graphs Γ of diameter at least 3 for which some subgroup G of automorphisms is transitive on the set of s-geodesics for each s≤3. If Γ has girth at least 6 then all 3-arcs are 3-geodesics so Γ is 3-arc-transitive, and such graphs have already been studied fruitfully; also graphs of girth 3 with these properties have been investigated successfully. We therefore focus on those of girth 4 or 5. We study their normal quotients ΓN modulo the orbits of a normal subgroup N of G and prove that, provided ΓN has diameter at least 3, then Γ is a cover of ΓN and ΓN,G/N have the same girth and transitivity properties as Γ,G (so if N≠1 we may reduce consideration to a smaller graph in the family).We then focus on the ‘degenerate case’ where ΓN has diameter at most 2. In these cases also, Γ is a cover of ΓN provided N has at least three vertex-orbits. If ΓN is a complete graph Kr (diameter 1), then we prove that Γ is either the complete bipartite graph Kr,r with the edges of a perfect matching removed, or a unique 6-fold-cover of K7. In the remaining case where ΓN has diameter 2, then ΓN is a 2-arc-transitive strongly regular graph. We classify all the 2-arc-transitive strongly regular graphs, and using this classification we describe all their finite (G,3)-geodesic-transitive covers of girth 4 or 5, except for a few difficult cases.
Published Version
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