Normal quotients of diameter at most two of finite three-geodesic-transitive graphs

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 Gamma 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 Gamma has girth at least 6 then all 3-arcs are 3-geodesics so Gamma 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 Gamma(N) modulo the orbits of a normal subgroup N of G and prove that, provided Gamma(N) has diameter at least 3, then Gamma is a cover of Gamma(N) and Gamma(N), G/N have the same girth and transitivity properties as Gamma, G (so if N not equal 1 we may reduce consideration to a smaller graph in the family). We then focus on the 'degenerate case' where Gamma(N) has diameter at most 2. In these cases also, Gamma is a cover of Gamma(N) provided N has at least three vertex-orbits. If Gamma(N) is a complete graph K-r (diameter 1), then we prove that Gamma is either the complete bipartite graph K-r,K-r with the edges of a perfect matching removed, or a unique 6-fold-cover of K-7. In the remaining case where Gamma(N) has diameter 2, then Gamma(N) 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. (C) 2020 Elsevier Inc. All rights reserved.