On geodesic transitive graphs

The main purpose of this paper is to investigate relationships between three graph symmetry properties: s-arc transitivity, s-geodesic transitivity, and s-distance transitivity. A well-known result of Weiss tells us that if a graph of valency at least 3 is s-arc transitive then s≤7. We show that for each value of s≤3, there are infinitely many s-arc transitive graphs that are t-geodesic transitive for arbitrarily large values of t. For 4≤s≤7, the geodesic transitive graphs that are s-arc transitive can be explicitly described, and all but two of these graphs are related to classical generalized polygons. Finally, we show that the Paley graphs and the Peisert graphs, which are known to be distance transitive, are almost never 2-geodesic transitive, with just three small exceptions.