6-Transitive graphs
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TL;DR
It is shown that a 6-transitive graph is complete multipartite, or complete bipartite with a matching deleted, or a cycle, or one of three special graphs on 9, 12 and 20 vertices.
Abstract
A connected graph is n-transitive if, whenever two n-tuples are isometric, there is an automorphism mapping the first to the second. It is shown that a 6-transitive graph is complete multipartite, or complete bipartite with a matching deleted, or a cycle, or one of three special graphs on 9, 12 and 20 vertices. These graphs are n-transitive for all n; but there are graphs (the smallest on 56 vertices) which are 5- but not 6-transitive.