Distance-transitive graph

A distance-transitive graph is interesting partly because it has a large automorphism group.

Some first examples of families of distance-transitive graphs include: After introducing them in 1971, Biggs and Smith showed that there are only 12 finite connected trivalent distance-transitive graphs.

These are: Every distance-transitive graph is distance-regular, but the converse is not necessarily true.

In 1969, before publication of the Biggs–Smith definition, a Russian group led by Georgy Adelson-Velsky showed that there exist graphs that are distance-regular but not distance-transitive.

Complete lists of distance-transitive graphs are known for some degrees larger than three, but the classification of distance-transitive graphs with arbitrarily large vertex degree remains open.