Distance-transitive graph

In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices "v" and "w" at any distance "i", and any other two vertices "x" and "y" at the same distance, there is an automorphism of the graph that carries "v" to "x" and "w" to "y". Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith.
A distance-transitive graph is interesting partly because it has a large automorphism group. Some interesting finite groups are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2.
Examples.
Some first examples of families of distance-transitive graphs include:
Classification of cubic distance-transitive graphs.
After introducing them in 1971, Biggs and Smith showed that there are only 12 finite trivalent distance-transitive graphs. These are:
Relation to distance-regular graphs.
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. The only graph of this type with degree three is the 126-vertex Tutte 12-cage. The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph. 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.