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:
 * The Johnson graphs.
 * The Grassmann graphs.
 * The Hamming Graphs (including Hypercube graphs).
 * The folded cube graphs.
 * The square rook's graphs.
 * The Livingstone graph.

Classification of cubic distance-transitive graphs
After introducing them in 1971, Biggs and Smith showed that there are only 12 finite connected 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 smallest distance-regular graph that is not distance-transitive is the Shrikhande graph, with 16 vertices and degree 6. The only graph of this type with degree three is the 126-vertex Tutte 12-cage. 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.