→
{displaystyle {oldsymbol {
ightarrow }}} distance-regular ←
{displaystyle {oldsymbol {leftarrow }}} t-transitive, t ≥ 2 | ←
{displaystyle {oldsymbol {leftarrow }}} strongly regular →
{displaystyle {oldsymbol {
ightarrow }}} edge-transitive | |
 | ||
distance-transitive →
{displaystyle {oldsymbol {
ightarrow }}} →
{displaystyle {oldsymbol {
ightarrow }}} edge-transitive and regular | ||
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.
A distance transitive graph is vertex transitive and symmetric as well as distance regular.
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.
Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith, who showed that there are only 12 finite trivalent distance-transitive graphs. These are:
Independently in 1969 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.
The simplest asymptotic family of examples of distance-transitive graphs is the Hypercube graphs. Other families are the folded cube graphs and the square rook's graphs. All three of these families have arbitrarily high degree.