# Distance-transitive graph

Graph families defined by their automorphisms
distance-transitive distance-regular strongly regular
symmetric (arc-transitive) t-transitive, t ≥ 2 skew-symmetric
(if connected)
vertex- and edge-transitive
edge-transitive and regular edge-transitive
vertex-transitive regular (if bipartite)
biregular
Cayley graph zero-symmetric asymmetric

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:

Graph name Vertex count Diameter Girth Intersection array
Tetrahedral graph or complete graph K4 4 1 3 {3;1}
complete bipartite graph K3,3 6 2 4 {3,2;1,3}
Petersen graph 10 2 5 {3,2;1,1}
Cubical graph 8 3 4 {3,2,1;1,2,3}
Heawood graph 14 3 6 {3,2,2;1,1,3}
Pappus graph 18 4 6 {3,2,2,1;1,1,2,3}
Coxeter graph 28 4 7 {3,2,2,1;1,1,1,2}
Tutte–Coxeter graph 30 4 8 {3,2,2,2;1,1,1,3}
Dodecahedral graph 20 5 5 {3,2,1,1,1;1,1,1,2,3}
Desargues graph 20 5 6 {3,2,2,1,1;1,1,2,2,3}
Biggs-Smith graph 102 7 9 {3,2,2,2,1,1,1;1,1,1,1,1,1,3}
Foster graph 90 8 10 {3,2,2,2,2,1,1,1;1,1,1,1,2,2,2,3}

## 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.