In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices and at any distance , and any other two vertices and at the same distance, there is an automorphism of the graph that carries to and to. 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 is2.

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.

References

1. Early works

2. .

3. .

4. .

5. .

6. .

7. Surveys

8. , chapter 20.

9. .

10. , chapter 7.

11. .

12. , section 4.5.

13. .

External links