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.
Some first examples of families of distance-transitive graphs include:
After introducing them in 1971, Biggs and Smith showed that there are only 12 finite connected 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} |
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.