Английская Википедия:Distance-transitive graph
Шаблон:Graph families defined by their automorphisms
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices Шаблон:Mvar and Шаблон:Mvar at any distance Шаблон:Mvar, and any other two vertices Шаблон:Mvar and Шаблон:Mvar at the same distance, there is an automorphism of the graph that carries Шаблон:Mvar to Шаблон:Mvar and Шаблон:Mvar to Шаблон:Mvar. 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.
- The folded cube graphs.
- The square rook's graphs.
- The hypercube 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:
| 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.
References
- Early works
- Surveys
- Шаблон:Citation, chapter 20.
- Шаблон:Citation.
- Шаблон:Citation, chapter 7.
- Шаблон:Citation.
- Шаблон:Citation, section 4.5.
- Шаблон:Citation.
External links
