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 is 2.
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: 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 , chapter 20. . , chapter 7. . , section 4.5. .
External links