Distance-transitive graph

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:

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 K_3,3	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

Adel'son-Vel'skii, G. M.; Veĭsfeĭler, B. Ju.; Leman, A. A.; Faradžev, I. A. (1969), "An example of a graph which has no transitive group of automorphisms", Doklady Akademii Nauk SSSR, 185: 975–976, MR 0244107.
Biggs, Norman (1971), "Intersection matrices for linear graphs", Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), London: Academic Press, pp. 15–23, MR 0285421.
Biggs, Norman (1971), Finite Groups of Automorphisms, London Mathematical Society Lecture Note Series, vol. 6, London & New York: Cambridge University Press, MR 0327563.
Biggs, N. L.; Smith, D. H. (1971), "On trivalent graphs", Bulletin of the London Mathematical Society, 3 (2): 155–158, doi:10.1112/blms/3.2.155, MR 0286693.
Smith, D. H. (1971), "Primitive and imprimitive graphs", The Quarterly Journal of Mathematics, Second Series, 22 (4): 551–557, doi:10.1093/qmath/22.4.551, MR 0327584.

Surveys

Biggs, N. L. (1993), "Distance-Transitive Graphs", Algebraic Graph Theory (2nd ed.), Cambridge University Press, pp. 155–163, chapter 20.
Van Bon, John (2007), "Finite primitive distance-transitive graphs", European Journal of Combinatorics, 28 (2): 517–532, doi:10.1016/j.ejc.2005.04.014, MR 2287450.
Brouwer, A. E.; Cohen, A. M.; Neumaier, A. (1989), "Distance-Transitive Graphs", Distance-Regular Graphs, New York: Springer-Verlag, pp. 214–234, chapter 7.
Cohen, A. M. Cohen (2004), "Distance-transitive graphs", in Beineke, L. W.; Wilson, R. J. (eds.), Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications, vol. 102, Cambridge University Press, pp. 222–249.
Godsil, C.; Royle, G. (2001), "Distance-Transitive Graphs", Algebraic Graph Theory, New York: Springer-Verlag, pp. 66–69, section 4.5.
Ivanov, A. A. (1992), "Distance-transitive graphs and their classification", in Faradžev, I. A.; Ivanov, A. A.; Klin, M.; et al. (eds.), The Algebraic Theory of Combinatorial Objects, Math. Appl. (Soviet Series), vol. 84, Dordrecht: Kluwer, pp. 283–378, MR 1321634.