Distance-transitive graph: Difference between revisions

The Biggs–Smith graph, the largest 3-regular distance-transitive graph.

In mathematics, 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.

A distance transitive graph is vertex transitive and symmetric as well as distance regular.

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.

Distance-transitive graphs were first defined in 1971 by Norman Biggs and D. H. Smith, who showed that there are only 12 finite trivalent distance-transitive graphs. These are:

Graph name	Vertex count	Diameter	Girth	Intersection array
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}
Graph of the cube	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}
Graph of the dodecahedron	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}

Independently in 1969 a Russian group led by Georgy Adelson-Velsky showed that there exist graphs that are distance-regular but not distance-transitive. 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.

Square rook's graphs provide examples of distance-transitive graphs of arbitrarily high degree.

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", Dokl. Akad. Nauk SSSR, 185: 975–976, MR0244107.

• Biggs, Norman (1971), "Intersection matrices for linear graphs", Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), London: Academic Press, pp. 15–23, MR0285421.

• Biggs, Norman (1971), Finite Groups of Automorphisms, London Mathematical Society Lecture Note Series, vol. 6, London & New York: Cambridge University Press, MR0327563.

• Biggs, N. L.; Smith, D. H. (1971), "On trivalent graphs", Bulletin of the London Mathematical Society, 3: 155–158, doi:10.1112/blms/3.2.155, MR0286693.

• Smith, D. H. (1971), "Primitive and imprimitive graphs", The Quarterly Journal of Mathematics, Oxford, Second Series, 22: 551–557, doi:10.1093/qmath/22.4.551, MR0327584.

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, MR2287450.

• 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.; Woldar, A. J. (eds.), The Algebraic Theory of Combinatorial Objects, Math. Appl. (Soviet Series), vol. 84, Dordrecht: Kluwer, pp. 283–378, MR1321634.

External links

• Weisstein, Eric W. "Distance-Transitive Graph". MathWorld.