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Distance-transitive graph

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Graph where any two nodes of equal distance are isomorphic
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The Biggs-Smith graph, the largest 3-regular distance-transitive graph.
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive 		→ edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric

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:

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
  • 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.

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