Distance-transitive graph

The Biggs-Smith graph, the largest 3-regular distance-transitive graph.
 Graph families defined by their automorphisms distance-transitive ${\displaystyle {\boldsymbol {\rightarrow }}}$ distance-regular ${\displaystyle {\boldsymbol {\leftarrow }}}$ strongly regular ${\displaystyle {\boldsymbol {\downarrow }}}$ symmetric (arc-transitive) ${\displaystyle {\boldsymbol {\leftarrow }}}$ t-transitive, t ≥ 2 skew-symmetric ${\displaystyle {\boldsymbol {\downarrow }}}$ (if connected) vertex- and edge-transitive ${\displaystyle {\boldsymbol {\rightarrow }}}$ edge-transitive and regular ${\displaystyle {\boldsymbol {\rightarrow }}}$ edge-transitive ${\displaystyle {\boldsymbol {\downarrow }}}$ ${\displaystyle {\boldsymbol {\downarrow }}}$ ${\displaystyle {\boldsymbol {\downarrow }}}$ vertex-transitive ${\displaystyle {\boldsymbol {\rightarrow }}}$ regular ${\displaystyle {\boldsymbol {\rightarrow }}}$ (if bipartite) biregular ${\displaystyle {\boldsymbol {\uparrow }}}$ Cayley graph ${\displaystyle {\boldsymbol {\leftarrow }}}$ 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.

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 L. 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. The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph. 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.

The simplest asymptotic family of examples of distance-transitive graphs is the Hypercube graphs. Other families are the folded cube graphs and the square rook's graphs. All three of these families have 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", 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, 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. Oxford. 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., Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications, 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., The Algebraic Theory of Combinatorial Objects, Math. Appl. (Soviet Series), 84, Dordrecht: Kluwer, pp. 283–378, MR 1321634.