# 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.

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 |

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.
- The folded cube graphs.
- The square rook's graphs.
- The hypercube graphs.

## Classification of cubic distance-transitive graphs

After introducing them in 1971, Biggs and Smith showed that there are only 12 finite trivalent distance-transitive graphs. These are:

Graph name | Vertex count | Diameter | Girth | Intersection array |
---|---|---|---|---|

complete graph K_{4} | 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} |

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} |

## 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 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.

## 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. (eds.),
*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. (eds.),
*The Algebraic Theory of Combinatorial Objects*, Math. Appl. (Soviet Series),**84**, Dordrecht: Kluwer, pp. 283–378, MR 1321634.