# Distance-transitive graph

## Graph where any two nodes of equal distance are isomorphic / From Wikipedia, the free encyclopedia

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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.

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (September 2021) |

**Quick Facts**Graph families defined by their automorphisms, → ...

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 |

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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.