# Distance-regular graph

In mathematics, a **distance-regular graph** is a regular graph such that for any two vertices *v* and *w*, the number of vertices at distance *j* from *v* and at distance *k* from *w* depends only upon *j*, *k*, and *i = d(v, w)*.

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 |

Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group.

## Intersection arrays

It turns out that a graph of diameter is distance-regular if and only if there is an array of integers such that for all , gives the number of neighbours of at distance from and gives the number of neighbours of at distance from for any pair of vertices and at distance on . The array of integers characterizing a distance-regular graph is known as its **intersection array**.

### Cospectral distance-regular graphs

A pair of connected distance-regular graphs are cospectral if and only if they have the same intersection array.

A distance-regular graph is disconnected if and only if it is a disjoint union of cospectral distance-regular graphs.

## Properties

Suppose is a connected distance-regular graph of valency with intersection array . For all : let denote the -regular graph with adjacency matrix formed by relating pairs of vertices on at distance , and let denote the number of neighbours of at distance from for any pair of vertices and at distance on .

### Graph-theoretic properties

- for all .
- and .

### Spectral properties

- for any eigenvalue multiplicity of , unless is a complete multipartite graph.
- for any eigenvalue multiplicity of , unless is a cycle graph or a complete multipartite graph.
- if is a simple eigenvalue of .
- has distinct eigenvalues.

If is strongly regular, then and .

## Examples

Some first examples of distance-regular graphs include:

- The complete graphs.
- The cycles graphs.
- The odd graphs.
- The Moore graphs.
- The collinearity graph of a regular near polygon.
- The Wells graph and the Sylvester graph.
- Strongly regular graphs of diameter .

## Classification of distance-regular graphs

There are only finitely many distinct connected distance-regular graphs of any given valency .[1]

Similarly, there are only finitely many distinct connected distance-regular graphs with any given eigenvalue multiplicity [2] (with the exception of the complete multipartite graphs).

### Cubic distance-regular graphs

The cubic distance-regular graphs have been completely classified.

The 13 distinct cubic distance-regular graphs are K_{4} (or tetrahedron), K_{3,3}, the Petersen graph, the cube, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the dodecahedron, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.

## References

- Bang, S.; Dubickas, A.; Koolen, J. H.; Moulton, V. (2015-01-10). "There are only finitely many distance-regular graphs of fixed valency greater than two".
*Advances in Mathematics*.**269**(Supplement C): 1–55. arXiv:0909.5253. doi:10.1016/j.aim.2014.09.025. - Godsil, C. D. (1988-12-01). "Bounding the diameter of distance-regular graphs".
*Combinatorica*.**8**(4): 333–343. doi:10.1007/BF02189090. ISSN 0209-9683.

## Further reading

- Godsil, C. D. (1993).
*Algebraic combinatorics*. Chapman and Hall Mathematics Series. New York: Chapman and Hall. pp. xvi+362. ISBN 978-0-412-04131-0. MR 1220704.