Distance-regular graph

In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and, the number of vertices at distance from and at distance from depends only upon,, and the distance between and.
Some authors exclude the complete graphs and disconnected graphs from this definition.
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

The intersection array of a distance-regular graph is the array in which is the diameter of the graph and for each, 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. There is also the number that gives the number of neighbours of at distance from. The numbers are called the intersection numbers of the graph. They satisfy the equation where is the valency, i.e., the number of neighbours, of any vertex.
It turns out that a graph of diameter is distance regular if and only if it has an intersection array in the preceding sense.

Cospectral and disconnected distance-regular graphs

A pair of connected distance-regular graphs are cospectral if their adjacency matrices have the same spectrum. This is equivalent to their having 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 each let denote the number of vertices at distance from any given vertex and let denote the -regular graph with adjacency matrix formed by relating pairs of vertices on at distance.

Graph-theoretic properties

for all.
and.
Spectral properties
has distinct eigenvalues.
The only simple eigenvalue of is or both and if is bipartite.
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 strongly regular, then and.

Association scheme

The -distance adjacency matrices for of a distance-regular graph form an association scheme.

Examples

Some first examples of distance-regular graphs include:

The complete graphs.
The cycle 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 are the distance-regular graphs of diameter 2.
Classification of distance-regular graphs

There are only finitely many distinct connected distance-regular graphs of any given valency.
Similarly, there are only finitely many distinct connected distance-regular graphs with any given eigenvalue multiplicity .

Cubic distance-regular graphs

The cubic distance-regular graphs have been completely classified.
The 13 distinct cubic distance-regular graphs are K₄, K_3,3, the Petersen graph, the Cubical graph, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the Dodecahedral graph, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.