Distance-regular graph

In the mathematical field of graph theory, 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 the distance between v and 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

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 $(b_{0},b_{1},\ldots ,b_{d-1};c_{1},\ldots ,c_{d})$ in which $d$ is the diameter of the graph and for each $1\leq j\leq d$, $b_{j}$ gives the number of neighbours of $u$ at distance $j+1$ from $v$ and $c_{j}$ gives the number of neighbours of $u$ at distance $j-1$ from $v$ for any pair of vertices $u$ and $v$ at distance $j$. There is also the number $a_{j}$ that gives the number of neighbours of $u$ at distance $j$ from $v$. The numbers $a_{j},b_{j},c_{j}$ are called the intersection numbers of the graph. They satisfy the equation $a_{j}+b_{j}+c_{j}=k,$ where $k=b_{0}$ is the valency, i.e., the number of neighbours, of any vertex.

It turns out that a graph $G$ of diameter $d$ 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 $G$ is a connected distance-regular graph of valency $k$ with intersection array $(b_{0},b_{1},\ldots ,b_{d-1};c_{1},\ldots ,c_{d})$. For each $0\leq j\leq d,$ let $k_{j}$ denote the number of vertices at distance $k$ from any given vertex and let $G_{j}$ denote the $k_{j}$-regular graph with adjacency matrix $A_{j}$ formed by relating pairs of vertices on $G$ at distance $j$.

Graph-theoretic properties

• ${\frac {k_{j+1}}{k_{j}}}={\frac {b_{j}}{c_{j+1}}}$ for all $0\leq j<d$.
• $b_{0}>b_{1}\geq \cdots \geq b_{d-1}>0$ and $1=c_{1}\leq \cdots \leq c_{d}\leq b_{0}$.

Spectral properties

• $G$ has $d+1$ distinct eigenvalues.
• The only simple eigenvalue of $G$ is $k,$ or both $k$ and $-k$ if $G$ is bipartite.
• $k\leq {\frac {1}{2}}(m-1)(m+2)$ for any eigenvalue multiplicity $m>1$ of $G,$ unless $G$ is a complete multipartite graph.
• $d\leq 3m-4$ for any eigenvalue multiplicity $m>1$ of $G,$ unless $G$ is a cycle graph or a complete multipartite graph.

If $G$ is strongly regular, then $n\leq 4m-1$ and $k\leq 2m-1$.

Examples

The degree 7 Klein graph and associated map embedded in an orientable surface of genus 3. This graph is distance regular with intersection array {7,4,1;1,2,7} and automorphism group PGL(2,7).

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 of diameter $2$.

Classification of distance-regular graphs

There are only finitely many distinct connected distance-regular graphs of any given valency $k>2$.

Similarly, there are only finitely many distinct connected distance-regular graphs with any given eigenvalue multiplicity $m>2$ (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 K₄ (or Tetrahedral graph), 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.

Further reading

• Godsil, C. D. (1993). Algebraic Combinatorics. Chapman and Hall Mathematics Series. New York: Chapman and Hall. ISBN 978-0-412-04131-0. MR 1220704.