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

It turns out that a graph ${\displaystyle G}$  of diameter ${\displaystyle d}$  is distance regular if and only if it has an intersection array in the preceding sense.

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.

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

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

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

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

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 ${\displaystyle 2}$ .

There are only finitely many distinct connected distance-regular graphs of any given valency ${\displaystyle k>2}$ .^[1]

Similarly, there are only finitely many distinct connected distance-regular graphs with any given eigenvalue multiplicity ${\displaystyle m>2}$ ^[2] (with the exception of the complete multipartite 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.

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