Distance regular graph

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

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).

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.

It turns out that a graph ${\displaystyle G}$ of diameter ${\displaystyle d}$ is distance-regular if and only if there is an array of integers ${\displaystyle \{b_{0},b_{1},\cdots ,b_{d-1};c_{1},\cdots ,c_{d}\}}$ such that for all ${\displaystyle 1\leq j\leq d}$, ${\displaystyle b_{j}}$ 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 ${\displaystyle j}$ on ${\displaystyle G}$. The array of integers characterizing a distance-regular graph is known as its intersection array.

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