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 $w$ .

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.

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, $t \geq 2$		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

Distance-regular graph

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