Distance-transitive graph

In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices $v$ and $w$ at any distance $i$ , and any other two vertices $x$ and $y$ at the same distance, there is an automorphism of the graph that carries $v$ to $x$ and $w$ to $y$ . Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith.

A distance-transitive graph is interesting partly because it has a large automorphism group. Some interesting finite groups are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2.

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