Distance-transitive graph

This article defines a property that can be evaluated to true/false for any undirected graph, and is invariant under graph isomorphism. Note that the term "undirected graph" as used here does not allow for loops or parallel edges, so there can be at most one edge between two distinct vertices, the edge is completely described by the vertices it joins, and there can be no edge from a vertex to itself.
View other such properties

Definition

An undirected graph $G$ is termed a distance-transitive graph if the following is true for every positive integer $i$: given vertices $v,w$ of $G$ at distance $i$ from each other and vertices $x,y$ of $G$ also at distance $i$ from each other, there is a graph automorphism of $G$ that sends $v$ to $x$ and $w$ to $y$.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
symmetric graph	automorphism group is transitive on ordered pairs of adjacent vertices			|FULL LIST, MORE INFO
edge-transitive graph	automorphism group is transitive on the edge set			|FULL LIST, MORE INFO
vertex-transitive graph	automorphism group is transitive on the vertex set			|FULL LIST, MORE INFO
regular graph	all vertices have the same degree			|FULL LIST, MORE INFO