Symmetric 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 is termed a symmetric graph, arc-transitive graph, or flag-transitive graph either if it is the empty graph or the induced action of the automorphism group on the set of ordered pairs of adjacent vertices is transitive.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| empty graph | ||||
| complete graph |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| edge-transitive graph | ||||
| vertex-transitive graph | ||||
| regular graph |