Graph Families Defined By Their Automorphisms
Several families of graphs are defined by having certain types of automorphisms:
- An asymmetric graph is an undirected graph without any nontrivial automorphisms.
- A vertex-transitive graph is an undirected graph in which every vertex may be mapped by an automorphism into any other vertex.
- An edge-transitive graph is an undirected graph in which every edge may be mapped by an automorphism into any other edge.
- A symmetric graph is a graph such that every pair of adjacent vertices may be mapped by an automorphism into any other pair of adjacent vertices.
- A distance-transitive graph is a graph such that every pair of vertices may be mapped by an automorphism into any other pair of vertices that are the same distance apart.
- A semi-symmetric graph is a graph that is edge-transitive but not vertex-transitive.
- A half-transitive graph is a graph that is vertex-transitive and edge-transitive but not symmetric.
- A skew-symmetric graph is a directed graph together with a permutation σ on the vertices that maps edges to edges but reverses the direction of each edge. Additionally, σ is required to be an involution.
Inclusion relationships between these families are indicated by the following table:
| distance-transitive | distance-regular | strongly regular | ||
| symmetric (arc-transitive) | t-transitive, t ≥ 2 | |||
| (if connected) | ||||
| vertex- and edge-transitive | edge-transitive and regular | edge-transitive | ||
| vertex-transitive | regular | |||
| Cayley graph |
Read more about this topic: Graph Automorphism
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