Graph Automorphism - Graph Families Defined By Their Automorphisms

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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