Vertex-transitive: automorphism group acts transitively on vertices. Every vertex 'looks the same'. All Cayley graphs are vertex-transitive. Vertex-transitive → regular (all degrees equal). Petersen graph is vertex-transitive but not a Cayley graph.
How to Use
Select graph:
- Test: VT?
- Orbits: One orbit
- Aut: Group size
Cayley Graphs
Cayley graph Cay(G,S): vertices = group elements, edges from generators S. Always vertex-transitive (group acts on itself). Not all VT graphs are Cayley: Petersen is VT but not Cayley. Characterizing Cayley graphs is hard.
Properties
VT → regular (constant degree). VT → α(G) ≥ n/χ(G) (Lovász bound). VT graphs have 'uniform' structure: same local neighborhoods everywhere. Used in network design for fault tolerance.
Step-by-Step Instructions
- 1Select graph.
- 2Compute orbits.
- 3Check single orbit.
- 4Find |Aut|.
- 5Classify symmetry.