Edge-transitive: automorphism group acts transitively on edges. Every edge 'looks the same'. K_n: both VT and ET. Semi-symmetric: ET but NOT VT (rare!). Smallest: Folkman graph (20 vertices). Edge-transitive ≠ vertex-transitive in general.
How to Use
Select graph:
- Test: ET?
- Orbits: Edge orbit 1
- Compare: ET vs VT
Semi-Symmetric
A graph is semi-symmetric if edge-transitive, regular, but NOT vertex-transitive. Very rare! Smallest: Folkman graph (1967, 20 vertices). Must be bipartite. Each partition looks different but edges between them are symmetric.
Symmetry Classification
VT+ET = symmetric (arc-transitivity sometimes needed). ET only: semi-symmetric on regular, otherwise 'edge-transitive but not VT'. VT only: vertex-transitive but not edge-transitive. Both independent properties.
Step-by-Step Instructions
- 1Select graph.
- 2Compute edge orbits.
- 3Check single orbit.
- 4Compare VT.
- 5Classify symmetry.