An orbital graph treats an ordered-pair orbit as directed edges, translating primitivity and subdegree questions into connectivity and valency. [@isaacs2008]
Definition
The paired orbital \(\Delta^*\) reverses every arc. If \(\Delta=\Delta^*\), the graph can be viewed as undirected.
Core result
Components of a disconnected graph form blocks under a vertex-transitive automorphism group. Conversely, choose two points in one block; every arc in their orbital remains inside blocks, so the graph is disconnected.
Structural properties
- \(G\) is a vertex-transitive automorphism group of every orbital graph.
- The out-valency is the corresponding subdegree.
- Connected components produce the smallest block containing a chosen suborbit.
- Paired orbitals exchange in- and out-valencies, which are equal in the finite transitive setting.
- Common-divisor graphs of subdegrees restrict possible local valency sets.
Orbital graphs let graph paths, components, and local degrees encode group-action structure.
Example and boundary
Knowledge network
Suborbits give neighborhoods, primitivity is characterized by connectivity, and the permutation core article constructs orbitals.
Proof and sources
Orbitals, connectivity, and subdegree-graph results are Theorems 8.34-8.43. [@isaacs2008]
