An orbital graph is a graph obtained from an orbital of a permutation group on pairs of points. More
precisely, let a transitive permutation
group 
act on a finite set 
. The induced diagonal group
action on the cartesian product 
sends 
to 
for 
. An orbit
 |
is an orbital, and the directed graph with vertex set 
and arc set 
is the corresponding orbital
digraph (Lauri and Scapellato 2016).
The paired orbital is 
. If 
, then the orbital is self-paired,
and opposite arcs can be identified to give the edges of an undirected
graph. Equivalently, an orbit of 
on unordered two-element subsets
of 
is the edge
set of an orbital graph. For an orbital that is not self-paired,
the union 
similarly gives an undirected graph, while retaining
only 
gives a directed graph.
The diagonal orbital 
is called trivial and is normally
omitted when constructing orbital graphs that are simple
graphs. Every nontrivial undirected orbital graph is vertex-transitive
and edge-transitive under the action of
. When its orbital is self-paired,
it is also arc-transitive under this action.
