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 uses the same 
in both coordinates. On the
Cartesian product 
, it sends 
to 
for 
. A group 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 group
action of 
.
When its orbital is self-paired,
it is also arc-transitive under this action.
