A graph automorphism of a graph 
is a permutation of its vertices that preserves the graph's
adjacency relation. Equivalently, it is a graph isomorphism from 
to itself. The automorphisms of 
form a permutation group
known as its graph automorphism group.
In the Wolfram Language, the automorphisms of a graph may be obtained by applying GroupElements to the result of GraphAutomorphismGroup[g]. Precomputed automorphisms for many named graphs can be obtained using GraphData[graph, "Automorphisms"].
For example, the grid graph 
has four automorphisms: (1, 2, 3, 4, 5, 6), (2, 1, 4,
3, 6, 5), (5, 6, 3, 4, 1, 2), and (6, 5, 4, 3, 2, 1). These correspond to the graph itself, the graph flipped
left-to-right, the graph flipped up-down, and the graph
flipped left-to-right and up-down, respectively, illustrated above. More generally,
as is clear from its symmetry, for positive
integers 
,
 |
(1)
|
Similarly, the star graph 
has six automorphisms: (1, 2, 3, 4), (1, 3, 2, 4), (2, 1,
3, 4), (2, 3, 1, 4), (3, 1, 2, 4), (3, 2, 1, 4), illustrated above. More generally,
as is clear from its symmetry, 
for 
.
