Graph Automorphism

An automorphism of a graph is a graph isomorphism with itself, i.e., a mapping from the vertices of the given graph  back to vertices of  such that the resulting graph is isomorphic with . The set of automorphisms defines a permutation group known as the graph's automorphism group. For every group , there exists a graph whose automorphism group is isomorphic to  (Frucht 1939; Skiena 1990, p. 185). The automorphism groups of a graph characterize its symmetries, and are therefore very useful in determining certain of its properties.

The group of graph automorphisms of a graph  may be computed in the Wolfram Language using GraphAutomorphismGroup[g], the elements of which may then be extracted using GroupElements. A number of software implementations exist for computing graph automorphisms, including nauty by Brendan McKay and SAUCY2, the latter of which performs several orders of magnitude faster than other implementations based on empirical tests (Darga et al. 2008).

Precomputed automorphisms for many named graphs can be obtained using GraphData[graph, "Automorphisms"], and the number of automorphisms using GraphData[graph, "AutomorphismCount"].

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,

(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 .

The following table summarizes  for various classes of graphs.

graph	OEIS	sequence
antiprism graph,	A124354	48, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, ...
complete graph	A000000	1, 2, 6, 24, 120, 720, ...
cycle graph ,	A000000	6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, ...
hypercube graph	A000165	2, 8, 48, 384, 3840, 46080, 645120, 10321920, ...
Möbius ladder,	A000000	72, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, ...
prism graph,	A124351	12, 48, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, ...
wheel graph ,	A000000	24, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, ...

The automorphism group of a graph complement is the same as that for the original graph. A graph possessing only a single automorphism is called an identity graph (Holton and Sheehan 1993, p. 24), or sometimes an asymmetric graph. The triangle of sorted lengths of the automorphism graphs on , 2, ... nodes is given by

(2)

(OEIS A075094).

The number of distinct orders for the automorphisms groups of simple graphs on , 2, ... are 1, 1, 2, 5, 8, 14, 19, 30, 45, ... (OEIS A095348).

The following table gives counts of the numbers of -node simple graphs having given automorphism group orders.

	OEIS	counts of graphs with 1, 2, ... nodes
1	A003400	0, 0, 0, 0, 0, 8, 152, 3696, 135004, ...
2	A075095	0, 2, 2, 3, 11, 46, 354, 4431, 89004, ...
3		0, 0, 0, 0, 0, 0, 0, 0, 4, ...
4	A075096	0, 0, 0, 2, 6, 36, 248, 2264, 31754, ...
6	A075097	0, 0, 2, 2, 2, 8, 38, 252, 3262, ...
8	A075098	0, 0, 0, 2, 4, 14, 74, 623, 7003, ...
10		0, 0, 0, 0, 1, 2, 2, 4, 16, ...
12	A095853	0, 0, 0, 0, 6, 18, 70, 446, 3924, ...
14		0, 0, 0, 0, 0, 0, 2, 4, 4, ...
16	A095854	0, 0, 0, 0, 0, 6, 20, 164, 1280, ...
18		0, 0, 0, 0, 0, 0, 0, 0, 4, ...
20		0, 0, 0, 0, 0, 0, 4, 12, 42, ...
24	A095855	0, 0, 0, 2, 2, 2, 24, 170, 1570, ...
32		0, 0, 0, 0, 0, 0, 0, 24, 176, ...
36	A095856	0, 0, 0, 0, 0, 2, 6, 22, 164, ...
48	A095857	0, 0, 0, 0, 0, 8, 28, 96, 660, ...
72	A095858	0, 0, 0, 0, 0, 2, 4, 28, 179, ...
120		0, 0, 0, 0, 2, 2, 2, 6, 26, ...
144		0, 0, 0, 0, 0, 0, 6, 24, 78, ...
240		0, 0, 0, 0, 0, 0, 6, 16, 70, ...
720		0, 0, 0, 0, 0, 2, 2, 8, 22, ...

The smallest nontrivial graph whose automorphism group is cyclic has nine nodes. The one illustrated by Harary (1994, p. 170; left top figure above) is implemented as GraphData["SmallestCyclicGroupGraph"]. However, there is at least one other graph on nine nodes whose automorphism group is isomorphic to the cyclic group , namely the graph obtained from the (9, 3)-configuration (second top figure). Other graphs whose automorphism groups are isomorphic to the cyclic group  include three of the Paulus graphs (each on 26 vertices), the 12th fullerene graph on 40 vertices, and Tutte's graph (on 46 vertices). These and other graphs whose automorphism groups are isomorphic to cyclic groups are illustrated in the remaining figures above.

The numbers of vertices of the minimal graph having an automorphism group of order  are 0, 2, 9, 4, 15, 3, 14, 4, 15, 5, ... (OEIS A080803). The graphs achieving these bounds are summarized in the following table, where  and  denote the empty graph and cyclic graph on  nodes, respectively. Let  denote graph union, and  denote the graph complement of . In addition, let  be the graph with vertices  and edges , where all indices are to be read modulo  (i.e.,  is made up of an -gon  with a rectangle drawn over each side plus one diagonal in each rectangle). Let  be the graph obtained from  by identifying  with every  where  is congruent  modulo , and likewise for the . Also let  be the graph with vertices  and edges , where all indices are taken modulo  (Voss 2003).

	graph
1		0
2		2
3		9
4		4
5		15
6		3
7		14
8		4
9		15
10		5
11		22
12		5
13		26
14		7
15		21
16		6
17		34
18		9
19		38
20		7
21		23
22		11
23		46
24		4
25		30
26		13
27		24
28		9
29		58
30		14
31		62

The following table gives the graph automorphisms groups for a number of common graphs.

complete graph	symmetric group
empty graph	symmetric group
cycle graph ,	dihedral group
	finite group C2×C2
,	finite group
as defined above	cyclic group
as defined above	cyclic group

REFERENCES

Darga, P. T.; Sakallah, K. A.; and Markov, I. L. "Faster Symmetry Discovery using Sparsity of Symmetries." Proceedings of the 45th Design Automation Conference, Anaheim, California, June 2008. 

 http://vlsicad.eecs.umich.edu/BK/SAUCY/saucy-dac08.pdf.Douglas, B. L. and Wang, J. B. "A Classical Approach to the Graph Isomorphism Problem Using Quantum Walks." J. Phys. A: Math. Theor. 41, 075303-1-15, 2008.

Duijvestijn, A. J. W. "Algorithmic Calculation of the Order of the Automorphism Group of a Graph." Memorandum No. 221. Enschede, Netherlands: Twente Univ. Technology, 1978.

Frucht, R. "Herstellung von Graphen mit vorgegebener abstrakter Gruppe." Compos. Math. 6, 239-250, 1939.

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.

Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, 1993.

Lauri, J. and Scapellato, R. Topics in Graph Automorphisms and Reconstruction. Cambridge, England: Cambridge University Press, 2003.

Lipton, R. J.; North, S. C.; and Sandberg, J. S. "A Method for Drawing Graphs." In Proc. First Annual Symposium on Computation Geometry (Ed. J. O'Rourke). New York: ACM Press, pp. 153-160, 1985.

McKay, B. "The Nauty Page." http://cs.anu.edu.au/~bdm/nauty/.Skiena, S. "Automorphism Groups." §5.2.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 184-187, 1990.

Sloane,N. J. A Sequences A000165/M1878, A003400/M4575, A075095, A075096, A075097, A075098, A080803, A095348, A124351, and A124354 in "The On-Line Encyclopedia of Integer Sequences."University of Michigan Electrical Engineering and Computer Science department. "Saucy: Fast Symmetry Discovery." http://vlsicad.eecs.umich.edu/BK/SAUCY/.Voss, J. "Re: RE: Graphs with automorphism groups of given order." seqfan@ext.jussieu.fr mailing list. March 27, 2003.