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,

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.

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

(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.

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).

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

A simple graph whose automorphism group is a cyclic group may be termed a cyclic group graph. The smallest nontrivial cyclic group graphs have nine nodes.

See also

Automorphism Group, Cyclic Group Graph, Edge Automorphism, Edge Automorphism Group, Edge-Transitive Graph, Frucht Graph, Graph Isomorphism, Isomorphic Graphs, Vertex-Transitive Graph

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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. 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.

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Graph Automorphism

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Weisstein, Eric W. "Graph Automorphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GraphAutomorphism.html

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