Number of automorphisms

Calculate the number of automorphisms of a graph, i.e. the number of isomorphisms to itself.

Usage

count_automorphisms(
  graph,
  colors = NULL,
  sh = c("fm", "f", "fs", "fl", "flm", "fsm")
)

Arguments

graph

The input graph, it is treated as undirected.

colors

The colors of the individual vertices of the graph; only vertices having the same color are allowed to match each other in an automorphism. When omitted, igraph uses the color attribute of the vertices, or, if there is no such vertex attribute, it simply assumes that all vertices have the same color. Pass NULL explicitly if the graph has a color vertex attribute but you do not want to use it.

sh

The splitting heuristics for the BLISS algorithm. Possible values are: ‘f’: first non-singleton cell, ‘fl’: first largest non-singleton cell, ‘fs’: first smallest non-singleton cell, ‘fm’: first maximally non-trivially connected non-singleton cell, ‘flm’: first largest maximally non-trivially connected non-singleton cell, ‘fsm’: first smallest maximally non-trivially connected non-singleton cell.

Value

A named list with the following members:

group_size

The size of the automorphism group of the input graph, as a string. This number is exact if igraph was compiled with the GMP library, and approximate otherwise.

nof_nodes

The number of nodes in the search tree.

nof_leaf_nodes

The number of leaf nodes in the search tree.

nof_bad_nodes

Number of bad nodes.

nof_canupdates

Number of canrep updates.

max_level

Maximum level.

Details

An automorphism of a graph is a permutation of its vertices which brings the graph into itself.

This function calculates the number of automorphism of a graph using the BLISS algorithm. See also the BLISS homepage at http://www.tcs.hut.fi/Software/bliss/index.html. If you need the automorphisms themselves, use automorphism_group() to obtain a compact representation of the automorphism group.

References

Tommi Junttila and Petteri Kaski: Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs, Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments and the Fourth Workshop on Analytic Algorithms and Combinatorics. 2007.

See also

canonical_permutation(), permute(), and automorphism_group() for a compact representation of all automorphisms

Other graph automorphism: automorphism_group()

Author

Tommi Junttila (http://users.ics.aalto.fi/tjunttil/) for BLISS and Gabor Csardi csardi.gabor@gmail.com for the igraph glue code and this manual page.

Related documentation in the C library

igraph_count_automorphisms().

Examples

## A ring has n*2 automorphisms, you can "turn" it by 0-9 vertices
## and each of these graphs can be "flipped"
g <- make_ring(10)
count_automorphisms(g)
#> $nof_nodes
#> [1] 6
#> 
#> $nof_leaf_nodes
#> [1] 4
#> 
#> $nof_bad_nodes
#> [1] 0
#> 
#> $nof_canupdates
#> [1] 1
#> 
#> $max_level
#> [1] 2
#> 
#> $group_size
#> [1] "20"
#> 

## A full graph has n! automorphisms; however, we restrict the vertex
## matching by colors, leading to only 4 automorphisms
g <- make_full_graph(4)
count_automorphisms(g, colors = c(1, 2, 1, 2))
#> $nof_nodes
#> [1] 5
#> 
#> $nof_leaf_nodes
#> [1] 3
#> 
#> $nof_bad_nodes
#> [1] 0
#> 
#> $nof_canupdates
#> [1] 1
#> 
#> $max_level
#> [1] 2
#> 
#> $group_size
#> [1] "4"
#>