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.

## 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"
#>
```