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Kevin Ryde
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# NAME

Graph::Maker::Petersen - create Petersen and generalized Petersen graphs

# SYNOPSIS

use Graph::Maker::Petersen;
\$graph = Graph::Maker->new ('Petersen');
\$graph = Graph::Maker->new ('Petersen', N=>7, K=>3);

# DESCRIPTION

Graph::Maker::Petersen creates a Graph.pm graph of the Petersen graph,

___1___
____/   |   \____
/        |        \
2__       6       __5           N => 5
|  ---7--/-\--10--  |           K => 2
|      \/   \/      |        (the defaults)
|      / \ / \      |
|   __8--/ \--9__   |
| --             -- |
3-------------------4

Parameters N and K give generalized Petersen graphs. For example

\$graph = Graph::Maker->new('Petersen', N=>7, K=>3);

N is the number of vertices in the outer cycle, and the same again in the inner circulant. K is how many steps between connections for the inner. In the default Petersen 5,2 for example at inner vertex 6 step by 2 for edge 6--8. The outer vertices are numbered 1 .. N and the inner N+1 .. 2*N.

Should have N >= 3 and K != 0 mod N. K=1 is an inner vertex cycle the same as the outer.

K will usually be in the range 1 <= K <= N/2. Other values are accepted but become the same as something in that range since the K steps wrap-around and go as both K and -K mod N. So for example 7,9 and 7,5 are both the same as 7,2.

N=4,K=1 is equivalent to the cube graph (Graph::Maker::Hypercube of 3 dimensions). N=4,K=2 is equivalent to a Mobius ladder of 4 rungs with 2 consecutive rungs deleted.

# FUNCTIONS

\$graph = Graph::Maker->new('Petersen', key => value, ...)

The key/value parameters are

N           => integer, number of outer vertices
(and corresponding inner)
K           => integer, step for inner circulant
graph_maker => subr(key=>value) constructor, default Graph->new

Other parameters are passed to the constructor, either graph_maker or Graph->new().

If the graph is directed (the default) then edges are added both ways between vertices. Option undirected => 1 creates an undirected graph and for it there is a single edge between vertices.

# HOUSE OF GRAPHS

House of Graphs entries for graphs here include

N=3, K=1, https://hog.grinvin.org/ViewGraphInfo.action?id=746, circular ladder 3 rungs
N=4, K=1, https://hog.grinvin.org/ViewGraphInfo.action?id=1022 (cube)
N=4, K=2, https://hog.grinvin.org/ViewGraphInfo.action?id=588
N=5, K=2, https://hog.grinvin.org/ViewGraphInfo.action?id=660 (Petersen)
N=7, K=2, https://hog.grinvin.org/ViewGraphInfo.action?id=28482
N=8, K=3, https://hog.grinvin.org/ViewGraphInfo.action?id=1229 (Mobius Kantor)
N=9, K=3, https://hog.grinvin.org/ViewGraphInfo.action?id=6700
N=10, K=2, https://hog.grinvin.org/ViewGraphInfo.action?id=1043 (Dodecahedral)
N=10, K=3, https://hog.grinvin.org/ViewGraphInfo.action?id=1036 (Desargues)
N=11, K=2, https://hog.grinvin.org/ViewGraphInfo.action?id=24052
N=12, K=2, https://hog.grinvin.org/ViewGraphInfo.action?id=27325
N=12, K=5, https://hog.grinvin.org/ViewGraphInfo.action?id=1234 (Nauru)

# OEIS

A few of the many entries in Sloane's Online Encyclopedia of Integer Sequences related to these graphs include

A077105   number of non-isomorphic K for given N,
K <= floor((N-1)/2), so not K=N/2 when N even
A182054   number of independent sets in N,2 for even N
A182077   number of independent sets in N,2 for odd N
A274047   diameter of N,2