Graph::Maker::Petersen - create Petersen and generalized Petersen graphs
use Graph::Maker::Petersen; $graph = Graph::Maker->new ('Petersen'); $graph = Graph::Maker->new ('Petersen', N=>7, K=>3);
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
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 vertices. 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.
N >= 3 and
K != 0 mod N. K=1 is an inner vertex cycle the same as the outer. N=4,K=1 is the cube graph, the same as Graph::Maker::Hypercube of 3 dimensions.
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. For example 7,9 is the same as 7,2, and 7,5 the same as 7,2 too, since steps are effectively forward and backward by K.
$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
If the graph is directed (the default) then edges are added both ways between vertices. Option
undirected => 1creates an undirected graph and for it there is a single edge between vertices.
House of Graphs entries for graphs here include
- N=3, K=1, https://hog.grinvin.org/ViewGraphInfo.action?id=746
- N=4, K=1, https://hog.grinvin.org/ViewGraphInfo.action?id=1022 (cube)
- 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=4, https://hog.grinvin.org/ViewGraphInfo.action?id=1229 (Moebius 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)
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
Copyright 2015, 2016, 2017 Kevin Ryde
This file is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
This file is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with This file. If not, see http://www.gnu.org/licenses/.