/ 
Graph-Maker-Other-19
River stage zero No dependents
/ Graph::Maker::Petersen

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

746     N=3, K=1    circular ladder 3 rungs
1022    N=4, K=1    cube
588     N=4, K=2
36274   N=5, K=1    circular ladder 5 rungs
660     N=5, K=2    Petersen
32798   N=6, K=1    cross-connected 6-cycles
34383   N=6, K=2
36287   N=7, K=1    circular ladder 7 rungs
28482   N=7, K=2
36292   N=8, K=1    circular ladder 8 rungs
1229    N=8, K=3    Mobius Kantor
36298   N=9, K=1    circular ladder 9 rungs
6700    N=9, K=3
36310   N=10, K=1   circular ladder 10 rungs
1043    N=10, K=2   Dodecahedral
1036    N=10, K=3   Desargues

24052   N=11, K=2
27325   N=12, K=2
1234    N=12, K=5   Nauru
36346   N=13, K=5
36361   N=15, K=4

(And a few more at bigger N.)

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

SEE ALSO

Graph::Maker, Graph::Maker::Cycle, Graph::Maker::Hypercube

HOME PAGE

http://user42.tuxfamily.org/graph-maker-other/index.html

LICENSE

Copyright 2015, 2016, 2017, 2018, 2019, 2020, 2021 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/.