# Constructing an I-graph

I am attempting to build I-graphs in MMA, i.e. the non-Generalized Petersen graphs, and I found some code from Eric Weisstein, but this is now outdated and doesn't run. I have attempted to switch the old commands from Combinatorica with the new versions, but it is not working. Can anyone help? Here is the old code:

Initialization

<< MathWorldGraphs


Code

IGraph[n_, j_, k_] := Module[{uu, u, vv, v},
uu = Array[u, n, 0];
vv = Array[v, n, 0];
{u, u}, Mod[#1[] - #2[] - j, n] == 0,
{v, v}, Mod[#1[] - #2[] - k, n] == 0,
{u, v}, #1[] == #2[]
] &,
Type -> Directed
], Flatten[Table[r Through[{Cos, Sin}[2 Pi s/n]], {r, 2}, {s, 0, n - 1}],
1]]
]
]

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• Could you show your updated version, and mention where and why the update is failing? – MarcoB May 12 '15 at 15:03
• @MarcoB, I changed MakeUndirected to UndirectedGraph. IGraph[n_, j_, k_] := Module[{uu, u, vv, v}, uu = Array[u, n, 0]; vv = Array[v, n, 0]; UndirectedGraph[ ChangeVertices[MakeGraph[Join[uu, vv], Switch[Head /@ {##}, {u, u}, Mod[#1[] - #2[] - j, n] == 0, {v, v}, Mod[#1[] - #2[] - k, n] == 0, {u, v}, #1[] == #2[] ] &, Type -> Directed ], Flatten[ Table[r Through[{Cos, Sin}[2 Pi s/n]], {r, 2}, {s, 0, n - 1}], 1]] ] ] IGraph[12, 3, 4] – Mat May 12 '15 at 23:51
• @MarcoB UndirectedGraph::graph: A graph object is expected at position 1 in UndirectedGraph[ChangeVertices(<<1>>,(1 0 Sqrt/2 1/2 1/2 Sqrt/2 0 1 -(1/2) Sqrt/2 -(Sqrt/2) 1/2 -1 0 -(Sqrt/2) -(1/2) -(1/2) -(Sqrt/2) 0 -1 1/2 -(Sqrt/2) Sqrt/2 -(1/2) 2 0 Sqrt 1 1 Sqrt 0 2 -1 Sqrt -Sqrt 1 -2 0 -Sqrt -1 -1 -Sqrt 0 -2 1 -Sqrt Sqrt -1 ))]. >> – Mat May 12 '15 at 23:52
• A temporary solution that might also help you in figuring out what changes to make would be to work with the old functions from Combinatorica. Let me see how that might work... – MarcoB May 13 '15 at 4:19

You can still use the old functions from Combinatorica, either to tide you over while you figure out the new ones, or as a semi-permanent solution. WRI has also provided a handy guide to transitioning from that package to the new built-in functions: "Upgrading from Combinatorica"

For example, in the case of the obsolete code you mentioned:

<< Combinatorica

IGraph[n_, j_, k_] := Module[
{uu, u, vv, v},
uu = Array[u, n, 0];
vv = Array[v, n, 0];
CombinatoricaMakeUndirected[
CombinatoricaChangeVertices[
CombinatoricaMakeGraph[
Join[uu, vv],
{u, u}, Mod[#1[] - #2[] - j, n] == 0,
{v, v}, Mod[#1[] - #2[] - k, n] == 0,
{u, v}, #1[] == #2[]
] &,
Type -> CombinatoricaDirected
],
Flatten[
Table[r Through[{Cos, Sin}[2 Pi s/n]], {r, 2}, {s, 0, n - 1}], 1]
]
]
]


When you load in Combinatorica, the system will warn you:

General::compat: Combinatorica Graph and Permutations functionality has been superseded by preloaded functionality. The package now being loaded may conflict with this. Please see the Compatibility Guide for details.

... but it will still function. In order to avoid conflicts, I added explicit contexts to the old functions from the Combinatorica package, so you know exactly what you are using.

Having done that, you can now obtain the sample graph you mentioned in the comments:

CombinatoricaShowGraph@IGraph[12, 3, 4] Just using builtin graph functions:

IGraph[n_, j_, k_, opts : OptionsPattern[]] :=
Block[{vert, u, v}, vert = Join[Array[u, n, 0], Array[v, n, 0]];
Boole[Table[
Switch[Head /@ {ii, jj}, {u, u},
Mod[ii[] - jj[] - j, n] == 0, {v, v},
Mod[ii[] - jj[] - k, n] == 0, {u, v},
ii[] == jj[], _, False], {ii, vert}, {jj, vert}]], opts,
DirectedEdges -> False,
VertexCoordinates ->
Flatten[Table[
r Through[{Cos, Sin}[2 Pi s/n]], {r, 2}, {s, 0, n - 1}], 1]]]

IGraph[12, 3, 4, PlotTheme -> "Web"] 