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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 HexagonalTorusGraph

Source Notebook

Get a skew graph on a torus with a given number of hexagonal cells

Contributed by: Ed Pegg Jr

ResourceFunction["HexagonalTorusGraph"][hexagons,{a,b}]

makes an (a,b)-skew torus graph with a given number of hexagons.

Details and Options

Some sextic circulant graphs have toroidal embeddings. Most of the time, the toroidal dual of circulant graph Cn(a,b,a+b) is the (a,b)-skew hexagonal torus graph.
Duals of circulant graphs C9(1,2,4), C12(1,2,5), C18(1,2,8) and C20(1,2,9) cannot be made with this method. The Shrikhande graph dual is another exception.

Examples

Basic Examples (2) 

There is a unique hexagonal torus graph with eight cells:

In[1]:=
ResourceFunction["HexagonalTorusGraph"][8, {1, 2}]
Out[1]=

A 30-hexagon toroidal graph in 3D:

In[2]:=
Graph3D[ResourceFunction["HexagonalTorusGraph"][30, {2, 5}]]
Out[2]=

Scope (1) 

When a hexagonal torus graph of this form exists, the toroidal dual is the circulant graph (a, b, a+b):

In[3]:=
Row[{ResourceFunction["HexagonalTorusGraph"][13, {1, 3}], CirculantGraph[13, {1, 3, 4}]}]
Out[3]=

Possible Issues (3) 

Some skew values are invalid:

In[4]:=
ResourceFunction["HexagonalTorusGraph"][13, {2, 9}]
Out[4]=

The torus graphs corresponding to the toroidal duals of the following sextic toroidal graphs up to 20 vertices are out of scope for this function:

In[5]:=
outofscope = {{"Circulant", {9, {1, 2, 4}}}, {"Circulant", {12, {1, 2, 5}}}, "ShrikhandeGraph", {"Sextic", {16, 2}}, {"Circulant", {18, {1, 2, 8}}}, {"Circulant", {20, {1, 2, 9}}}};

The number of distinct torus graphs with 7 to 100 hexagons is given by the following code (see A129033):

In[6]:=
T2[n_] := Piecewise[{{DivisorSigma[0, n] - 2 - T6[n], Mod[n, 2] == 1}, {DivisorSigma[0, n/2] - 2, Mod[n, 4] == 2}}, DivisorSigma[0, n/2] + DivisorSigma[0, n/4] - 4 - T6[n]]; T3[n_] := Length[{ToRules[ Reduce[n == p^2 + p q + q^2 && 0 < q < p, {p, q}, Integers]]}]; T6[n_] := Boole[n >= 9 && (IntegerQ[Sqrt[n]] || IntegerQ[Sqrt[n/3]])] T[n_] := Piecewise[{{DivisorSigma[1, n]/6 + T2[n]/2 + 2/3 T3[n] + 5/6 T6[n] - (2 - Mod[n, 2]), n > 6}}]; Table[T[n], {n, 7, 100}]
Out[6]=

Neat Examples (1) 

The 33 distinct hexagonal torus graphs up to 20 hexagons that can be made with this method:

In[7]:=
distinctgraphs = {{7, {1, 2}}, {8, {1, 2}}, {9, {1, 2}}, {10, {1, 3}}, {11, {1, 2}}, {12, {1, 2}}, {12, {1, 3}}, {12, {1, 4}}, {13, {1, 2}}, {13, {1, 3}}, {14, {1, 2}}, {14, {1, 3}}, {15, {1, 2}}, {15, {1, 3}}, {15, {1, 4}}, {15, {2, 3}}, {16, {1, 2}}, {16, {1, 3}}, {16, {2, 3}}, {17, {1, 2}}, {17, {1, 3}}, {18, {1, 2}}, {18, {1, 3}}, {18, {1, 4}}, {18, {1, 5}}, {19, {1, 2}}, {19, {1, 3}}, {19, {2, 3}}, {20, {1, 2}}, {20, {1, 3}}, {20, {1, 4}}, {20, {2, 3}}, {20, {3, 4}}}; Column[Row /@ Partition[ ResourceFunction["HexagonalTorusGraph"] @@ # & /@ distinctgraphs, UpTo[5]]]
Out[7]=

Version History

  • 2.0.0 – 28 February 2020
  • 1.0.0 – 05 February 2020

License Information