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graph recursion of a cube sort

  • To: mathgroup at smc.vnet.net
  • Subject: [mg101035] graph recursion of a cube sort
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Sun, 21 Jun 2009 07:07:55 -0400 (EDT)
  • References: <h13vq8$8q5$1@smc.vnet.net>

http://www.geocities.com/rlbagulatftn/5d_cube.gif
I figured out a way to make a recursive square-cube like
graph product:
here are the matrices :
Line:
{{0, 1},
{1, 0}},
Square:
{{0, 1, 1, 0},
{1, 0, 0, 1},
{1, 0, 0, 1},
{0, 1, 1, 0}},
Cube:
{{0, 1, 1, 0, 1, 0,0, 0},
{1, 0, 0, 1, 0, 1, 0, 0},
{1, 0, 0, 1, 0, 0, 1, 0},
{0, 1, 1, 0, 0, 0, 0, 1},
{1, 0, 0, 0, 0,1, 1, 0},
{0, 1, 0, 0, 1, 0, 0, 1},
{0, 0, 1, 0, 1, 0, 0, 1},
{0, 0, 0, 1, 0, 1, 1, 0}},
4d cube ( Tesseract):
{{0, 1, 1, 0, 1, 0, 0, 0,1, 0, 0, 0, 0, 0, 0, 0},
{1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0},
{1, 0, 0, 1, 0, 0, 1, 0, 0,  0, 1, 0, 0, 0, 0, 0},
{0, 1, 1, 0, 0, 0, 0, 1, 0, 0,0, 1, 0, 0, 0, 0},
{1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0},
{0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0},
{0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0},
{0, 0, 0, 1,0, 1, 1, 0, 0, 0, 0, 0,  0, 0, 0, 1},
{1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0},
{0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0},
{0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0},
{0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1},
{0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0},
{0, 0, 0,  0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1},
{0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1},
{0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0}}}

As I said before they are a lot like Hadamards.
But not orthogonal or block form that I can find.
Triangle from the polynomial is:
{{-1, 0, 1},
{0, 0, -4, 0, 1},
{9, 0, -28, 0, 30, 0, -12, 0,1},
{0, 0, 0, 0, 0, 0, -4096, 0, 4352, 0, -1792, 0, 352, 0, -32, 0, 1}}
{1476225, 0, -15641424, 0, 75436920, 0, -218887920, 0,425462940, 0, 
-583700560, 0, 580113224, 0, -421986160, 0, 224447430, 0, -86417520, 0, 
23674440, 0, -4516176, 0, 586140, 0, -50160, 0, 2680, 0, -80, 0, 1}}
As you can see it is the odd levels that don't have a lot of zeros.
Mathematica:
<< DiscreteMath`GraphPlot`;
<< DiscreteMath`ComputationalGeometry`
<< DiscreteMath`Combinatorica`
Clear[g, n, gm]
g[0] := CompleteGraph[2]
g[1] := GraphProduct[CompleteGraph[2], g[0]]
g[n_] := g[n] = GraphProduct[g[n - 1], g[0]]
Table[g[n], {n, 0, 3}]
gm = Table[ToAdjacencyMatrix[g[n]], {n, 0, 4}]
Table[CoefficientList[CharacteristicPolynomial[ToAdjacencyMatrix[g[n]],
   x], x], {n, 0, 4}]
Table[ToAdjacencyMatrix[g[n]].Transpose[ToAdjacencyMatrix[g[n]]], {n, 0, 
3}]
Table[Dimensions[gm[[n]]], {n, 1, 4}]
Table[GraphPlot3D[gm[[n]]], {n, 1, 5}]

-- 
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: rlbagula at sbcglobal.net

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