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MathGroup Archive 2009

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double pentagon bridge graph polygons

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100811] double pentagon bridge graph polygons
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Sun, 14 Jun 2009 21:20:06 -0400 (EDT)

The idea of adding two more rows of pentagons
occurred to me.
Vertex types:
{n, 2*n, 4*n, 2*n, n}
{2, 4, 8, 4, 2},
{3, 6, 12, 6, 3},
{4, 8, 16, 8, 4},
{5, 10, 20, 10, 5}, all pentagons
{6, 12, 24, 12, 6}, somewhat like the Buckyball
{7, 14, 28, 14, 7},
{8, 16, 32, 16, 8},
{9, 18, 36, 18, 9},
{10, 20, 40, 20, 10}
The Euler data are in general:
{V,E,F}={10*n, 16*n, 6*n + 2}
{30, 48, 20},
{40, 64, 26},
{50, 80, 32},
{60, 96, 38},
{70, 112, 44},
{80, 128, 50},
{90, 144, 56},
{100, 160, 62}
I did the Schlegel diagrams for 2 and 3
and they appear to work at least as graphs.
The one of immediate interest is
{V,E,F}={50, 80, 32}
{5, 10, 20, 10, 5}, all pentagons

http://www.geocities.com/rlbagulatftn/5_10_20_10_5graph.gif
{5, 10, 20, 10, 5}
{V,E,F}={50, 80, 32}
Not all the pentagons are flat as faces ( so it isn't an ideal  Platonic 
solid),
but it exists as a graph.

http://www.geocities.com/rlbagulatftn/3_6_12_6_3graph.gif
Just like in the Schlegel diagram:
{3,6,12,6,3}
{V,E,F}={30,48,20}

So two of the cases work fine.
All the graphs of the pentagon bridge type are here ( single and double):
http://www.flickr.com/photos/fractalmusic/sets/72157619727734302/

Mathematica:{30, 48, 20}/{3, 6, 12, 6, 3}
<< DiscreteMath`GraphPlot`;
<< DiscreteMath`ComputationalGeometry`
<< DiscreteMath`Combinatorica`

m24 = {{0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, \
0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 0,
     0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,
        0, 0}, {1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
        0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0,
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0, 0,
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0, 0,
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1, 0,
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0, 0,
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CharacteristicPolynomial[m24, x]
NSolve[CharacteristicPolynomial[m24, x] == 0, x]
GraphPlot3D[m24]
g = FromAdjacencyMatrix[m24]
ShowGraph[Contract[FromAdjacencyMatrix[m24], {6, 19}]]
ShowGraph[RankedEmbedding[FromAdjacencyMatrix[m24], {24}]]
ShowGraph[SpringEmbedding[FromAdjacencyMatrix[m24]]]
ShowGraph[LineGraph[FromAdjacencyMatrix[m24]]]
ShowGraph[OrientGraph[FromAdjacencyMatrix[m24]]]
ShowGraph[FromAdjacencyMatrix[m24], VertexNumber -> True]
EdgeConnectivity[FromAdjacencyMatrix[m24]]
3
M[FromAdjacencyMatrix[m24]]
48
V[FromAdjacencyMatrix[m24]]
30
M[FromAdjacencyMatrix[m24]] - V[FromAdjacencyMatrix[m24]] + 2
20

Mathematica:{V,E,F}={50, 80, 32}
{5, 10, 20, 10, 5};
<< DiscreteMath`GraphPlot`;
<< DiscreteMath`ComputationalGeometry`
<< DiscreteMath`Combinatorica`

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0, 0,
        0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0,
        0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
        0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 
0}, {
        0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
        0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0}};
CharacteristicPolynomial[m24, x]
NSolve[CharacteristicPolynomial[m24, x] == 0, x]
GraphPlot3D[m24]
g = FromAdjacencyMatrix[m24]
ShowGraph[Contract[FromAdjacencyMatrix[m24], {6, 19}]]
ShowGraph[RankedEmbedding[FromAdjacencyMatrix[m24], {24}]]
ShowGraph[SpringEmbedding[FromAdjacencyMatrix[m24]]]
ShowGraph[LineGraph[FromAdjacencyMatrix[m24]]]
ShowGraph[OrientGraph[FromAdjacencyMatrix[m24]]]
ShowGraph[FromAdjacencyMatrix[m24], VertexNumber -> True]
EdgeConnectivity[FromAdjacencyMatrix[m24]]
3
M[FromAdjacencyMatrix[m24]]
80
V[FromAdjacencyMatrix[m24]]
50
M[FromAdjacencyMatrix[m24]] - V[FromAdjacencyMatrix[m24]] + 2
32


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