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a boolean algebra graph drawing question

• To: mathgroup at smc.vnet.net
• Subject: [mg90717] a boolean algebra graph drawing question
• From: Manal Helal <mhelal at usyd.edu.au>
• Date: Mon, 21 Jul 2008 04:28:23 -0400 (EDT)

Hello

I need to customize the graph in this link, which is for 5 dimensions
with 6 levels (I consider each level as a one distance away from the
origin or the level before it):

http://www.cs.sunysb.edu/~skiena/combinatorica/animations/ham.html

how can I make the function in /Combinatorica :/
ShowGraph[ a = BooleanAlgebra[2]] for 2D, or
ShowGraph[ a = BooleanAlgebra[3]] for 3D

work on more levels? An example of my understanding of a 2D and 3D
tensors is provided here:
http://www.cse.unsw.edu.au/~mhelal/2DDepNetwork.pdf
http://www.cse.unsw.edu.au/~mhelal/3DDepNetwork.pdf

Each node has k-children where k is the rank of the tensor. The nodes in 
a level have overlapping
parents from a previous level.  This keeps growing till the middle
level, then the number of nodes will start to decay, till it becomes one 
at the last level.

There are equations for the relationship between the tensor shape (upper
bounds of each dimension, but I can assume they are all the same) and 
the total waves
(levels in the graph), and the number of nodes in each level. Can I use 
these equations to
customize the graph?

I have done these graphics manually, but I need to do more analysis and
draw more graphs for higher dimensions and more levels in the graph, and
I am sure mathematica can help. There is an obvious symmetry in the 
dependency arrows between the nodes,
and I am not sure whether there is an equation that can scale well with 
dimensionality, where I can
parameterize it with the dimensionality, level in the graph, node order, 
and it can give me the node
orders of its parents on the lower level, and its children on the higher 
level?

It is a research question, and if someone is willing to collaborate, we 
can publish a paper together,
because I come from computer science background, not really math. 
However, finding an answer for this
question can significantly speed up my application.

I appreciate your help a lot,

Kind Regards,

Manal