[Paper Review] Four Cubes

Szymon Łukaszyk|arXiv (Cornell University)|Jul 7, 2020

Graph theory and applications32 citations

TL;DR

A short survey of four graphs in {0,1}^n, detailing degenerate spectra of cotan Laplacians and adjacency matrices, with connections to the Buckminster Fuller vector equilibrium.

ABSTRACT

A short survey on the properties of four graphs constructed in $\{0, 1\}^n$ Boolean space is presented. Flexible activation function of an artificial neuron in a sparse distributed memory model is defined on the basis of the Ugly duckling theorem. Cotan Laplacian on 2-face triangulation of $n$-cube has degenerate spectrum of eigenvalues corresponding to the Hamming distance distribution of $\{0, 1\}^n$ space. Degenerate spectrum of eigenvalues of the cotan Laplacian defined on the graph comprising $2^n$ 2-face triangulated $n$-cubes sharing common origin includes all integers from 0 to 3$n$, without the eigenvalue of 3$n$-1 (multiplicities of the same eigenvalues form A038717 OEIS sequence), while the multiplicities of the same eigenvalues $[-n\sqrt{2}, n\sqrt{2}]$ of the adjacency matrix of $2^n$-cube form trinomial triangle. The distance matrix of this graph, providing further OEIS sequences, as well as its relation with Buckminster Fuller vector equilibrium is also discussed.

Motivation & Objective

• Motivate and survey properties of four graphs constructed in Boolean space {0,1}^n.
• Explore spectral properties of cotan Laplacians on these graphs.
• Examine how eigenvalue degeneracies relate to Hamming distance distributions.
• Discuss relationships between distance matrices and OEIS sequences and vector equilibrium concepts.

Proposed method

• Define and analyze four graphs in {0,1}^n with respect to their metric and spectral properties.
• Use cotan Laplacian and adjacency matrices to study eigenvalue distributions.
• Identify degeneracies and their combinatorial interpretations via Hamming distance distributions.
• Relate distance matrices to OEIS sequences and Buckminster Fuller vector equilibrium.

Experimental results

Research questions

• RQ1 What are the spectral properties (degeneracies) of the cotan Laplacian on the four-cube constructions?
• RQ2 How do eigenvalue distributions of the cotan Laplacian and adjacency matrices relate to Hamming distance in {0,1}^n?
• RQ3 What combinatorial or geometric interpretations emerge from these spectra (e.g., OEIS sequences, vector equilibrium)?

Key findings

• Cotan Laplacian on 2-face triangulation of an n-cube has a degenerate spectrum linked to the Hamming distance distribution.
• Cotan Laplacian on the graph of 2^n 2-face triangulated n-cubes sharing origin has eigenvalues 0 through 3n, excluding 3n-1, with multiplicities forming OEIS A038717.
• Adjacency eigenvalues for the 2^n cube form the trinomial triangle.
• The distance matrix yields additional OEIS sequences and relates to Buckminster Fuller vector equilibrium.