Four Cubes

TL;DR

This paper explores the spectral properties of specific graphs derived from the n-cube in Boolean space, revealing connections to OEIS sequences, eigenvalue degeneracies, and geometric structures like Buckminster Fuller’s vector equilibrium.

Contribution

It provides a detailed analysis of the eigenvalues and spectra of graphs constructed from the n-cube, introducing new insights into their mathematical properties and relations to known sequences.

Findings

Eigenvalues of cotan Laplacian on 2-face triangulation are degenerate and relate to Hamming distances.

Eigenvalue spectrum of the graph includes all integers from 0 to 3n, excluding 3n-1.

Distance matrix relates to OEIS sequences and 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.