
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 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 -cube has degenerate spectrum of eigenvalues corresponding to the Hamming distance distribution of space. Degenerate spectrum of eigenvalues of the cotan Laplacian defined on the graph comprising 2-face triangulated -cubes sharing common origin includes all integers from 0 to 3, without the eigenvalue of 3-1 (multiplicities of the same eigenvalues form A038717 OEIS sequence), while the multiplicities of the same eigenvalues of the adjacency matrix of -cube form trinomial triangle. The distance matrix of this graph, providing further OEIS…
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Taxonomy
TopicsGraph theory and applications · Cellular Automata and Applications · Neural Networks and Applications
