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Four Cubes
Szymon Łukaszyk
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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 2ⁿ 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√(2), n√(2)] of the adjacency matrix of 2ⁿ-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.
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