The spectrum of the hypercube
The eigenvalues of Q6 are the yang levels
Walk the seven eigenvalues with their multiplicity and compare them with the lattice levels and the random walk.
The hypercube Q6 is the graph of the 64 hexagrams with one edge per single-line mutation. Its eigenvalues (of the adjacency matrix) are a classical theorem: 6 − 2k with multiplicity C(6,k), because the eigenvectors are the characters χ(v) = (−1)^⟨w,v⟩, with eigenvalue 6 − 2·(lines of w). And the multiplicities are no accident: they are the yang levels of the lattice B6, C(6,k) hexagrams with k yang lines.
eigenvalue 6 − 2k · multiplicity C(6,k) = number of hexagrams with k yang lines
The simple random walk on Q6 has as its transition matrix the adjacency divided by 6, so its spectrum is this same one divided by 6: 1.00, 0.67, 0.33, 0.00, −0.33, −0.67, −1.00. The second eigenvalue, 4/6 ≈ 0.67, fixes its mixing speed. The spectral theorem turns a dynamical question into a count of yang lines.
The eigenvalues of the adjacency of Q6 are 6−2k with multiplicity C(6,k): the multiplicities are the yang levels of the lattice B6, and the spectrum of the simple walk is this one divided by 6, which fixes its mixing speed.
verificar_espectro_q6The types of claim and the full bibliography (APA) are in Foundations.