Q3_vertices_eq

formal source

  28theorem Q3_edge_count : Q3_edges = Q3_degree * Q3_vertices / 2 := by
  29  unfold Q3_edges Q3_degree Q3_vertices; omega
  30
  31theorem Q3_vertices_eq : Q3_vertices = 2 ^ Q3_degree := by
  32  unfold Q3_vertices Q3_degree; omega
  33
  34/-! ## Q₃ Laplacian Spectrum
  35
  36The graph Laplacian L = D − A of Q₃ has eigenvalues determined by the
  37Hamming weight of binary vectors in {0,1}³. For vertex v with Hamming
  38weight w(v), the eigenvalue is 2·w(v). The spectrum is:
  39- λ₀ = 0 (multiplicity 1, the all-ones eigenvector)
  40- λ₁ = 2 (multiplicity 3, the three coordinate functions)
  41- λ₂ = 4 (multiplicity 3, the three pairwise products)
  42- λ₃ = 6 (multiplicity 1, the parity function)
  43-/
  44
  45def Q3_laplacian_eigenvalues : List ℕ := [0, 2, 2, 2, 4, 4, 4, 6]
  46
  47def Q3_spectral_gap : ℕ := 2
  48def Q3_max_eigenvalue : ℕ := 6
  49
  50theorem Q3_eigenvalue_count : Q3_laplacian_eigenvalues.length = Q3_vertices := by
  51  unfold Q3_laplacian_eigenvalues Q3_vertices; native_decide
  52
  53theorem Q3_trace : Q3_laplacian_eigenvalues.sum = Q3_degree * Q3_vertices := by
  54  unfold Q3_laplacian_eigenvalues Q3_degree Q3_vertices; native_decide
  55
  56theorem Q3_max_eigenvalue_eq : Q3_max_eigenvalue = 2 * Q3_degree := by
  57  unfold Q3_max_eigenvalue Q3_degree; omega
  58
  59/-- The multiplicities are {1, 3, 3, 1} = binomial coefficients C(3,k). -/
  60def Q3_multiplicities : List ℕ := [1, 3, 3, 1]
  61