The Laplacian Spectrum of the Truncated Octahedron Face Adjacency Graph

We compute the exact spectrum of the graph Laplacian on the face adjacency graph of the truncated octahedron (Kelvin cell). The 14 faces (6 squares, 8 hexagons) form a graph where two faces are adjacent if they share an edge. The Laplacian eigenvalues are: 0 (×1), (9−√17)/2 (×3), 4 (×2), (9+√17)/2 (×3), 7 (×4), 9 (×1). The characteristic polynomial factors completely: p(λ) = λ(λ²−9λ+16)³(λ−4)²(λ−7)⁴(λ−9). All eigenvalues are algebraic numbers over Q(√17). Using the full octahedral symmetry group O_h (order 48), each eigenspace is identified with an irreducible representation: A1g, T1u, Eg, T1u, A1g⊕T2g, A2u. All results verified by trace identities, numerical computation, and character-theoretic decomposition.…