Theorem: trace of cycle-Laplacian for C_4 equals 8 (Chebyshev-sum / root-of-unity identity)

Theorem (C_4 Laplacian-trace canonical): for the cycle graph C_4 with Laplacian L = 2 I - A, the trace tr(L) = sum over k in {0, 1, 2, 3} of (2 - 2 cos(2 pi k / 4)) evaluates to exactly 8. Canonical sympy pins: k = sp.symbols('k',…