Cycle-graph Laplacian spectrum: eigenvalues 2 - 2 cos(2 pi k/N); trace equals 2N (Chebyshev sum identity)

Cycle-graph Laplacian framework via Chebyshev polynomials. Setup: the graph Laplacian L of the cycle C_N is the N x N matrix L = 2I - A where A is the cycle adjacency matrix (A_{ij} = 1 if |i - j| = 1 mod N, else 0). L is diagonalised by…