Spectral Equivalence of the Laplacian and the Squared Complement Adjacency Matrix in Regular Graphs

In this paper, we investigate the spectral relationship between the discrete Laplacian matrix of a finite graph and the square of the adjacency matrix of its complement. We rigorously prove that for any d-regular graph, these two matrices commute. Consequently, by the Spectral Theorem, they are simultaneously diagonalizable. This resolves the problem of finding a graph class where the spectral basis of the gradient-derived operator exhibits perfect collinearity (unit cosine similarity) with the spectral basis of the squared complement graph.