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

Key Points

• This investigation aims to clarify the spectral equivalence between the discrete Laplacian and the squared adjacency matrix of the complement graph.
• Developed a mathematical proof demonstrating the commutation of the Laplacian and complement adjacency matrices for d-regular graphs.
• Utilized the Spectral Theorem to establish simultaneous diagonalizability.
• Showed that the spectral bases of the gradient-derived operator and the squared complement graph exhibit unit cosine similarity.
• Resolved a long-standing question regarding the spectral characterization of a specific class of graphs.

Abstract