GraphenLaplaceOperator

GraphenLaplaceOperator is a Laplacian-type operator defined on graphs, commonly used to model diffusion, vibrational modes, and quantum-like processes on discrete structures. In its standard form it acts on functions defined on the vertices of a graph and encodes the connectivity through edge weights.

For a simple undirected graph G with adjacency matrix A and degree matrix D (where D is diagonal with entries d_i equal to the sum of weights incident to vertex i), the basic GraphenLaplaceOperator L is defined as L = D − A. When weights are present, w_ij, the same formula applies with A being the weighted adjacency matrix, and d_i = sum_j w_ij. The operator is symmetric and positive semidefinite, and its eigenvalues provide information about graph structure, such as connectivity and clustering.

A normalized variant, often used in data analysis, is L_norm = I − D^−1/2 A D^−1/2, or equivalently L_rw = I − D^−1 A, known as the random-walk Laplacian. The spectrum of L_norm lies in [0, 2], with the smallest eigenvalue 0 corresponding to constant or nearly uniform signals on the graph. The eigenvectors (modes) reveal clusters and diffusion patterns.

In the context of graphene, the GraphenLaplaceOperator is frequently applied to the honeycomb lattice, a bipartite graph representing carbon atoms. Its discrete Laplacian serves as a tight-binding-like operator, and the low-energy spectrum near the Dirac points reflects relativistic-like behavior described by the Dirac equation in the continuum limit.

Applications include spectral clustering, graph signal processing, diffusion, and modeling of vibrational and electronic properties on graphene and other networks.