The generalized Laplacian is defined as: Notice the ordinary Laplacian is a generalized Laplacian. Magnetic Laplacian Entries of the adjacency matrix can be complex-valued, in which case the notion of the matrix symmetry needs to be replaced with a Hermitian matrix. Magnetic Laplacian for a directed graph with real weights is constructed as the Hadamard product of the real symmetric matrix of the symmetrized Laplacian and the Hermitian phase matrix with the complex entries which encode the edge direction into the phase in the complex plane. In the context of quantum physics, the magnetic Laplacian can be interpreted as the operator that describes the phenomenology of a free charged particle on a graph, which is subject to the action of a magnetic field and the parameter is called electric charge. In the following example : Deformed Laplacian The deformed Laplacian is commonly defined as where I is the identity matrix, A is the adjacency matrix, D is the degree matrix, and s is a (complex-valued) number. The standard Laplacian is just and is the signless Laplacian. Signless Laplacian The signless Laplacian is defined as where is the degree matrix, and is the adjacency matrix. Like the signed Laplacian , the signless Laplacian also is positive semi-definite as it can be factored as where is the incidence matrix. has a 0-eigenvector if and only if it has a bipartite connected component other than isolated vertices. This can be shown as This has a solution where if and only if the graph has a bipartite connected component. Directed multigraphs An analogue of the Laplacian matrix can be defined for directed multigraphs. In this case the Laplacian matrix L is defined as where D is a diagonal matrix with Di,i equal to the outdegree of vertex i and A is a matrix with Ai,j equal to the number of edges from i to j (including loops). Open source software implementations SciPy NetworkX Application software scikit-learn Spectral Clustering PyGSP: Graph Signal Processing in Python megaman: Manifold Learning for Millions of Points smoothG Laplacian Change Point Detection for Dynamic Graphs (KDD 2020) LaplacianOpt (A Julia Package for Maximizing Laplacian's Second Eigenvalue of Weighted Graphs) LigMG (Large Irregular Graph MultiGrid) Laplacians.jl See also Stiffness matrix Resistance distance Transition rate matrix Calculus on finite weighted graphs Graph Fourier transform References Algebraic graph theory Matrices Numerical differential equations