As An Approximation To The Negative Continuous Laplacian
The graph Laplacian matrix can be further viewed as a matrix form of an approximation to the negative Laplacian operator obtained by the finite difference method. In this interpretation, every graph vertex is treated as a grid point; the local connectivity of the vertex determines the finite difference approximation stencil at this grid point, the grid size is always one for every edge, and there are no constraints on any grid points, which corresponds to the case of the homogeneous Neumann boundary condition, i.e., free boundary.
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