Its cool how the Laplacian matrix of a graph is actually the same idea as the Laplacian operator that we see in calculus and differential equations since they seem like totally different concepts at first.
keenan
@0x484884 Yes, though you need to be careful: a graph Laplacian with edge weights 1 will not in general give you a good approximation of the Laplacian (e.g., on a triangle mesh with irregular edge lengths, triangle sizes, etc.). But yeah, a lot of the intuition you gain from studying PDEs can and does provide insight into more discrete/computational problems involving the graph Laplacian.
The formula for the triangle mesh is kind of bizarre, any further reading on that? Also are there any other kind of meshes?
@Arthas007 Yep! See Chapter 6 of my course notes from discrete differential geometry, which explain how the cotan formula is actually quite natural.
Other common kinds of meshes include general polygonal meshes (not necessarily triangles, hence not necessarily planar), and tetrahedral meshes (one dimension higher). There's been a lot of work on how to define "good" Laplace operators for such meshes. E.g., Discrete Laplacians on General Polygonal Meshes and Properties of Laplace Operators for Tetrahedral Meshes.
Its cool how the Laplacian matrix of a graph is actually the same idea as the Laplacian operator that we see in calculus and differential equations since they seem like totally different concepts at first.
@0x484884 Yes, though you need to be careful: a graph Laplacian with edge weights 1 will not in general give you a good approximation of the Laplacian (e.g., on a triangle mesh with irregular edge lengths, triangle sizes, etc.). But yeah, a lot of the intuition you gain from studying PDEs can and does provide insight into more discrete/computational problems involving the graph Laplacian.