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I would like to construct the heat equation for a finite graph $G$ using basic math concepts. For example, if $G = \mathbb{Z}/m\mathbb{Z}$ then I think of $G$ as a discrete $x$-axis with $\delta_x = 1$ and use calculus differentials to approximate $u_{xx}$. This eventually gives me the matrix form of the heat equation for $G$, $u_{t} + A*u = 0$. If $G = \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ then I look at $G$ as a discrete $xy$-plane with $\delta_x = \delta_y = 1$ and approximate $u_{xx}$ and $u_{yy}$. Suppose $G$ is now an arbitrary graph like for example, a weighted directional bipartite graph with $6$ vertices. Is there a way for me to construct the heat equation for $G$ using calculus differentials and perhaps other basic math concepts?

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I suspect that what you're looking for is the Laplacian of a graph or perhaps a weighted version of it. See, for example, http://en.wikipedia.org/wiki/Laplacian_matrix and the references there for information about this. It should be easy to modify the ideas there to take into account the possibility that different edges of your graph are regarded as having different lengths, so that they contribute with different weights to a "weighted Laplacian" of the graph.

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Thanks. My audience are basic Calculus students. I need to use simple concepts. Here's my idea. Let G be a 4-regular graph, e.g., Z/mZ x Z/nZ or the bipartite graph with 8 vertices. Choose a vertex v. View one pair of edges at v as the "x-axis". Construct the differential "u_{xx}". View the other pair of edges as the "y-axis". Construct the differential "u_{yy}". When you compute "u_{xx}+u_{yy}", how you pair the edges no longer seems to matter. The resulting Laplacian looks O.K. Is my approach O.K. or is there a better one out there? What do I do if the graph has say, 5 edges per vertex? –  Marvin Nov 24 '13 at 1:55

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