# Heat equation for a finite graph

Thank you for an interesting website.

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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