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In what way are the Laplacian operator (defined on functions from $\Bbb{R}^n$ to $\Bbb{R}$ which are twice-differentiable) and the Laplacian matrix (defined on simple graphs) similar or related?

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They are analogues of each other. The continuous Laplacian operator measures how a function changes “on average” as you move away from a given point $\mathbf p$. You can think of it as finding the average value of a function $u$ over all points a small fixed distance away from $\mathbf p$. For a function $u$ defined on the nodes of a graph, the discrete Laplacian operator similarly computes a weighted average of the values of $u$ at a node’s nearest neighbors. Formally, this operator is a linear map from $C^0$, the space of $0$-cochains of the graph, to $C_0$, the space of $0$-chains of the graph, and its matrix relative to a particular pair of bases for these spaces is the Laplacian matrix of the graph.

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    But the Laplacian matrix is not defined for a function $u$ on a graph, it's defined for the graph itself (as the rank matrix minus the adjacency matrix) – 2017-01-25
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    @Lior An element $u$ of $C^0$ *is* a function on the graph: it associates a number with each node. – 2017-01-25