Discrete harmonic functions
A (continuous) harmonic function f on an open set a function that is twice differentiable and satisfied Laplace's equation: a2f = 0. Such functions have a mean value property: at a point x interior to the set, f(x) equals the average value of f over any ball around x that fits inside the region. It turns out that the converse is true: a locally-integrable function satisfying the mean value property is harmonic.
We can take the mean value property from the continuous case as the definition in the discrete case. A function on a graph is harmonic at a node x if f(x) equals the average of f's values on the nodes connected to x. A function is said to have a singularity at points of a graph where it is not harmonic.
A non-constant function on a finite connected graph cannot be harmonic at every node. It has to have at least two singularities, namely the points where it takes on its maximum and its minimum. This is analogous to Liouville's theorem for (continuous) harmonic functions: a bounded harmonic function on all of Euclidean space must be constant. Some infinite graphs have non-constant functions that are harmonic everywhere. For example, linear functions on the integers are harmonic.
In the continuous case, we want to find solutions to Laplace's equation satisfying a boundary condition. We want to find a function that has the specified values on the boundary and is harmonic in the interior. With the right regularity conditions, this solution exists and is unique.
We can do the same for discrete harmonic functions. If we specify the values of a function on some non-empty subset of nodes, there is a unique discrete harmonic function that extends the function to the rest of a graph.
Discrete harmonic functions come up in applications such as random walks and electrical networks.
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