An undirected graph may be regarded as a resistor network where each edge corresponds to a resistor of unit resistance. This paper covers such an approach.


* [*On electric resistances for distance-regular graphs*](https://arxiv.org/abs/1103.2810) by Jack Koolen, Greg Markowsky, and Jongyook Park


I paraphrase some statements from Chapter 3 of this paper.




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Consider a graph $G$ as a resistor network where each edge is a unit resistance. 
To compute the effective resistance between two vertices $u$ and $v$, assume that a battery of voltage $V$ is attached to $u$ and $v$ producing a current through the graph.
The current flowing through the circuit can be determined by calculating the voltage at each point on the graph.
A function $f$ on the vertex set $V$ of $G$ is **harmonic** at a point $z\in V$ if $f(z)$ is the average of neighboring values of f; that is,
\begin{equation\*}
\sum\_{\text{for any $x$ adjacent to $z$}}\big(f(x)-f(z)\big) = 0.
\end{equation\*}
The voltage function on V can be characterized as the *unique* function which is harmonic on $V − \{u, v\}$ having the prescribed values on $u$ and $v$.




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I'd like to use this characterization on my research. But I cannot find a reference for this 'harmonic characterization' on any other textbook. Even the above paper doesn't cite any reference for this statement. I skim through many graph theory textbooks (such as Bollobas) but in vain.


Is this characterization true? Which textbooks or papers can I use as a reference?



reference request: voltage in a resistor network is a unique harmonic function
If I am reading correctly, this appears as Theorem 1.15 of Geoffrey Grimmett's book *Probability on Graphs*. 


It is available for free on [Grimmett's web site](http://www.statslab.cam.ac.uk/~grg/books/USpgs-rev3.pdf) and has also been [published by Cambridge University Press](http://www.cambridge.org/us/academic/subjects/statistics-probability/probability-theory-and-stochastic-processes/probability-graphs-random-processes-graphs-and-lattices?format=PB&isbn=9780521147354).


