# reference request: voltage in a resistor network is a unique harmonic function

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.

I paraphrase some statements from Chapter 3 of this paper.

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 $v$.

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?

• Kemeny and Snell: Denumerable Markov Chains? – Anthony Quas Oct 1 '16 at 9:50
• @AnthonyQuas Doyle and Snell is probably an easier read. – Igor Rivin Oct 1 '16 at 10:09
• @IgorRivin: I think that is the book that I was thinking of. – Anthony Quas Oct 1 '16 at 10:31
• Nice question. But be careful: see xkcd.com/356 – Joël Oct 6 '16 at 19:07