Let $G$ be a simple, undirected, connected graph on $n$ vertices, and let $A$ be an abelian group. A function $f:V(G)\rightarrow A$, on the vertices of the graph $V(G)$, is said to be *$A$-harmonic* if for any $v\in V(G)$,
$$
\deg(v) f(v) = \sum_{w\in N(v)} f(w)~,
$$
where $N(v)$ is the set of neighbours of $v$ and $\deg(v) = |N(v)|$.

**Question 1:** Let $A = \mathbb Z_k$ be the ring of integers modulo $k$. Given two distinct vertices $v,w\in V(G)$, I want to know when there is a $\mathbb Z_k$-harmonic function $f$ such that $f(v)\ne f(w)$.

When $A=U(1)$, I know the complete answer using the injectivity and universal property of the *Abel-Jacobi map* on the graph. (See Chapter 3.5 of [1] for definition and properties of this map.) The answer is as follows: *there is a $U(1)$-harmonic function $f$ with $f(v)\ne f(w)$ if and only if there are at least two paths from $v$ to $w$ in $G$.*

I am not sure what to do when $A=\mathbb Z_k$ except for some very special graphs. For example, say $G$ is the cycle graph on $n$ vertices. Then such an $f$ exists if and only if the number of edges in either path from $v$ to $w$ is not divisible by $\gcd(n,k)$.

I have looked for any literature on this question and found that a $\mathbb Z_k$-harmonic function is also called a *balanced vertex-weighting* [2], or a *conservative vertex colouring* [3]. However, these notions don't seem to be commonly studied, and they don't discuss when two vertices can be separated by a $\mathbb Z_k$-harmonic function.

**Question 2:** Is there an equivalent way of phrasing the above question that's more commonly studied, say, a $k$-colouring-type problem?

[1]: Corry and Parkinson, *Divisors and Sandpiles* (2010), http://people.reed.edu/~davidp/divisors_and_sandpiles/mbk_draft.pdf

[2]: Berman, *Bicycles and Spanning Trees*, SIAM Journal on Algebraic Discrete Methods 1986 7:1, 1-12

[3]: Lamey, Silver, and Williams, *Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure* (2014), https://arxiv.org/abs/1408.6570