Let $G$ be some outdegree-regular graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ from $\mathbb{Z}^n$ to $\mathbb{Z}/k\mathbb{Z}$ with the property that performing a chip-firing move on a vector in $\mathbb{Z}^n$ increases the value of $f$ by 1 mod $k$ (I call such a function $f$ a chip-firing “clock”). Such an $f$ will exist only for certain values of $k$. This feels like it’s related to the structure of the cokernel of the Laplacian but I’m having trouble seeing exactly what's going on. I’m interested in knowing about all such functions $f$ for a given $G$.

Let $G$ be some outdegree-regular directed graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ from $\mathbb{Z}^n$ to $\mathbb{Z}/k\mathbb{Z}$ with the property that performing a chip-firing move on a vector in $\mathbb{Z}^n$ increases the value of $f$ by 1 mod $k$ (I call such a function $f$ a chip-firing “clock”). Such an $f$ will exist only for certain values of $k$. This feels like it’s related to the structure of the cokernel of the Laplacian but I’m having trouble seeing exactly what's going on. I’m interested in knowing about all such functions $f$ for a given $G$.