Let $X$ be a graph. Let $V(X)$ and $E(X)$ be the sets of vertices and edges of the graph respectively. If $f:V(X) \rightarrow G$ where $G$ is an abelian group, then one can define a graph Laplacian as follows:
$$\Delta f(i) = \sum_{j \mid (ij) \in E(X)} \big(f(i) - f(j)\big)$$
where the sum involves all vertices $j \in V(X)$ such that there is an edge between $i$ and $j$.
In spectral graph theory, one then usually considers the Smith decomposition of the Laplacian. If we represent the Laplacian $\Delta$ by a matrix $L$. Then, there's a pair of invertible integer-valued matrices $P$ and $G$ such that:
$$L = PRQ$$ where $R$ is diagonal: $R = \operatorname{Diag}(r_{1}, \dotsc)$ and:
$$r_{1} \mid r_{2} \mid \dotsb.$$
Now, I am interested in understanding more about the invariant factors $r_a$ and the matrices $P$ and $Q$ in the special case when there is a notion of locality that can be defined for the graph $X$. To be more clear, I am mostly interested in graphs $X$ that are regular lattices embedded on Riemann surfaces. Say $X \subset \Sigma_g$ where $\Sigma_g$ is a genus $g$ Riemann surface.
What can we say about $P$, $Q$, and $R$ in this case?
