  Let $M \in M_n (\mathbb Z)$ denote any matrix. Thinking of $M$ as a linear
map $M:{\mathbb Z}^n \to {\mathbb Z}^n$, we denote by ${\Image}(M)$ the
$\mathbb Z$-span of the column vectors of $M$. Let $e_1, ..., e_n,$ denote the
standard basis of ${\mathbb Z}^n$, and let $E_{ij}: = e_i - e_j$, $ (i \neq
j)$. In this article, we are interested in the group ${\mathbb Z}^n
/{\Image}(M)$, and in particular in the elements of this group defined by the
images $\tau_{ij}$ of the vectors $E_{ij}$ under the quotient ${\mathbb Z}^n
\to {\mathbb Z}^n / {\Image} (M)$. Most of this article is devoted to the study
of the case where $M$ is the laplacian of a graph. In this case, the elements
$\tau_{ij}$ have finite order, and we study how the geometry of the graph
relates to these orders. Applications to the theory of semistable reduction of
curves will appear in a forthcoming article.
