Arithmetical properties of Laplacians of graphs

Abstract

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 $τ_{ij}$ 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 $τ_{ij}$ 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.