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 $E_{ij}:=e_i-e_j$, $(i\ne j)$. 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.