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图的拉普拉斯算子的算术性质

Arithmetical properties of Laplacians of graphs

Dino J. Lorenzini

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摘要：

令$M \in M_n (\mathbb Z)$表示任何矩阵。把$M$看作线性映射$M:{\mathbb Z}^n \to {\mathbb Z}^n$，我们用${\Image}(M)$表示$M$的列向量的$\mathbb Z$ -跨度.令$e_1, ..., e_n,$表示${\mathbb Z}^n$的标准基，并令$E_{ij}: = e_i - e_j$、$ (i \neq j)$。在这篇文章中，我们感兴趣的是组${\mathbb Z}^n /{\Image}(M)$，特别是在该组的元素定义的图像$τ_{ij}$的矢量$E_{ij}$下的商${\mathbb Z}^n \to {\mathbb Z}^n / {\Image} (M)$。本文主要研究$M$是图的拉普拉斯算子的情况。在这种情况下，元素$τ_{ij}$有有限的顺序，我们研究如何与这些订单的图形的几何。应用到理论的半稳定减少曲线将出现在即将到来的文章。

英文摘要：

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.

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