Arithmetical properties of Laplacians of graphs

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Let M ∈ Mₙ (Z) denote any matrix. Thinking of M as a linear map M:Zⁿ → Zⁿ, we denote by (M) the Z-span of the column vectors of M. Let e₁,..., eₙ, denote the standard basis of Zⁿ, and let Eij: = eᵢ - eⱼ, (i ≠ j). In this article, we are interested in the group Zⁿ /(M), and in particular in the elements of this group defined by the images τij of the vectors Eij under the quotient Zⁿ → Zⁿ / (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 τij 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.