Arithmetical properties of laplacians of graphs

Let M is an element of M-n(Z) denote any matrix. Thinking of M as a linear map M : Z(n) --> Z(n), we denote by Im(M) the Z-span of the column vectors of M. Let e(1), ..., e(n), denote the standard basis of Z(n), and let E-y = e(i) - e(j), (i not equal j). In this article, we are interested in the group Z(n)/Im(M), and in particular in the elements of this group defined by the images tau (ij) of the vectors E-ij under the quotient Z(n) --> Z(n)/Im(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. We give in particular a criterion in terms of the topology of the graph to determine when such an element has order 1 or 2.