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Let G be a simple graph. Denote by Q the vertex-edge incidence matrix corresponding to some orientation of the edges of G. If Q' is the transpose of Q, then the Laplacian matrix, L(G) = QQ'. The matrix K(G) = Q'Q is its so-called edge version. Viewed as integer matrices, the Smith Normal Form of L(G) is complicated, but the Smith Normal Form of K(G) is completely determined by the numbers of vertices, edges, and components of G. There is an application to digraph flows with constant resultants over abelian groups.

Graphs G and H are Laplacian congruent if there is an integer matrix E of determinant +1 such that E'L(G)E = L(H). Using an old result of H. Whitney and a new result of W. Watkins, it is shown that Laplacian congruent graphs have the same chromatic polynomial.

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