Arithmetical properties of Laplacians of graphs

Dino J. Lorenzini

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 ve…