Let $G$ be a connected, undirected multigraph, without loops. Let $L_G = D_G - A_G$, where $D_G= diag (val (v_1), \ldots , val (v_n) )$ where $n$ is the no. of vertices of $G$ and $val (v_i)$ denotes the valence (i.e. degree ) of $v_i$, and where $A_G$ is the adjacency matrix of $G$, i.e. the $ij$-th entry of $A_G$ is $1$ if vertices $v_i, v_j$ are adjacent the entry is $0$ otherwise. That is, $L_G$ is the Laplacian matrix of $G$.
Then let $L(G)$ denote the subgroup of $\mathbb Z^n$ generated by the columns of $L_G$, let us call it the Laplacian lattice of the graph $G$. So given a graph, we can naturally associate a group with it.
Now given two connected, undirected multigraphs without loops $G,H$;
Can we define a suitable subset $Mor (G,H)$ of the set of all graph homomorphisms between the graphs such that the collection of all connected, undirected graphs, without loops, as objects along with the suitable set of morphisms, form a category such that the Laplacian group of the graphs is a functorial construction?
(the target space being the category of abelian groups and group homomorphisms ... of course given a morphism between graphs as in the category, we will need to define a suitable morphism between the Laplacian groups of the two graphs ... that's part of the problem too ... and I would be happy if we can even get a contravariant functor ) ?