# On the laplacian of connected, undirected, multigraphs without loops

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 ) ?

[...] there is some literature relating critical groups for different graphs in a functorial fashion, having roots in an early paper$^1$ of Berman , work of Lorenzini , and unpublished work of Treumann; see [6, 7] for some notable applications of this functoriality; the theory is reviewed in Section 2.2 below. More recently, the role of critical groups in the analogy between graphs and algebraic curves, discussed originally by Bacher, de la Harpe and Nagnibeda  (see also Biggs ) has been extended further to chip-firing or sandpile groups of metric graphs and the theory of tropical geometry, where some of these notions of functoriality appear also in work of Urakawa  and Baker and Norine [4, 5] under the name of harmonic morphisms; for a recent survey and references, see Perkinson et al .