The divisor theory of graphs views a finite connected graph G as a discrete version of a Riemann surface. Divisors on G are formal integral combinations of the vertices of G, and linear equivalence of divisors is determined by the discrete Laplacian operator for G. As in the case of Riemann surfaces, we are interested in the complete linear system |D| of a divisor D-the collection of nonnegative divisors linearly equivalent to D. Unlike the case of Riemann surfaces, the complete linear system of a divisor on a graph is always finite. We compute generating functions encoding the sizes of all complete linear systems on G and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the (dual of the) Jacobian group of G. If G is a cycle graph, our results lead to a bijection between complete linear systems and binary necklaces. The final section generalizes our results to a model based on integral M -matrices.2010 Mathematics Subject Classification. primary 05C30, secondary 05C25. For the combinatorial structure of |D| in the case of a metric graph (tropical curve), see [13]. 1 Divisor theory preliminariesLet G = (V, E) be a connected, undirected multigraph with finite vertex set V and finite edge multiset E. Many of our constructions will depend on fixing a vertex q ∈ V , which we do now, once and for all. Loops are allowed but our results are not affected if they are removed. We let N := Z ≥0 denote the natural numbers.We recall some of the theory of divisors on graphs, referring readers unfamiliar with this theory to [3] or to the textbooks [8] and [15]. A divisor on G is an element of the free abelian group on the vertices of G,The degree of a divisor D is the sum of its coefficients: deg(D) := v∈V D(v). For instance, if we consider v ∈ V as a divisor, then deg(v) = 1. We use the notation deg G (v) to refer to the ordinary degree of a vertex-the number of edges incident on v. The set of divisors of degree k is denoted by Div k (G). The (discrete) Laplacian operator of G is the function L : Z V → Z V given by

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A discrete group is matricially stable if every function from the group to a complex unitary group that is "almost multiplicative" in the point-operator norm topology is "close" to a genuine unitary representation. It follows from a recent result due to Dadarlat that all amenable, groups with non-torsion integral 2-cohomology are not matricially stable, but the proof does not lead to explicit examples of asymptotic representations that are not perturbable to genuine representations. The purpose of this paper is to give an explicit formula, in terms of cohomological data, for asymptotic representations that are not perturbable to genuine representations for a class of groups that contains all finitely generated groups with a non-torsion 2-cohomology class that corresponds to a central extension where the middle group is residually finite. This class includes polycyclic groups with non-torsion 2-cohomology.

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