Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Enumerating linear systems on graphs   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.

26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
9
The categories of this paper are: nlin