Symmetries of Linear Systems on Graphs
In the divisor theory of graphs, a finite, connected graph is viewed as a discrete analog of a Riemann surface. A divisor D on a graph is an assignment of integers to each vertex of the graph. An important statistic in this setting is the complete linear system of D, which is the collection of effective (e.g. non-negative) divisors linearly equivalent to D via the discrete Laplacian operator. Recently, S. Brauner, F. Glebe, and D. Perkinson characterized all complete linear systems on a finite graph G using generating sets called primary and secondary divisors. By counting the integer points in a cone defined by a bijection with the primary and secondary divisor sets, this gives the number of divisors in each complete linear system. We extend their results by computing the stable sets of divisors in any complete linear system under action by a symmetry of G and define an action of the symmetry on the set of cones that allows us to count the number of stable divisors.
