The Hilbert Series of Paths, Cycles, and Related Graphs

The Hilbert Series of a finitely-generated graded R-module, M, is a series which is often given in the form a rational function in the variable, t. This series encodes a great many invariant properties of the module M. In this dissertation, I study the Hilbert series and related invariants of the graph rings for paths and cycles. By utilizing a result of Kyle Trainor, I am able to examine the Hilbert series and the related invariants of these graph rings recursively through second and higher-order difference equations. This technique allows me to extract information about the Hilbert series that has not been previously well known, and for which other techniques are limited.