Betti numbers of edge ideals of cyclic graphs

Abstract

The goal of this Honors thesis is to analyze the Betti numbers of the edge ideals of cyclic graphs. In particular, our aim is to prove the values of the Betti numbers corresponding to the minimal linear first syzygies and the minimal quadratic first syzygies of the edge ideal of a cyclic graph. The proofs of these claims rely on a theorem of Bayer, Charalambous, and Popescu relating combinatorial topology to commutative algebra and other combinatorial techniques. In this paper we also conjecture formulas to determine the Betti table for the edge ideal of a cyclic graph on any number of vertices. We developed our conjectures by examining the Betti tables for the edge of ideals of cyclic graphs on three to thirty vertices and used regression to create formulas describing the patterns we observed.