| Abstract: Given a matrix, a determinantal ideal is an ideal generated by all minors of a specified size of that matrix. These ideals in general are unwieldy in the sense that they have many generators. However, they have a rich combinatorial structure that makes it feasible to study them. An open question by Eisenbud, Huneke, and Ulrich, concerns the heights of these determinantal ideals. They ask if the height of determinantal ideals of symmetric matrices can be bounded above by the height in the generic case. Many cases of this question remain open. In joint work with Hunter Simper, we prove a very specific case of this conjecture by translating the problem in terms of edge covers of a graph. In our special setting our symmetric matrices have 0's along their diagonal, and we look at the ideal of their 3x3 minors. In this setting, it is possible to associate to these ideals a graph that comes from a certain type of edge coverings. The structure of these graphs shed light on the structure of the matrices. We prove a structure result for these graphs and in this way we prove a special case of the question in a purely combinatorial fashion. In this talk, I’ll provide some background on the conjecture, and then discuss the graph theory used in tackling a special case of it. The graph theory is elementary in nature and hence the talk, I hope, will be fairly introductory. |