The Inverse Characteristic Polynomial Problem for Graphs over Finite Fields

Let $\mathbb{F}$ be a finite field and let $G$ be a graph on $n$ vertices. We study the possible characteristic polynomials that may be realized by matrices $A$ whose graph is $G$. We primarily focus on the case $G$ is a tree $T$. Computation is used extensively, and several interesting structures emerge. This should pave the way for future study. Prior to this work, little was known. For some graphs, such as the path, it is thought that all monic polynomials are realized over any field with more than two elements, although showing this has so far proved elusive.