Tuva Hirschberg: On the Ehrhart theory of toric Gaussian graphical models

Supervisor: Liam Solus
Abstract: Recent research has classified Gaussian graphical models with toric vanishing ideals. These ideals are associated with convex polytopes. However, the combinatorics of these polytopes has largely been unexplored. This thesis explores the Ehrhart theory of Gaussian graphical models with toric vanishing ideals. Analyzing the vanishing ideal induced by the path we find a bijection between the maximal faces of an initial complex and Dyck paths. Furthermore, we find that h*-polynomial of the polytope induced by the vanishing ideal is the Narayana polynomial. Lastly, we conjecture a characterization of the maximal faces of the polytope associated with the vanishing ideal of families of block graphs we call chain graphs.