Generalizing Parking Polytopes to Non-Complete Graphs by Felix Nusbaum ’25

Generalizing Parking Polytopes to Non-Complete Graphs by Felix Nusbaum ’25, Friday May 16, 1:00 – 1:50pm, North Science Building 015, Mathematics Thesis Defense

In this thesis, we explore some of the combinatorial and geometric structures underlying chip-firing games on graphs, and their relation to parking functions. Specifically, we investigate the convex hulls of superstable configurations on graphs. We show that these objects encode extensive structural information about the graph, including some or all edge relations between vertices, depending on certain connectivity properties of the graph. Building on foundational work in chip-firing and parking functions, we develop tools for partial and complete reconstruction of graphs from the convex hulls of their configurations, and answer the question of when these objects are unique for a given graph.