Fry, Andy J., authorCavalieri, Renzo, advisorShoemaker, Mark, committee memberWilson, James, committee memberTavani, Daniele, committee member2021-09-062021-09-062021https://hdl.handle.net/10217/233859We use a graph to define a new stability condition for the algebraic and tropical moduli spaces of rational curves. Tropically, we characterize when the moduli space has the structure of a balanced fan by proving a combinatorial bijection between graphically stable tropical curves and chains of flats of a graphic matroid. Algebraically, we characterize when the tropical compactification of the compact moduli space agrees with the theory of geometric tropicalization. Both characterization results occur only when the graph is complete multipartite.born digitaldoctoral dissertationsengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.Text