Date of Award
5-11-2025
Date Published
June 2025
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
Advisor(s)
Lee Kennard
Keywords
3-regular;cages;cubic;graph;systole bounds;weighted girth
Subject Categories
Mathematics | Physical Sciences and Mathematics
Abstract
Systole bounds on graphs provide a combinatorial approach to establishing geometric bounds on a special class of torus representations. This implication, which arises through a connection with invariants of regular matroids, is explored in recent work by Kennard, Wiemeler, and Wilking, where it is applied to problems in Riemannian geometry. In this thesis, we analyze the systoles of cubic graphs with small Betti number and prove optimal upper bounds that depend on the Betti number and girth of the graph, refining previous results. Cubic graphs are particularly important in this work because they represent the extreme cases. Our approach uses improved recursive estimates and also includes direct computation and classification arguments in cases where the recursive bounds are not optimal. Our method is also self-contained and independent of non-trivial results from topological graph theory that were applied in the work of Kennard, Wiemeler, and Wilking. Our detailed calculations also result in several rigidity results for cubic graphs.
Access
Open Access
Recommended Citation
Sato, Chelsea, "Systoles of Graphs with Small Betti Number" (2025). Dissertations - ALL. 2125.
https://surface.syr.edu/etd/2125
