Open Access Senior Thesis
Bachelor of Science
Francis Edward Su
2020 Savana Jade Ammons
In this thesis, we develop a discrete analogue to the Poincaré–Hopf Theorem. We define the notion of a vector field on a graph, and establish an index theory for such a field. Specifically, we create well-defined indices for the nodes and “cells" formed by a planar graph. Then, we show that the sum of these indices remains constant for certain types of planar graphs, regardless of the discrete vector fields they have.
Ammons, Savana, "A Discrete Analogue for the Poincaré-Hopf Theorem" (2020). HMC Senior Theses. 237.