Tutte-Equivalent Matroids

• Main Author: Rocha, Maria Margarita
• Format: Others
• Published: CSUSB ScholarWorks 2018
• Subjects: Matroid Theory; Tutte Polynomail; Invariants; graph theory; linear algebra; Discrete Mathematics and Combinatorics; Set Theory
• Online Access: https://scholarworks.lib.csusb.edu/etd/759; https://scholarworks.lib.csusb.edu/cgi/viewcontent.cgi?article=1842&context=etd

We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a well-defined two-variable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte polynomial gives us significant information about a matroid, it does not uniquely determine a matroid. This thesis will focus on non-isomorphic matroids that have the same Tutte polynomial. We call such matroids Tutte-equivalent, and we will study the characteristics needed for two matroids to be Tutte-equivalent. Finally, we will demonstrate methods to construct families of Tutte-equivalent matroids.