Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

This dissertation is a collection of work concerning the structure and enumeration of circuits and other distinguished sets in a matroid. The Tutte polynomial, recognized as the universal deletion-contraction invariant in matroid and graph theory, is a natural starting point when considering enumeration problems for matroids. The main result in Chapter 2 generalizes a theorem of Dean Lucas concerning the Tutte polynomial and its behavior under rank-preserving weak maps. This generalization provides an avenue for comparing the numbers of circuits, bases, rank-k flats, and hyperplanes of a matroid containing an element given two distinct, yet related, elements.

Chapter 3 addresses extremal problems for the number of circuits containing a fixed element. In 2019, Oxley and Wang determined lower bounds for the number of circuits in a connected matroid and for the number of circuits in a connected matroid containing some fixed element. We determine that the connected matroids whose set of circuits attain equality in the latter bound are precisely those whose set of circuits attain equality in the former bound. We conclude Chapter 3 by determining a lower bound for the number of circuits containing a fixed element in a 3-connected matroid and by describing the 3-connected matroids that attain this bound.

Finally, Chapter 4 presents a symmetric variant of the strong circuit elimination axiom. As this symmetric variant is not a property of all matroids, the main result of this chapter provides a characterization of the connected matroids that satisfy this property.

Date

4-8-2026

Committee Chair

Oxley, James

LSU Acknowledgement

1

LSU Accessibility Acknowledgment

1

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