Activities and coefficients of the Tutte polynomial

Abstract

Matroids are combinatorial objects that capture abstractly the essence of dependence. The Tutte polynomial, defined for matroids and graphs, has a numerous amount of information about these structures. In this thesis, we will introduce matroids, define them and give the most important properties they have. Then we will define the Tutte polynomial and give interesting results found in this area of research. After that we will start interpreting Tutte coefficients. First we will discuss about some specific Tutte coefficients and the relation between Tutte coefficients and parallel and series classes in a matroid. These relations were found recently for graphs, and we worked out in applying them for matroids and adding to the given results. Then we will discuss will be about a theorem linking matroid connectivity and the coefficient of x, where we have also our contribution in proving it using the activities. Finally, we will introduce Brylawski's equations. We will discuss the proof of two cases of this equation. The proof of the second equation in this section is a result of our work on this thesis.