Tutte Polynomials, Chromatic Polynomials and Matroids

Abstract

In this thesis we study two polynomials associated with matroids, namely, the characteristic polynomial and the Tutte polynomial. We define an operation called H-lift on restrictions of Dowling group geometries. Then we find an expression for the characteristic polynomial on an H-lift. We then use this expression of the characteristic polynomial of an H-lift to show that if a certain sequence of H-lifts is done on a special type of tangential k-block over GF(q) the resulting matroid is a tangential (k + n)-block over GF(q). We also use the known characteristic polynomials of projective and affine geometries to give explicit expressions of Tutte polynomials of projective geometries and affine geometries. We give a theorem on cocircuit partitions of binary affine matroids. Furthermore, we give polymatroids associated with binary, affine matroids and extend Tutte polynomials to polymatroids.