Expansions of Chromatic Polynomials in Different Bases

In this thesis, we give an introduction to chromatic polynomials and explore alternative representations of the chromatic polynomial to its standard form, specifically the falling factorial and closely related homogeneous falling factorial basis expansions. Expressions are derive for chromatic polynomials of certian graphs in both the standard and falling factorial basis. We then prove non-negativity of the coefficients in the homogeneous expansion for certain families of graphs. However, a numerical search exhaustively listing the homogeneous expansion coefficients of the chromatic polynomials of m-partite graphs of order n <= 42 reveals several cases where the coefficients fail to be non-negative.