Symmetric Function Generalizations of Graph Polynomials

Motivated by certain conjectures regarding immanants of Jacobi-Trudi matrices, Stanley has recently defined and studied a symmetric function generalization XG of the chromatic polynomial of a graph G. Independently, Chung and Graham have defined and studied a directed graph invariant called the cover polynomial. The cover polynomial is closely related to the chromatic polynomial and to the rook polynomial, so it is natural to ask if one can mimic Stanley's construction and generalize the cover polynomial to a symmetric function. The answer is yes, and the bulk of this thesis is devoted to the study of this generalization, which we call the path-cycle symmetric function. We obtain analogues of some of Stanley's theorems about XG and we generalize some of the theory of the cover polynomial and the rook polynomial. In addition, we are led to define a symmetric function basis that seems to be a "natural" generalization of the polynomial basis d ) k=0,1,...,d and we prove a combinatorial reciprocity theorem that gives an affirmative answer to Chung and Graham's question of whether the cover polynomial of a digraph determines the cover polynomial of -its complement. The reciprocity theorem also ties together several scattered results in the literature that previously seemed unrelated. In the remainder of the thesis, we prove some miscellaneous results about Stanley's function XG and we also sketch briefly in the introduction a possible approach to generalizing other graph polynomials (and a few other combinatorial polynomials) to symmetric functions. Thesis supervisor: Richard P. Stanley Title: Professor of Applied Mathematics