Extensions of the Tutte Polynomial and Results on the Interlace Polynomial

Abstract

In graph theory, graph polynomials are an important tool to encode information from a graph. The Tutte polynomial, first introduced in 1947, is one of the most important graph polynomials due to its universality. Here, we present three classic definitions of the Tutte polynomial via a deletion-contraction recursion, via rank and nullity, and via activities. We will touch on the significance of this polynomial to the field of mathematics to motivate an extension to signed graphs. Extending the polynomial to retain deletion contraction and inactivity information, we introduce an extended Tutte polynomial to allow for the construction of a Tutte like polynomial on signed graphs. Using the extended information, we examine the monomials of these polynomials as grid walks. Using grid walking and the extended Tutte polynomial, we investigate the relationship between the Tutte polynomial of a graph and that of its bipartite representation. This is done with a view toward the construction of a Tutte like polynomial for oriented hypergraphs.

While many graph polynomials are directly related to the Tutte polynomial, there are also a wide variety of polynomials related in special cases only. One such polynomial is the Martin polynomial and, related to it, the interlace polynomial. Here, we discuss how these two polynomials are related and how results on the Martin polynomial can be extended to the interlace polynomial. The Martin invariant, a specific evaluation of the Martin polynomial, obeys the symmetries of the Feynman period. The Feynman period of a graph is useful in quantum field theory, but difficult to compute and thus there is interest in finding graph invariants that have the same symmetries. It was established that the interlace polynomial on interlace graphs was equal to the Martin polynomial on the associated 4-regular graph. While only graphs that do not contain a set of forbidden vertex minors are interlace graphs, the interlace polynomial is defined over all graphs. We discuss how this provides a way to try and extend the notion of Feynman symmetries via the interlace polynomial and some specific classes of graphs with formulas. Additionally, the interlace polynomial is only equal to the Martin polynomial for interlace graphs of 4-regular graphs, but the Martin polynomial is defined for 2k-regular graphs. Thus, we work toward creating an interlace-like polynomial for graphs derived from 2k-regular cases of the Martin polynomial.