Zeros of Chromatic and Flow Polynomials of Graphs

We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.Comment: 21 pages, corrected statement of Theorem 34 and some typo

Graph polynomials and their applications I: The Tutte polynomial

In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized, and how they can encode information relevant to physical applications. We conclude with a brief discussion of computational complexity considerations.Comment: This is a preliminary version of one of two book chapters for inclusion in a volume on graph structures. Minor change

Zero-Free Intervals for Flow Polynomials of Near-Cubic Graphs

Let P(G,t) and F(G,t) denote the chromatic and flow polynomials of a graph G. G.D. Birkhoff and D.C. Lewis showed that, if G is a plane near triangulation, then the only zeros of P(G,t) in (−∞,2] are 0, 1 and 2. We will extend their theorem by showing that a stonger result to the dual statement holds for both planar and non-planar graphs: if G is a bridgeless graph with at most one vertex of degree other than three, then the only zeros of F(G,t) in (−∞,α] are 1 and 2, where α ≈ 2.225... is the real zero in (2,3) of the polynomial t 4 − 8t 3 + 22t 2 − 28t + 17. In addition we construct a sequence of ‘near-cubic ’ graphs whose flow polynomials have zeros converging to α from above.