Zeros of chromatic and flow polynomials of graphs

Abstract

We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.

Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids

“…Unimodality (χ-1) was conjectured by Read [34] in late 1960s (for graphs), and logarithmic concavity in the form (χ-2) was conjectured by Hoggar [23] in early 1970s (also for graphs). The literature on zeros or coefficients of chromatic polynomials is extensive - [8,25,34,38] provide a starting point.…”

Discrimination power of graph measures based on complex zeros of the partial Hosoya polynomial

“…Related work focuses on the distribution of the zeros of graph polynomials. For example, the zeros of flow and chromatic polynomials have been studied by Jackson [3] and Woodall [5]. They discuss regions or zero-free intervals of these polynomials.…”

“…Various graph polynomials have proved useful in mathematical chemistry and related disciplines [1][2][3][4]. Counting polynomials are of particular importance, and the Hosoya polynomial [1] is a prime example.…”

Planar Triangulations with Real Chromatic Roots Arbitrarily Close to 4

“…One of the unresolved problems described by Jackson [11] was a conjecture of Beraha which (among other things) asserted the existence of planar graphs with real chromatic roots arbitrarily close to 4. At the time of Jackson's survey the largest known real chromatic root of a planar graph was 3.8267 · · · obtained from a 21-vertex graph found by Woodall.…”