On the real roots of $\sigma$-Polynomials

TL;DR

This paper investigates the real roots of $\sigma$-polynomials, revealing that unlike chromatic polynomials, their roots have a closure of $(-\infty,0]$, indicating a fundamentally different root distribution.

Contribution

It establishes that the only maximal zero-free interval for $\sigma$-polynomials is $[0,\infty)$, contrasting with chromatic polynomial root intervals.

Findings

Real roots of $\sigma$-polynomials are contained in $(-\infty,0]$

Maximal zero-free interval for $\sigma$-polynomials is $[0,\infty)$

Difference in root distributions compared to chromatic polynomials

Abstract

The $\sigma$-polynomial is given by $\sigma(G,x) = \sum_{i=\chi(G)}^{n} a_{i}(G)\, x^{i}$, where $a_{i}(G)$ is the number of partitions of the vertices of $G$ into $i$ nonempty independent sets. These polynomials are closely related to chromatic polynomials, as the chromatic polynomial of $G$ is given by $\sum_{i=\chi(G)}^{n} a_{i}(G)\, x(x-1) \cdots (x-(i-1))$. It is known that the closure of the real roots of chromatic polynomials is precisely $\{0,~1\} \bigcup [32/27,\infty)$, with $(-\infty,0)$, $(0,1)$ and $(1,32/27)$ being maximal zero-free intervals for roots of chromatic polynomials. We ask here whether such maximal zero-free intervals exist for $\sigma$-polynomials, and show that the only such interval is $[0,\infty)$ -- that is, the closure of the real roots of $\sigma$-polynomials is $(-\infty,0]$.