On the Real Roots of $\sigma$-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 $\sum _{i=\chi (G)}^{n} a_{i}(G)\, x(x-1) \ldots (x-(i-1))$. 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 $(-\infty ,0]$.