On the real roots of $σ$-Polynomials

The $σ$-polynomial is given by $σ(G,x) = \sum_{i=χ(G)}^{n} a_{i}(G)\, x^{i}$, 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 $\sum_{i=χ(G)}^{n} a_{i}(G)\, x(x-1) \cdots (x-(i-1))$. 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 $σ$-polynomials, and show that the only such interval is $[0,\infty)$ -- that is, the closure of the real roots of $σ$-polynomials is $(-\infty,0]$.

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