On the real roots of $\ddot{I}ƒ$-Polynomials

Abstract

The $\ddot{I}ƒ$-polynomial is given by $\ddot{I}ƒ(G,x)={\sum }_{i=\ddot{I}‡(G)}^n{a}_i(G)x^i$, 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 ${\sum }_{i=\ddot{I}‡(G)}^n{a}_i(G)x(x-1)\cdots (x-(i-1))$. 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 $\ddot{I}ƒ$-polynomials, and show that the only such interval is $[0,\infty )$ -- that is, the closure of the real roots of $\ddot{I}ƒ$-polynomials is $(-\infty ,0]$.

9 pages, 6 figures