On the location of chromatic zeros of series-parallel graphs

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, showing density outside the disk $\mathrm{\mid }q-1\mathrm{\mid }\le 1$, we show density of these zeros in the half plane $\mathrm{\Re }(q)>3\mathrm{/}2$ and we show there exists an open region $U$ containing the interval $(0,32\mathrm{/}27)$ such that $U\setminus \{1\}$ does not contain zeros of the chromatic polynomial of series-parallel graphs. We also disprove a conjecture of Sokal by showing that for each large enough integer $\mathrm{\Delta }$ there exists a series-parallel graph for which all vertices but one have degree at most $\mathrm{\Delta }$ and whose chromatic polynomial has a zero with real part exceeding $\mathrm{\Delta }$.