  The $\textit{Herschel}$ DEBRIS, DUNES and GT programmes observed 37 exoplanet
host stars within 25 pc at 70, 100 and 160 $\mu$m with the sensitivity to
detect far-infrared excess emission at flux density levels only an order of
magnitude greater than that of the Solar system's Edgeworth-Kuiper belt. Here
we present an analysis of that sample, using it to more accurately determine
the (possible) level of dust emission from these exoplanet host stars and
thereafter determine the links between the various components of these
exoplanetary systems through statistical analysis. We have fitted the flux
densities measured from recent \textit{Herschel} observations with a simple two
parameter ($T_{d}$, $L_{\rm IR}/L_{\star}$) black body model (or to the
3-$\sigma$ upper limits at 100 $\mu$m). From this uniform approach we calculate
the fractional luminosity, radial extent, dust temperature and disc mass. We
then plotted the calculated dust luminosity or upper limits against the stellar
properties, e.g. effective temperature, metallicity, age, and identified
correlations between these parameters. A total of eleven debris discs are
identified around the 37 stars in the sample. An incidence of ten cool debris
discs around the Sun-like exoplanet host stars (29 $\pm$ 9 %) is consistent
with the detection rate found by DUNES (20.2 $\pm$ 2.0 %). For the debris disc
systems, the dust temperatures range from 20 to 80 K, and fractional
luminosities ($L_{\rm IR}/L_{\star}$) between 2.4 $\times$10$^{-6}$ and 4.1
$\times$10$^{-4}$. In the case of non-detections, we calculated typical
3-$\sigma$ upper limits to the dust fractional luminosities of a few
$\times10^{-6}$. We recover the previously identified correlation between
stellar metallicity and hot Jupiter planets in our data set. We find a
correlation between the increased presence of dust, lower planet masses and
lower stellar metallicities. (abridged)
