Correlations between the stellar, planetary and debris components of exoplanet systems observed by $\textit{Herschel}$
The $\textit{Herschel}$ DEBRIS, DUNES and GT programmes observed 37 exoplanet host stars within 25 pc at 70, 100 and 160 $μ$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-$σ$ upper limits at 100 $μ$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}$. In the case of non-detections, we calculated typical 3-$σ$ 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)