  Using a sample of the $Spitzer$ SWIRE-field galaxies whose optical spectra
are taken from the Data Release 4 of the Sloan Digital Sky Survey, we study
possible correlations between the Mid-InfraRed (MIR) dust emission from these
galaxies and both their metallicities and AGN activities. We find that both
metallicity and AGN activity are well correlated with the following ratios:
PAH(Polycyclic Aromatic Hydrocarbon)-to-star, VSG(Very Small Grain)-to-star and
PAH-to-VSG, which can be characterized by $\nu L_{\nu}[8\mu m({\rm dust})]/\nu
L_{\nu}[3.6\mu m]$, $\nu L_{\nu}[24\mu m]/\nu L_{\nu}[3.6\mu m]$, and$\nu
L_{\nu}[8\mu m({\rm dust})]/\nu L_{\nu}[24\mu m]$ respectively. We argue that
our MIR-metallicity correlation could be explained by either the amount of dust
(ongoing dust formation) or dust destruction (PAHs and VSGs could be destroyed
by hard and intense radiation fields), and that the MIR-AGN correlation could
arise due to either PAH destruction or an enhanced VSG continuum by the central
AGN.
