We explore the relationships between the Polycyclic Aromatic Hydrocarbon (PAH) feature strengths, mid-infrared continuum luminosities, far-infrared spectral slopes, optical spectroscopic classifications, and silicate optical depths within a sample of 107 ULIRGs observed with the Infrared Spectrograph on the Spitzer Space Telescope. The detected 6.2 micron PAH equivalent widths (EQWs) in the sample span more than two orders of magnitude (0.006-0.8 micron), and ULIRGs with HII-like optical spectra or steep far-infrared spectral slopes (S_{25} / S_{60} < 0.2) typically have 6.2 micron PAH EQWs that are half that of lower-luminosity starbursts. A significant fraction (~40-60%) of HII-like, LINER-like, and cold ULIRGs have very weak PAH EQWs. Many of these ULIRGs also have large (tau_{9.7} > 2.3) silicate optical depths. The far-infrared spectral slope is strongly correlated with PAH EQW, but not with silicate optical depth. In addition, the PAH EQW decreases with increasing rest-frame 24 micron luminosity. We argue that this trend results primarily from dilution of the PAH EQW by continuum emission from dust heated by a compact central source, probably an AGN. High luminosity, high-redshift sources studied with Spitzer appear to have a much larger range in PAH EQW than seen in local ULIRGs, which is consistent with extremely luminous starburst systems being absent at low redshift, but present at early epochs.
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