  To better understand the mechanism or mechanisms that lead to AGN activity
today, we measure the X-ray AGN fraction in a new sample of nearby clusters and
examine how it varies with galaxy properties, projected cluster-centric radius,
and cluster velocity dispersion. We present new wide-field Chandra X-ray
Observatory observations of Abell 85, Abell 754 and the background cluster
Abell 89B out to their virial radii. Out of seventeen X-ray sources associated
with galaxies in these clusters, we classify seven as X-ray AGN with L_{X,B} >
10^{41} erg/s. Only two of these would be classified as AGN based on their
optical spectra. We combine these observations with archival data to create a
sample of X-ray AGN from six z < 0.08 clusters and find that 3.4+1.1/-0.8% of
M_R < -20 galaxies host X-ray AGN with L_{X,B} > 10^{41} erg/s. We find that
more X-ray AGN are detected in more luminous galaxies and attribute this to
larger spheriods in more luminous galaxies and increased sensitivity to lower
Eddington-rate accretion from black holes in those spheroids. At a given X-ray
luminosity limit, more massive black holes can be accreting less efficiently,
yet still be detected. If interactions between galaxies are the principal
drivers of AGN activity, then the AGN fraction should be higher in lower
velocity dispersion clusters and the outskirts of clusters. However, the
tendency of the most massive and early-type galaxies to lie in the centers of
the richest clusters could dilute such trends. While we find no variation in
the AGN fraction with projected cluster-centric radius, we do find that the AGN
fraction increases significantly from 2.6+1.0/-0.8% in rich clusters to
10.0+6.2/-4.3% in those with lower velocity dispersions.
