Exploring Relations Between BCG \& Cluster Properties in the
  SPectroscopic IDentification of eROSITA Sources Survey from $0.05 < z < 0.3$
  We present a sample of 329 low to intermediate redshift ($0.05 < z < 0.3$)
brightest cluster galaxies (BCGs) in X-ray selected clusters from the
SPectroscopic IDentification of eRosita Sources (SPIDERS) survey, a
spectroscopic survey within Sloan Digital Sky Survey-IV (SDSS-IV). We define
our BCGs by simultaneous consideration of legacy X-ray data from ROSAT, maximum
likelihood outputs from an optical cluster-finder algorithm and visual
inspection. Using SDSS imaging data, we fit S\'ersic profiles to our BCGs in
three bands (\textit{g}, \textit{r}, \textit{i}) with \textsc{SIGMA}, a
\textsc{GALFIT}-based software wrapper. We examine the reliability of our fits
by running our pipeline on ${\sim}10^{4}$ psf-convolved model profiles injected
into 8 random cluster fields, we then use the results of this analysis to
create a robust subsample of 198 BCGs. We outline three cluster properties of
interest: overall cluster X-ray luminosity ($L_{X}$), cluster richness as
estimated by \textsc{redMaPPer} ($ \lambda $) and cluster halo mass
($M_{200}$), which is estimated via velocity dispersion. In general, there are
significant correlations with BCG stellar mass between all three environmental
properties, but no significant trends arise with either S\'ersic index or
effective radius. There is no major environmental dependence on the strength of
the relation between effective radius and BCG stellar mass. Stellar mass
therefore arises as the most important factor governing BCG morphology. Our
results indicate that our sample consists of a large number of relaxed, mature
clusters containing broadly homogeneous BCGs up to $z \sim 0.3$, suggesting
that there is little evidence for much ongoing structural evolution for BCGs in
these systems.
