  Given a projective plane $\Sigma$ and a polarity $\theta$ of $\Sigma$, the
corresponding polarity graph is the graph whose vertices are the points of
$\Sigma$, and two distinct points $p_1$ and $p_2$ are adjacent if $p_1$ is
incident to $p_2^{ \theta}$ in $\Sigma$. A well-known example of a polarity
graph is the Erd\H{o}s-R\'{e}nyi orthogonal polarity graph $ER_q$, which
appears frequently in a variety of extremal problems. Eigenvalue methods
provide an upper bound on the independence number of any polarity graph. Mubayi
and Williford showed that in the case of $ER_q$, the eigenvalue method gives
the correct upper bound in order of magnitude. We prove that this is also true
for other families of polarity graphs. This includes a family of polarity
graphs for which the polarity is neither orthogonal nor unitary. We conjecture
that any polarity graph of a projective plane of order $q$ has an independent
set of size $\Omega (q^{3/2})$. Some related results are also obtained.
