  We study subgraphs of Paley graphs of prime order $p$ induced on the sets of
vertices extending a given independent set of size $a$ to a larger independent
set. Using a sufficient condition proved in the author's recent companion work,
we show that a family of character sum estimates would imply that, as $p \to
\infty$, the empirical spectral distributions of the adjacency matrices of any
sequence of such subgraphs have the same weak limit (after rescaling) as those
of subgraphs induced on a random set including each vertex independently with
probability $2^{-a}$, namely, a Kesten-McKay law with parameter $2^a$. We prove
the necessary estimates for $a = 1$, obtaining in the process an alternate
proof of a character sum equidistribution result of Xi (2022), and provide
numerical evidence for this weak convergence for $a \geq 2$. We also conjecture
that the minimum eigenvalue of any such sequence converges (after rescaling) to
the left edge of the corresponding Kesten-McKay law, and provide numerical
evidence for this convergence. Finally, we show that, once $a \geq 3$, this
(conjectural) convergence of the minimum eigenvalue would imply bounds on the
clique number of the Paley graph improving on the current state of the art due
to Hanson and Petridis (2021), and that this convergence for all $a \geq 1$
would imply that the clique number is $o(\sqrt{p})$.
