  Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded
degree graphs have sublinear second eigenvalue multiplicity (always referring
to the adjacency matrix). This result was a key step in the solution to the
problem of equiangular lines with fixed angles. It led to the natural question:
what is the maximum second eigenvalue multiplicity of a connected bounded
degree $n$-vertex graph? The best known upper bound is $O(n/\log\log n)$. The
previously known best known lower bound is on the order of $n^{1/3}$ (for
infinitely many $n$), coming from Cayley graphs on $\text{PSL}(2,q)$.
  Here we give constructions showing a lower bound on the order of
$\sqrt{n/\log n}$. We also construct Cayley graphs with second eigenvalue
multiplicity at least $n^{2/5}-1$.
  Earlier techniques show that there are at most $O(n/\log\log n)$ eigenvalues
(counting multiplicities) within $O(1/\log n)$ of the second eigenvalue. We
give a construction showing this upper bound on approximate second eigenvalue
multiplicity is tight up to a constant factor. This demonstrates a barrier to
earlier techniques for upper bounding eigenvalue multiplicities.
