  As a discrete analogue of Kac's celebrated question on "hearing the shape of
a drum", and towards a practical graph isomorphism test, it is of interest to
understand which graphs are determined up to isomorphism by their spectrum (of
their adjacency matrix). A striking conjecture in this area, due to van Dam and
Haemers, is that "almost all graphs are determined by their spectrum", meaning
that the fraction of unlabelled $n$-vertex graphs which are determined by their
spectrum converges to $1$ as $n\to\infty$.
  In this paper we make a step towards this conjecture, showing that there are
exponentially many $n$-vertex graphs which are determined by their spectrum.
This improves on previous bounds (of shape $e^{c\sqrt{n}}$), and appears to be
the limit of "purely combinatorial" techniques. We also propose a number of
further directions of research.
