Exponentially many graphs are determined by their spectrum

Illya Koval, Matthew Kwan

Published 2023-09-18Version 1

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.