Exponentially many graphs are determined by their spectrum

TL;DR

This paper demonstrates that exponentially many graphs are uniquely identified by their spectrum, advancing understanding of spectral graph characterization and supporting the conjecture that almost all graphs are spectrum-determined.

Contribution

It proves that the number of spectrum-determined graphs grows exponentially with the number of vertices, improving previous subexponential bounds.

Findings

Exponential growth in spectrum-determined graphs with respect to number of vertices

Improved bounds from exponential in sqrt(n) to exponential in n

Supports the conjecture that almost all graphs are determined by their spectrum

Abstract

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 $e^{c\sqrt{n}}$). We also propose a number of further directions of research.