Haemers’ determination-by-spectrum conjecture for random graphs

Establish that, for almost all labeled graphs on n vertices (in the usual Erdős–Rényi sense of “almost all”), the multiset of eigenvalues of the adjacency matrix uniquely determines the graph up to isomorphism; equivalently, show that the probability a random graph is determined by its adjacency spectrum tends to 1 as n → ∞.

Background

The paper situates this conjecture as a discrete analogue of Kac’s question on hearing the shape of a drum and references extensive prior work on cospectral graphs and graphs determined by spectrum (DS graphs). Despite many constructed examples, the overarching behavior for almost all graphs remains unproven.

The authors’ results on cospectral mates of bounded height provide evidence toward spectral uniqueness phenomena but do not resolve the conjecture. They also discuss related generalizations to other graph matrices beyond the adjacency matrix.