Haemers' conjecture on spectral determination of almost all graphs

Determine whether almost all graphs are determined by the spectrum of their adjacency matrix; equivalently, prove or disprove that if f(n) denotes the fraction of non-isomorphic graphs on n vertices uniquely determined by their adjacency spectrum, then f(n) tends to 1 as n tends to infinity.

Background

Haemers' conjecture concerns the asymptotic prevalence of graphs determined by their adjacency spectrum. The paper defines f(n) as the fraction of non-isomorphic n-vertex graphs with no non-isomorphic cospectral mate and states that the conjecture predicts f(n)→1. The conjecture is directly relevant to the paper because switching constructs cospectral mates and therefore provides a mechanism for showing that graphs are not determined by their spectrum.

Although computational evidence supports the conjecture, the paper emphasizes that it remains unresolved. In particular, proving that an individual graph is determined by its spectrum generally requires ruling out every possible non-isomorphic cospectral mate, which is difficult without exhaustive enumeration.

References

One of the major open problems in spectral graph theory is Haemers' conjecture (2003), which states that almost all graphs are determined by their adjacency spectrum. More precisely, if $f(n)$ is the fraction of non-isomorphic graphs on $n$ vertices that are uniquely determined by the spectrum of the their adjacency matrix, then the conjecture says that $f(n)\rightarrow 1$ if $n\rightarrow \infty$.

— Counting cospectral graphs obtained via switching  (2503.08627 - Abiad et al., 11 Mar 2025) in Section 1, Introduction