Van Dam–Haemers Conjecture: are almost all graphs determined by their adjacency spectrum?

Two graphs are *cospectral* if their adjacency matrices have the same multiset of eigenvalues. A graph is *determined by its spectrum* (DS) if no non-isomorphic graph is cospectral with it. Van Dam & Haemers (2003) conjectured that **almost all graphs are DS**, i.e. the fraction of (unlabeled) $n$-vertex graphs that are DS tends to $1$ as $n \to \infty$. Determine whether this holds. Computational sub-question: compute, for $n$ as large as feasible, the fraction $q_n$ of $n$-vertex graphs that have a cospectral mate (equivalently $1 - q_n$ is the DS fraction), and study its trend.

Acceptance. FULLY RESOLVES: a proof or disproof of the asymptotic statement. PARTIAL PROGRESS (computational, checkable): using $\texttt{nauty}$/$\texttt{geng}$ to enumerate all graphs on $n$ vertices and grouping them by characteristic polynomial, report the exact count of graphs with a cospectral mate for $n \ge 12$ (extending the published tables), with the eigenvalue/char-poly code and the cospectral-pair census. Any such new exact $q_n$ is independently reproducible and advances the empirical picture.