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Almost all graphs are determined by their generalized spectrum

Ziqing Xiang

Palomar Registrypalomar/PALOMAR-2026-10-23-000011Oct 23, 2026math.COscore 0

A complete Lean 4 and Mathlib formalization of the theorem that almost all graphs are determined by their generalized spectrum, which is Haemers' conjecture for the generalized spectrum. The generalized spectrum of a graph is the spectrum of its adjacency matrix A together with the spectrum of the adjacency matrix J - I - A of its complement, and a graph is determined by its generalized spectrum (DGS) if every graph with the same generalized spectrum is isomorphic to it. The main theorem, DGSMain.almost_all_graphs_are_DGS, states that the proportion of the graphs on the vertex set {0, ...,…

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