发表机构
Southern University of Science and Technology（南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了Haemers关于几乎所有图由广义谱确定的猜想，通过结合多种数学理论，证明随机图G(n,1/2)以趋于1的概率满足该性质。
AI 中文摘要
Haemers关于广义谱的猜想断言：几乎所有图都由其广义谱确定，即对于几乎所有图G，每个与G具有相同谱且补图与G的补图具有相同谱的图都与G同构。我们证明了该猜想，具体是证明随机图G(n,1/2)以趋于1的概率由其广义谱确定。该证明结合了：关联广义共谱图的有理正交矩阵分母的局部理论、有限环上随机对称矩阵的傅里叶分析、Seidel变换、有限域上的逆Littlewood-Offord理论、模素数的随机对称矩阵的秩估计，以及零对角随机±1对称矩阵特征多项式的有理因子研究。
英文摘要
Haemers' conjecture for the generalized spectrum asserts that almost all graphs are determined by their generalized spectrum, that is, for almost all graphs $G$, every graph with the same spectrum as $G$ whose complement has the same spectrum as the complement of $G$ is isomorphic to $G$. We prove this conjecture by showing that the random graph $G(n,1/2)$ is determined by its generalized spectrum with probability tending to one. The proof combines a local theory of the denominators of the rational orthogonal matrices that relate generalized cospectral graphs, Fourier analysis of random symmetric matrices over finite rings, Seidel switching, inverse Littlewood--Offord theory over finite fields, rank estimates for random symmetric matrices modulo primes, and a study of the rational factors of the characteristic polynomial of a random symmetric $\pm1$ matrix with zero diagonal.
