稀疏图通常由其谱确定吗？
Are sparse graphs typically determined by their spectrum?
AI总结：
研究稀疏图是否通常由其谱确定，发现悬垂树可能是障碍，通过修剪2-核可提高谱确定性。
摘要：
我们研究稀疏图是否通常由其邻接谱唯一确定。我们的第一个结果表明，在平均度足够小时，Erdős-Rényi图的巨连通分量是同谱的。证明依赖于存在特定的悬垂树，结合Schwenk的方法，通过交换树来构造同谱的伴侣。似乎悬垂树是本质上的障碍，这意味着如果通过考虑2-核来修剪这些树，巨连通分量应以高概率由谱确定。本文大部分内容致力于理论和数值证据支持这一概念。我们在此方向的主要定理表明，局部交换方法不能导致2-核同谱。我们还讨论了R-同谱性和固定水平的有理同谱性。
英文摘要：
We investigate whether it is typical for a sparse graph to be uniquely characterized by its adjacency spectrum up to isomorphism. Our first result shows that the giant component of an Erdős-Rényi graph is cospectral when the average degree is sufficiently small. The proof relies on the existence of a specific pendant tree, combined with a method by Schwenk that swaps trees to construct a cospectral mate. It seems possible that pendant trees are essentially the only obstruction, meaning that the giant should become characterized by spectrum with high probability if one prunes these by considering the 2-core. The majority of the paper is devoted to theoretical and numerical evidence supporting this concept. Our main theorem in this direction establishes that local switching methods can not cause the 2-core to be cospectral. We also discuss R-cospectrality and rational cospectrality at fixed level.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
