{"paper":{"title":"Square-free Discriminants of Matrices and the Generalized Spectral Characterizations of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tao Yu, Wei Wang","submitted_at":"2016-08-03T10:54:14Z","abstract_excerpt":"Let $S_n(\\mathbb{Z})$ and $O_n(\\mathbb{Q})$ denote the set of all $n\\times n$ symmetric matrices over the ring of integers $\\mathbb{Z}$ and the set of all $n\\times n$ orthogonal matrices over the field of rational numbers $\\mathbb{Q}$, respectively. The paper is mainly concerned with the following problem: Given a matrix $A\\in {S_n(\\mathbb{Z})}$. How can one find all rational orthogonal matrices $Q\\in{O_n(\\mathbb{Q})}$ such that $Q^TAQ\\in {S_n(\\mathbb{Z})}$, and in particular, when does $Q^TAQ\\in {S_n(\\mathbb{Z})}$ with $Q\\in{O_n(\\mathbb{Q})}$ imply that $Q$ is \\emph{a signed permutation matri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01144","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}