{"paper":{"title":"Cayley Graph on Symmetric Group Generated by Elements Fixing $k$ Points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheng Yeaw Ku, Kok Bin Wong, Terry Lau","submitted_at":"2014-05-26T05:16:25Z","abstract_excerpt":"Let $\\mathcal{S}_{n}$ be the symmetric group on $[n]=\\{1, \\ldots, n\\}$. The $k$-point fixing graph $\\mathcal{F}(n,k)$ is defined to be the graph with vertex set $\\mathcal{S}_{n}$ and two vertices $g$, $h$ of $\\mathcal{F}(n,k)$ are joined if and only if $gh^{-1}$ fixes exactly $k$ points. In this paper, we derive a recurrence formula for the eigenvalues of $\\mathcal{F}(n,k)$. Then we apply our result to determine the sign of the eigenvalues of $\\mathcal{F}(n,1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6462","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"}