{"paper":{"title":"On the Spectra of Simplicial Rook Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jennifer D. Wagner, Jeremy L. Martin","submitted_at":"2012-09-16T15:23:28Z","abstract_excerpt":"The \\emph{simplicial rook graph} SR(d,n) is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\\mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of SR(3,n) have integral spectrum for every $n$. The proof proceeds by calculating an explicit eigenbasis. We conjecture that SR(d,n) is integral for all $d$ and $n$, and present evidence in support of this conjecture. For $n<\\binom{d}{2}$, the evidence indicates that the smallest eigenvalue of the adjacency matrix is $-n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3493","kind":"arxiv","version":2},"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"}