{"paper":{"title":"Spectral properties of generalized Paley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Denis E. Videla, Ricardo A. Podest\\'a","submitted_at":"2023-10-23T21:32:10Z","abstract_excerpt":"We study the spectrum of generalized Paley graphs $\\Gamma(k,q)=Cay(\\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\\{x^k:x\\in \\mathbb{F}_q^*\\}$ where $q=p^m$ with $p$ prime and $k\\mid q-1$. We first show that the eigenvalues of $\\Gamma(k,q)$ are given by the Gaussian periods $\\eta_{i}^{(k,q)}$ with $0\\le i\\le k-1$. Then, we explicitly compute the spectrum of $\\Gamma(k,q)$ with $1\\le k \\le 4$ and of $\\Gamma(5,q)$ for $p\\equiv 1\\pmod 5$ and $5\\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $\\Gamma(k,q)$ is integral if and only if $p$ divides $(q-1)/(p-1)$. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.15378","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2310.15378/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}