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We prove that the spectrum of the undirected prefix-reversal graph $\\mathbb{P}_m(n)$ contains all even integers in the interval $[0,2n]\\setminus\\{2\\lfloor n/2\\rfloor\\}$ and if $m\\equiv0\\pmod4$, we then show that the spectrum contains all even intege"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.08345","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2025-06-10T01:59:13Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"398c4f6962ea7a2fff23bab37099817825e3940f21a3fded404cb0854913b6f5","abstract_canon_sha256":"972591ddcf1811323a396ac8d547d569dfcc67420e99d0a67efba7e14dad293c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:48.334967Z","signature_b64":"acUmB8dq6BIoG7Hrn+YHglCRGBe63OtQyP9QathOhl6B5iKXkemYLESqHvDCLQNw1IPavd5eWsI2gdYD+SLuBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8b14acf77f3e073c108b33b85c82fb36bf6769c3848ebe8fca4935e00e30cd9","last_reissued_at":"2026-07-05T11:18:48.334496Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:48.334496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the spectra of prefix-reversal graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Charles Buehrle, Sa\\'ul A. 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