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Suppose that $G$ is not a complete bipartite graph, and $e-1$ and $e+1$ are not twin primes. We determine the maximal spectral radius of $G$. As a byproduct of our study, we obtain a spectral characterization of a pair $(e-1, e+1)$ of integers to be a pair of twin primes."},"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":"1509.07586","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-25T05:07:22Z","cross_cats_sorted":[],"title_canon_sha256":"f7e71e1c08b29251234894e63ce860d367cac043a4a3e9de3f2b86d1ebb99864","abstract_canon_sha256":"6b43ce5cfd49797000cc63e839a867e8ce4916b1cbedada48bbfd5d86a637351"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:53.505545Z","signature_b64":"E5lKwUueWyUU0P8flVbjJDa1mJHR8UbBAXbg/R81sAIYgrHAH3GyrbLYZ+0FTGb3scKPgYcLTnkhvarCCgb0AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5d7ab84e2fd28303609faaf16ec8b67195dbf29c55095680cc9c71b7751a0b4","last_reissued_at":"2026-05-18T01:27:53.504842Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:53.504842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An extending result on spectral radius of bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chih-wen Weng, Feng-lei Fan, Yen-Jen Cheng","submitted_at":"2015-09-25T05:07:22Z","abstract_excerpt":"Let $G$ denote a bipartite graph with $e$ edges without isolated vertices. 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