{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:JW7AZSYDMFTIP5EYI3MRPOZLVJ","short_pith_number":"pith:JW7AZSYD","schema_version":"1.0","canonical_sha256":"4dbe0ccb03616687f49846d917bb2baa48c331e091bd581b067130ce8a535fa6","source":{"kind":"arxiv","id":"1601.03511","version":1},"attestation_state":"computed","paper":{"title":"The Randi\\'{c} index and signless Laplacian spectral radius of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Xing Peng","submitted_at":"2016-01-14T07:48:36Z","abstract_excerpt":"Given a connected graph $G$, the Randi\\'c index $R(G)$ is the sum of $\\tfrac{1}{\\sqrt{d(u)d(v)}}$ over all edges $\\{u,v\\}$ of $G$, where $d(u)$ and $d(v)$ are the degree of vertices $u$ and $v$ respectively. Let $q(G)$ be the largest eigenvalue of the singless Laplacian matrix of $G$ and $n=|V(G)|$. Hansen and Lucas (2010) made the following conjecture: \\[ \\frac{q(G)}{R(G)} \\leq \\begin{cases}\n  \\frac{4n-4}{n} & 4 \\leq n\\leq 12\n  \\frac{n}{\\sqrt{n-1}} & n\\geq 13\n  \\end{cases} \\] with equality if and only if $G=K_{n}$ for $4\\leq n\\leq 12$ and $G=S_n$ for $n\\geq 13$, respectively. Deng, Balachandr"},"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":"1601.03511","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T07:48:36Z","cross_cats_sorted":[],"title_canon_sha256":"71a082f20541be06efb49eda88aea532bb524f5787b8269d21c36acdd76f311b","abstract_canon_sha256":"f11d883869df9cb56d6b1b1e41d4498cfeb50bd6c55e492856fd17e935dbf9c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:26.784014Z","signature_b64":"OuRzkagl/F7u/8mXscfGFLoYvi1ng3FE07wNbqJrDRJaBa9axHQjyY5VAXS34GFo6i/JkbMnXoUqlNZrqUlBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4dbe0ccb03616687f49846d917bb2baa48c331e091bd581b067130ce8a535fa6","last_reissued_at":"2026-05-17T23:59:26.783433Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:26.783433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Randi\\'{c} index and signless Laplacian spectral radius of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Xing Peng","submitted_at":"2016-01-14T07:48:36Z","abstract_excerpt":"Given a connected graph $G$, the Randi\\'c index $R(G)$ is the sum of $\\tfrac{1}{\\sqrt{d(u)d(v)}}$ over all edges $\\{u,v\\}$ of $G$, where $d(u)$ and $d(v)$ are the degree of vertices $u$ and $v$ respectively. Let $q(G)$ be the largest eigenvalue of the singless Laplacian matrix of $G$ and $n=|V(G)|$. Hansen and Lucas (2010) made the following conjecture: \\[ \\frac{q(G)}{R(G)} \\leq \\begin{cases}\n  \\frac{4n-4}{n} & 4 \\leq n\\leq 12\n  \\frac{n}{\\sqrt{n-1}} & n\\geq 13\n  \\end{cases} \\] with equality if and only if $G=K_{n}$ for $4\\leq n\\leq 12$ and $G=S_n$ for $n\\geq 13$, respectively. 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