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This enables us to strengthen various known lower and upper bounds for $R_\\alpha$ and to generalise a non-spectral bound due to Bollob\\'as \\emph{et al}. We also prove that the zeroth-order general Randi\\'c index, $Q_\\alpha = \\sum_{i \\in V} d_i^\\alpha \\ge n\\lambda^\\alpha$ for $\\alpha < 0$."},"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":"1508.07950","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-31T18:36:09Z","cross_cats_sorted":[],"title_canon_sha256":"1aba2235ef25e2b2441b15d3a070e4f48747e2f81e03933ba4388850c5d7d8af","abstract_canon_sha256":"1c0bc0a8876e1c8659fb2aa849f56e07e06de664e561f2948d7896f9f1c0c396"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:31.150370Z","signature_b64":"iGi+Nu7O7aBnpRLjbf4XKT1lUPmkvEEcdzJEC5J8UKBajme8N33mwd1NMoAQgwfMqOzPPcjZvvyTbQYEIvOJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48c4f191edde73cd4f2f629620b372846f8da483fbddc53715498429bbd1a2e3","last_reissued_at":"2026-05-18T01:34:31.149917Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:31.149917Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds and power means for the general Randic index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Clive Elphick, Pawel Wocjan","submitted_at":"2015-08-31T18:36:09Z","abstract_excerpt":"We review bounds for the general Randi\\'c index, $R_{\\alpha} = \\sum_{ij \\in E} (d_i d_j)^\\alpha$, and use the power mean inequality to prove, for example, that $R_\\alpha \\ge m\\lambda^{2\\alpha}$ for $\\alpha < 0$, where $\\lambda$ is the spectral radius of a graph. 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