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In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \\ge \\frac{\\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex $(d,D)$-biregular. Furthermore, we show that if $G$ is an $n$-vertex connected graph with minimum degree $d$ and maximum degree $D$, then $R(G) \\"},"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":"1705.05963","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T00:50:24Z","cross_cats_sorted":[],"title_canon_sha256":"d66b47b7ae7055ca04b6212f54365ac5c7b13b7863585995667cf53925fbd479","abstract_canon_sha256":"3d62a26ddb879f6bbc196d976205f2db6ec6224c2fd6f46c86870cbd533cf945"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:19.329513Z","signature_b64":"KHoqQn04Fy9AV9lWJwHwhzi0y0i6tqhHP4lzPIQL9TvAq5gT6xb3eHl5kRo/e2IoKqDbYGMR1xZIC3zcDPz/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"506ba93ee88a0f7c61284f2af370c4fbaf6f0ce0debccbb354e30b5c8854fadc","last_reissued_at":"2026-05-18T00:44:19.329016Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:19.329016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp bounds for the Randic index of graphs with given minimum and maximum degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Suil O, Yongtang Shi","submitted_at":"2017-05-17T00:50:24Z","abstract_excerpt":"The Randi{\\' c} index of a graph $G$, written $R(G)$, is the sum of $\\frac 1{\\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\\sum_{uv \\in E(G)} \\frac 1{\\sqrt{d(u)d(v)}}$, which is called the Randi{\\' c} index of it. Let $d$ and $D$ be positive integers $d < D$. In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \\ge \\frac{\\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex $(d,D)$-biregular. 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