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Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \\sim 1 \\sim 2 \\sim ... \\sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\\in\\{1,2,...,t\\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. Cioab\\v{a}-van Dam-Koolen-Lee \\cite{CDK} conjectured for fi"},"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":"1110.2444","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-11T17:30:09Z","cross_cats_sorted":[],"title_canon_sha256":"e31fb2558f70a66dc6840503d2321c0f9c1af1ed5174272fa2a83728b3d36910","abstract_canon_sha256":"bdeb29759fbcb4480c098fea83050ce74902d75f9d4bd29402b21b6757aea8aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:09.027986Z","signature_b64":"0HGCk8xUExu9A+DymyEKy8hbF1rysBwavYNOGEaRgMaf4NAAZ4FBkv3vq+nX80bDR80pa03I9E4rNNVA138RDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a027dcd161812194e955bd70de256d14e29f198e0c113e29e8feef02f74eb0c","last_reissued_at":"2026-05-18T04:11:09.027370Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:09.027370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Graphs with Diameter $n-e$ Minimizing the Spectral Radius","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jingfen Lan, Lingsheng Shi, Linyuan Lu","submitted_at":"2011-10-11T17:30:09Z","abstract_excerpt":"The spectral radius $\\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \\sim 1 \\sim 2 \\sim ... \\sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\\in\\{1,2,...,t\\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. 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