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We prove that the collection of real Wiener roots of trees is dense in $(-\\infty, 0]$, and the collection of complex Wiener roots of trees is dense in $\\mathbb C$. We also prove that the maximum modulus among all Wiener roots of trees of order $n \\ge 31$ is between $2n-15$ and $2n-16$, and we determine the unique tre"},"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":"1807.10967","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-28T18:50:21Z","cross_cats_sorted":[],"title_canon_sha256":"e7972fb1d1b107683d6b173dfe933d567acb58be8bb0dfc8e3c709687253fe84","abstract_canon_sha256":"2534587753e44687eacdc79e0b16ad25a57506b08dd10450b4b8063916ec39a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:34.879470Z","signature_b64":"8Dww/abCVaICs1oGdCdbcPaDX0upgM7NAAk4otf3BoV0O3Eev8xCe5NrxrPM4qczVIPj0MSbaF2WnAGRpAjACw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02beca467a0b3684bc5a6050a04ecd0e197a6e54803956486ea188bb2ddf982a","last_reissued_at":"2026-05-18T00:09:34.879015Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:34.879015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On roots of Wiener polynomials of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Danielle Wang","submitted_at":"2018-07-28T18:50:21Z","abstract_excerpt":"The \\emph{Wiener polynomial} of a connected graph $G$ is the polynomial $W(G;x) = \\sum_{i=1}^{D(G)} d_i(G)x^i$ where $D(G)$ is the diameter of $G$, and $d_i(G)$ is the number of pairs of vertices at distance $i$ from each other. We examine the roots of Wiener polynomials of trees. We prove that the collection of real Wiener roots of trees is dense in $(-\\infty, 0]$, and the collection of complex Wiener roots of trees is dense in $\\mathbb C$. We also prove that the maximum modulus among all Wiener roots of trees of order $n \\ge 31$ is between $2n-15$ and $2n-16$, and we determine the unique tre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10967","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1807.10967","created_at":"2026-05-18T00:09:34.879085+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.10967v1","created_at":"2026-05-18T00:09:34.879085+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10967","created_at":"2026-05-18T00:09:34.879085+00:00"},{"alias_kind":"pith_short_12","alias_value":"AK7MURT2BM3I","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"AK7MURT2BM3IJPC2","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"AK7MURT2","created_at":"2026-05-18T12:32:13.499390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY","json":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY.json","graph_json":"https://pith.science/api/pith-number/AK7MURT2BM3IJPC2MBIKATWNBY/graph.json","events_json":"https://pith.science/api/pith-number/AK7MURT2BM3IJPC2MBIKATWNBY/events.json","paper":"https://pith.science/paper/AK7MURT2"},"agent_actions":{"view_html":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY","download_json":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY.json","view_paper":"https://pith.science/paper/AK7MURT2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.10967&json=true","fetch_graph":"https://pith.science/api/pith-number/AK7MURT2BM3IJPC2MBIKATWNBY/graph.json","fetch_events":"https://pith.science/api/pith-number/AK7MURT2BM3IJPC2MBIKATWNBY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY/action/storage_attestation","attest_author":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY/action/author_attestation","sign_citation":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY/action/citation_signature","submit_replication":"https://pith.science/pith/AK7MURT2BM3IJPC2MBIKATWNBY/action/replication_record"}},"created_at":"2026-05-18T00:09:34.879085+00:00","updated_at":"2026-05-18T00:09:34.879085+00:00"}