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We bound the maximum modulus, $\\mbox{maxmod}(n)$, of an independence root over all graphs on $n$ vertices and the maximum modulus, $\\mbox{maxmod}_{T}(n)$, of an independence root over all trees on $n$ vertices in terms of $n$. In particular, we show that\n  $$\\frac{\\log_3(\\mbox{maxmod}(n))}{n}=\\frac{1}{3}+o(1)$$\n  and $$\\frac{\\log_2(\\mbox{maxmod}_{T}(n))}{n}=\\frac{1}{2}+o(1).$$"},"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":"1812.09775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-23T21:17:30Z","cross_cats_sorted":[],"title_canon_sha256":"20ced4422c44483ef058a7502b12d45fb799e76a53452941dfbf0411a00787ca","abstract_canon_sha256":"87c93f73267fccdcf16446d012dc92890989b6bb12d62ea2679e9d4057fbdfc5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:26.820195Z","signature_b64":"BYTp0l8TH8FTWqRtA3IZelgnh9SOFXliZAaQx9qkvt6g8XjN4c2qw8XuLT1JTRDKXaca4ltMxFLSL8X+eNR6BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46a5e5b64dbfaa846ca870d33559715fca82b34f7b95882e6ce26f7940aad7be","last_reissued_at":"2026-05-17T23:57:26.819519Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:26.819519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximum Modulus of Independence Roots of Graphs and Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Cameron, Jason I.Brown","submitted_at":"2018-12-23T21:17:30Z","abstract_excerpt":"The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. 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In particular, we show that\n  $$\\frac{\\log_3(\\mbox{maxmod}(n))}{n}=\\frac{1}{3}+o(1)$$\n  and $$\\frac{\\log_2(\\mbox{maxmod}_{T}(n))}{n}=\\frac{1}{2}+o(1).$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.09775","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":"1812.09775","created_at":"2026-05-17T23:57:26.819627+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.09775v1","created_at":"2026-05-17T23:57:26.819627+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.09775","created_at":"2026-05-17T23:57:26.819627+00:00"},{"alias_kind":"pith_short_12","alias_value":"I2S6LNSNX6VI","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"I2S6LNSNX6VII3FI","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"I2S6LNSN","created_at":"2026-05-18T12:32:28.185984+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/I2S6LNSNX6VII3FIODJTKWLRL7","json":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7.json","graph_json":"https://pith.science/api/pith-number/I2S6LNSNX6VII3FIODJTKWLRL7/graph.json","events_json":"https://pith.science/api/pith-number/I2S6LNSNX6VII3FIODJTKWLRL7/events.json","paper":"https://pith.science/paper/I2S6LNSN"},"agent_actions":{"view_html":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7","download_json":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7.json","view_paper":"https://pith.science/paper/I2S6LNSN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.09775&json=true","fetch_graph":"https://pith.science/api/pith-number/I2S6LNSNX6VII3FIODJTKWLRL7/graph.json","fetch_events":"https://pith.science/api/pith-number/I2S6LNSNX6VII3FIODJTKWLRL7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7/action/storage_attestation","attest_author":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7/action/author_attestation","sign_citation":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7/action/citation_signature","submit_replication":"https://pith.science/pith/I2S6LNSNX6VII3FIODJTKWLRL7/action/replication_record"}},"created_at":"2026-05-17T23:57:26.819627+00:00","updated_at":"2026-05-17T23:57:26.819627+00:00"}