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By the reduction operation for a graph (defined by Tian and Wong, 2026), we could turn to the reduced graphs with each quasi-pendant vertex of degree 2 to investigate $m_{L(G)}(1)$. Then let $T$ be a reduced tree on $n(\\geq 7)$ vertices with each quasi-pendant vertex of degree 2 and"},"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":"2606.11547","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-10T01:14:07Z","cross_cats_sorted":[],"title_canon_sha256":"1f9826f278e089dd676fcd57cc800271123ab7f1c8637857a72496b5c105d201","abstract_canon_sha256":"4a11bd7477c89388f90155826a45a4a1f99732be19354b23261106cac2cb278f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:09:55.444639Z","signature_b64":"nlKEPsNZcn9oB8r5V2b+N4uG8bO8yVKLh6li9gRWpOmQ802ezNHWHw/zbu/QhktHehXisw43L7tyhxeVc0deBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6eca1bf182f76fb60b93861bd303ff639463dcd4b50aa0bf3cd7233881a64cfc","last_reissued_at":"2026-06-11T01:09:55.443795Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:09:55.443795Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicity of Laplacian eigenvalue 1 of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fenglei Tian, Yuhao Zhou","submitted_at":"2026-06-10T01:14:07Z","abstract_excerpt":"Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. 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Then let $T$ be a reduced tree on $n(\\geq 7)$ vertices with each quasi-pendant vertex of degree 2 and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11547","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11547/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.11547","created_at":"2026-06-11T01:09:55.443930+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.11547v1","created_at":"2026-06-11T01:09:55.443930+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11547","created_at":"2026-06-11T01:09:55.443930+00:00"},{"alias_kind":"pith_short_12","alias_value":"N3FBX4MC65X3","created_at":"2026-06-11T01:09:55.443930+00:00"},{"alias_kind":"pith_short_16","alias_value":"N3FBX4MC65X3MC4T","created_at":"2026-06-11T01:09:55.443930+00:00"},{"alias_kind":"pith_short_8","alias_value":"N3FBX4MC","created_at":"2026-06-11T01:09:55.443930+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/N3FBX4MC65X3MC4TQYN5GA77MO","json":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO.json","graph_json":"https://pith.science/api/pith-number/N3FBX4MC65X3MC4TQYN5GA77MO/graph.json","events_json":"https://pith.science/api/pith-number/N3FBX4MC65X3MC4TQYN5GA77MO/events.json","paper":"https://pith.science/paper/N3FBX4MC"},"agent_actions":{"view_html":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO","download_json":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO.json","view_paper":"https://pith.science/paper/N3FBX4MC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.11547&json=true","fetch_graph":"https://pith.science/api/pith-number/N3FBX4MC65X3MC4TQYN5GA77MO/graph.json","fetch_events":"https://pith.science/api/pith-number/N3FBX4MC65X3MC4TQYN5GA77MO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO/action/storage_attestation","attest_author":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO/action/author_attestation","sign_citation":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO/action/citation_signature","submit_replication":"https://pith.science/pith/N3FBX4MC65X3MC4TQYN5GA77MO/action/replication_record"}},"created_at":"2026-06-11T01:09:55.443930+00:00","updated_at":"2026-06-11T01:09:55.443930+00:00"}