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Our main result is:\n  Theorem Let $\\Gamma$ be a non-bipartite distance-regular graph with valency $k \\geq "},"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":"1511.05239","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-17T01:09:34Z","cross_cats_sorted":[],"title_canon_sha256":"86a95f316104db3a1f027bdebdc19096712d41d3087c600560759b949abdc69b","abstract_canon_sha256":"cf2c43eec16369d04b366d188a14ab4cfafd822c34a6b4b496e17fbcb0096f81"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:49.683459Z","signature_b64":"Ow0bZ5OQG7cWliswYSI8i4My6ymKucMmwPze+BqeVayF3GpdqqCjsSCH4O2fXiL3dkXjZKz9U76VEgvWMOZ6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b6d18b688180fec84ee0b71d76a8425b31680ce2ddfbd74b40455be254ba0e8","last_reissued_at":"2026-05-18T01:26:49.682747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:49.682747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Light tails and the Hermitian dual polar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jack Koolen, Zhi Qiao","submitted_at":"2015-11-17T01:09:34Z","abstract_excerpt":"Juri\\'{s}i\\v{c} et al. conjectured that if a distance-regular graph $\\Gamma$ with diameter $D$ at least three has a light tail, then one of the following holds:\n  1.$a_1 =0$;\n  2.$\\Gamma$ is an antipodal cover of diameter three;\n  3.$\\Gamma$ is tight;\n  4.$\\Gamma$ is the halved $2D+1$-cube;\n  5.$\\Gamma$ is a Hermitian dual polar graph $^2A_{2D-1}(r)$ where $r$ is a prime power.\n  In this note, we will consider the case when the light tail corresponds to the eigenvalue $-\\frac{k}{a_1 +1}$. Our main result is:\n  Theorem Let $\\Gamma$ be a non-bipartite distance-regular graph with valency $k \\geq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05239","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":"1511.05239","created_at":"2026-05-18T01:26:49.682854+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.05239v1","created_at":"2026-05-18T01:26:49.682854+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05239","created_at":"2026-05-18T01:26:49.682854+00:00"},{"alias_kind":"pith_short_12","alias_value":"TNWRRNUIDAH6","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TNWRRNUIDAH6ZBHO","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TNWRRNUI","created_at":"2026-05-18T12:29:42.218222+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/TNWRRNUIDAH6ZBHOBNY5O2UEEW","json":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW.json","graph_json":"https://pith.science/api/pith-number/TNWRRNUIDAH6ZBHOBNY5O2UEEW/graph.json","events_json":"https://pith.science/api/pith-number/TNWRRNUIDAH6ZBHOBNY5O2UEEW/events.json","paper":"https://pith.science/paper/TNWRRNUI"},"agent_actions":{"view_html":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW","download_json":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW.json","view_paper":"https://pith.science/paper/TNWRRNUI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.05239&json=true","fetch_graph":"https://pith.science/api/pith-number/TNWRRNUIDAH6ZBHOBNY5O2UEEW/graph.json","fetch_events":"https://pith.science/api/pith-number/TNWRRNUIDAH6ZBHOBNY5O2UEEW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW/action/storage_attestation","attest_author":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW/action/author_attestation","sign_citation":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW/action/citation_signature","submit_replication":"https://pith.science/pith/TNWRRNUIDAH6ZBHOBNY5O2UEEW/action/replication_record"}},"created_at":"2026-05-18T01:26:49.682854+00:00","updated_at":"2026-05-18T01:26:49.682854+00:00"}