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A graph $G$ is called a $(\\lambda, 1)$-{\\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\\lambda$ triangles, and $(c2)$ every two non-adjacent vertices in $G$ are the end-vertices of exactly one two-edge path in $G$. It turns out that there are infinitely many feasible 4-tuples $(v, d, \\lambda, 1)$ with $\\lambda \\ge 1$. On the other hand (and this is our main result), there is no $(v, d, \\lambda, 1)$-graphs with $\\lambda \\ge 1$. As a byproduct, we obtain"},"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":"1806.06315","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-16T23:58:47Z","cross_cats_sorted":[],"title_canon_sha256":"9055a992ee103271d2c7d41ff2294827d043f6d1af45acec1c78014cdebec7de","abstract_canon_sha256":"c61054293b261ee433dd3345f6aa8d80e5ddebfdac8f9a855e20e505297eb3cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:31.855079Z","signature_b64":"3n2V/N5aKrve3aTTlztB1sGdz955Km0vivoFKST3+eE2bw8YW6cWI367YNchMSDgiYhUBzu78YUfpWug39dICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f011210106de3d1a91bcf91757a5c16e096c3a9a377f1e777e968a6cfd0d43c6","last_reissued_at":"2026-05-18T00:03:31.854448Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:31.854448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"ON $(\\triangle, 1)$-GRAPHS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Kelmans, Rafael Aparicio","submitted_at":"2018-06-16T23:58:47Z","abstract_excerpt":"Let $G = (V, E)$ be a graph and $\\lambda $ a non-negative integer. A graph $G$ is called a $(\\lambda, 1)$-{\\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\\lambda$ triangles, and $(c2)$ every two non-adjacent vertices in $G$ are the end-vertices of exactly one two-edge path in $G$. It turns out that there are infinitely many feasible 4-tuples $(v, d, \\lambda, 1)$ with $\\lambda \\ge 1$. On the other hand (and this is our main result), there is no $(v, d, \\lambda, 1)$-graphs with $\\lambda \\ge 1$. As a byproduct, we obtain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06315","kind":"arxiv","version":5},"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":"1806.06315","created_at":"2026-05-18T00:03:31.854549+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.06315v5","created_at":"2026-05-18T00:03:31.854549+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06315","created_at":"2026-05-18T00:03:31.854549+00:00"},{"alias_kind":"pith_short_12","alias_value":"6AISCAIG3Y6R","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"6AISCAIG3Y6RVEN4","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"6AISCAIG","created_at":"2026-05-18T12:32:08.215937+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/6AISCAIG3Y6RVEN47ELVPJOBNY","json":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY.json","graph_json":"https://pith.science/api/pith-number/6AISCAIG3Y6RVEN47ELVPJOBNY/graph.json","events_json":"https://pith.science/api/pith-number/6AISCAIG3Y6RVEN47ELVPJOBNY/events.json","paper":"https://pith.science/paper/6AISCAIG"},"agent_actions":{"view_html":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY","download_json":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY.json","view_paper":"https://pith.science/paper/6AISCAIG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.06315&json=true","fetch_graph":"https://pith.science/api/pith-number/6AISCAIG3Y6RVEN47ELVPJOBNY/graph.json","fetch_events":"https://pith.science/api/pith-number/6AISCAIG3Y6RVEN47ELVPJOBNY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/action/storage_attestation","attest_author":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/action/author_attestation","sign_citation":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/action/citation_signature","submit_replication":"https://pith.science/pith/6AISCAIG3Y6RVEN47ELVPJOBNY/action/replication_record"}},"created_at":"2026-05-18T00:03:31.854549+00:00","updated_at":"2026-05-18T00:03:31.854549+00:00"}