{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:AIKH4MBSZLNBAVFOS7RJKHCFNY","short_pith_number":"pith:AIKH4MBS","schema_version":"1.0","canonical_sha256":"02147e3032cada1054ae97e2951c456e368e5b7a61dc315df52dd86b11a9c8c7","source":{"kind":"arxiv","id":"1306.0093","version":1},"attestation_state":"computed","paper":{"title":"On a conjecture for the signless Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jieshan Yang, Lihua You","submitted_at":"2013-06-01T10:37:03Z","abstract_excerpt":"Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\\geq q_2(G)\\geq\\cdots\\geq q_n(G)\\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \\ldots, n.$ F. Ashraf et al. conjectured that $S_k^+(G)\\leq e(G)+\\binom{k+1}{2}$ for $k=1, 2, \\ldots, n.$ In this paper, we give various upper bounds for $S_k^+(G),$ and prove that this conjecture is true for the following cases: connected graph with sufficiently large $k,$ unicyclic graphs and bicyclic graphs for all $k,$ and tricyclic graphs when $k\\neq 3.$"},"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":"1306.0093","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-01T10:37:03Z","cross_cats_sorted":[],"title_canon_sha256":"3224c9d5dc5f8c1c6be3c006f1e56b281c85feb4df0ad72de68cfd0d517d3de5","abstract_canon_sha256":"f2afc3dbc4ecc4b8f6c5c7de2bd864d6163801d31d44a19ed5de8d65e651ae98"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:56.157510Z","signature_b64":"vgRlX9+mvfOCYJveLRZ3cjTFPy7gjAn6+V5Pzn+GCS6jB+PyXzku7VBkDQsZtUajOOisy/HAUOuelVy4rExSCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02147e3032cada1054ae97e2951c456e368e5b7a61dc315df52dd86b11a9c8c7","last_reissued_at":"2026-05-18T03:21:56.157094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:56.157094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a conjecture for the signless Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jieshan Yang, Lihua You","submitted_at":"2013-06-01T10:37:03Z","abstract_excerpt":"Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\\geq q_2(G)\\geq\\cdots\\geq q_n(G)\\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \\ldots, n.$ F. Ashraf et al. conjectured that $S_k^+(G)\\leq e(G)+\\binom{k+1}{2}$ for $k=1, 2, \\ldots, n.$ In this paper, we give various upper bounds for $S_k^+(G),$ and prove that this conjecture is true for the following cases: connected graph with sufficiently large $k,$ unicyclic graphs and bicyclic graphs for all $k,$ and tricyclic graphs when $k\\neq 3.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0093","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":"1306.0093","created_at":"2026-05-18T03:21:56.157149+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.0093v1","created_at":"2026-05-18T03:21:56.157149+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0093","created_at":"2026-05-18T03:21:56.157149+00:00"},{"alias_kind":"pith_short_12","alias_value":"AIKH4MBSZLNB","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"AIKH4MBSZLNBAVFO","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"AIKH4MBS","created_at":"2026-05-18T12:27:38.830355+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/AIKH4MBSZLNBAVFOS7RJKHCFNY","json":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY.json","graph_json":"https://pith.science/api/pith-number/AIKH4MBSZLNBAVFOS7RJKHCFNY/graph.json","events_json":"https://pith.science/api/pith-number/AIKH4MBSZLNBAVFOS7RJKHCFNY/events.json","paper":"https://pith.science/paper/AIKH4MBS"},"agent_actions":{"view_html":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY","download_json":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY.json","view_paper":"https://pith.science/paper/AIKH4MBS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.0093&json=true","fetch_graph":"https://pith.science/api/pith-number/AIKH4MBSZLNBAVFOS7RJKHCFNY/graph.json","fetch_events":"https://pith.science/api/pith-number/AIKH4MBSZLNBAVFOS7RJKHCFNY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY/action/storage_attestation","attest_author":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY/action/author_attestation","sign_citation":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY/action/citation_signature","submit_replication":"https://pith.science/pith/AIKH4MBSZLNBAVFOS7RJKHCFNY/action/replication_record"}},"created_at":"2026-05-18T03:21:56.157149+00:00","updated_at":"2026-05-18T03:21:56.157149+00:00"}