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We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\\mathcal{B}(n)$ of all $n$-vertex bicyclic graphs. $\\mathcal{B}^1(n)$ denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that $B_n^1$ (obtained from $C_4$ by adding one edge between two non-adjacent vertices and adding $n-4$ pendent vertices at the vertex of degree 3) minimizes all the signless La"},"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":"1212.5261","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-20T12:51:17Z","cross_cats_sorted":[],"title_canon_sha256":"0406f3260e4893e00be2d30667512054b76734968b807f37b44e6407cd303644","abstract_canon_sha256":"f21c8c154da8e884137b0fb334ef85d9633c0f076044b33277a821a6c8720010"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:02.341855Z","signature_b64":"zGhRppRdeLK1sUiGB7Y/tiePEDCdoWHR/EuxI1xsEB7UlkkeCvEMWUvz5H62alDQkuSwTSdK8cIG9pFTEVSBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e33b25aabf9b6047f209c01996f072d22ae8f264f80c7c77ddfea8d16392e41","last_reissued_at":"2026-05-18T03:38:02.341105Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:02.341105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On signless Laplacian coefficients of bicyclic graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Zhang, Xiao-Dong Zhang","submitted_at":"2012-12-20T12:51:17Z","abstract_excerpt":"Let $G$ be a graph of order $n$ and $Q_G(x)= det(xI-Q(G))= \\sum_{i=1}^n (-1)^i \\varphi_i x^{n-i}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\\mathcal{B}(n)$ of all $n$-vertex bicyclic graphs. $\\mathcal{B}^1(n)$ denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that $B_n^1$ (obtained from $C_4$ by adding one edge between two non-adjacent vertices and adding $n-4$ pendent vertices at the vertex of degree 3) minimizes all the signless La"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5261","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":"1212.5261","created_at":"2026-05-18T03:38:02.341224+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.5261v1","created_at":"2026-05-18T03:38:02.341224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5261","created_at":"2026-05-18T03:38:02.341224+00:00"},{"alias_kind":"pith_short_12","alias_value":"JYZ3EWVL7G3A","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"JYZ3EWVL7G3AI7ZA","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"JYZ3EWVL","created_at":"2026-05-18T12:27:11.947152+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/JYZ3EWVL7G3AI7ZATQAZS3YHFU","json":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU.json","graph_json":"https://pith.science/api/pith-number/JYZ3EWVL7G3AI7ZATQAZS3YHFU/graph.json","events_json":"https://pith.science/api/pith-number/JYZ3EWVL7G3AI7ZATQAZS3YHFU/events.json","paper":"https://pith.science/paper/JYZ3EWVL"},"agent_actions":{"view_html":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU","download_json":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU.json","view_paper":"https://pith.science/paper/JYZ3EWVL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.5261&json=true","fetch_graph":"https://pith.science/api/pith-number/JYZ3EWVL7G3AI7ZATQAZS3YHFU/graph.json","fetch_events":"https://pith.science/api/pith-number/JYZ3EWVL7G3AI7ZATQAZS3YHFU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU/action/storage_attestation","attest_author":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU/action/author_attestation","sign_citation":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU/action/citation_signature","submit_replication":"https://pith.science/pith/JYZ3EWVL7G3AI7ZATQAZS3YHFU/action/replication_record"}},"created_at":"2026-05-18T03:38:02.341224+00:00","updated_at":"2026-05-18T03:38:02.341224+00:00"}