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Then, by using techniques from electrical networks, we obtain the following formula: $$ t(L(G)) = \\frac{1}{\\prod_{v\\in V}d^2(v)}\\sum_{T\\subseteq \\mathcal{T}(G)}\\big[\\prod_{e = xy\\in T}d(x)d(y)\\big]\\big[\\prod_{e = uv\\in E\\backslash T}[d(u)+d(v)]\\big]. $$ As a result, we provide a very simple and different proof of the formula on the number of spanning trees of some irregular line graphs, and give a "},"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":"1507.06389","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-23T05:50:15Z","cross_cats_sorted":[],"title_canon_sha256":"b2b1b78a03188567a929bc0190b50a9d074ee156c17cb4313c091f49e087160d","abstract_canon_sha256":"474a74f9a4f6ef7b4eeb174f61b12281caeca8e1376e7f45ecc57b180242dcff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:07.866300Z","signature_b64":"pr8oNRUy+P+X17guO22VpMvnBxvqUjcrL2ksPcYKzMLA7BhgXivLWH7qyGbM3LV4tVREwVDkQQa5Vgu/56CqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5fe3d35e7360bc94cedbdd132783872b38286d39e79512fef1e22dbca9b9d3d9","last_reissued_at":"2026-05-18T01:36:07.865470Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:07.865470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A formula for the number of the spanning trees of line graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Helin Gong, Xian'an Jin","submitted_at":"2015-07-23T05:50:15Z","abstract_excerpt":"Let $G=(V,E)$ be a loopless graph and $\\mathcal{T}(G)$ be the set of all spanning trees of $G$. Let $L(G)$ be the line graph of the graph $G$ and $t(L(G))$ be the number of spanning trees of $L(G)$. Then, by using techniques from electrical networks, we obtain the following formula: $$ t(L(G)) = \\frac{1}{\\prod_{v\\in V}d^2(v)}\\sum_{T\\subseteq \\mathcal{T}(G)}\\big[\\prod_{e = xy\\in T}d(x)d(y)\\big]\\big[\\prod_{e = uv\\in E\\backslash T}[d(u)+d(v)]\\big]. $$ As a result, we provide a very simple and different proof of the formula on the number of spanning trees of some irregular line graphs, and give a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06389","kind":"arxiv","version":2},"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":"1507.06389","created_at":"2026-05-18T01:36:07.865640+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.06389v2","created_at":"2026-05-18T01:36:07.865640+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06389","created_at":"2026-05-18T01:36:07.865640+00:00"},{"alias_kind":"pith_short_12","alias_value":"L7R5GXTTMC6J","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"L7R5GXTTMC6JJTW3","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"L7R5GXTT","created_at":"2026-05-18T12:29:29.992203+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/L7R5GXTTMC6JJTW33UJSPA4HFM","json":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM.json","graph_json":"https://pith.science/api/pith-number/L7R5GXTTMC6JJTW33UJSPA4HFM/graph.json","events_json":"https://pith.science/api/pith-number/L7R5GXTTMC6JJTW33UJSPA4HFM/events.json","paper":"https://pith.science/paper/L7R5GXTT"},"agent_actions":{"view_html":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM","download_json":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM.json","view_paper":"https://pith.science/paper/L7R5GXTT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.06389&json=true","fetch_graph":"https://pith.science/api/pith-number/L7R5GXTTMC6JJTW33UJSPA4HFM/graph.json","fetch_events":"https://pith.science/api/pith-number/L7R5GXTTMC6JJTW33UJSPA4HFM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM/action/storage_attestation","attest_author":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM/action/author_attestation","sign_citation":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM/action/citation_signature","submit_replication":"https://pith.science/pith/L7R5GXTTMC6JJTW33UJSPA4HFM/action/replication_record"}},"created_at":"2026-05-18T01:36:07.865640+00:00","updated_at":"2026-05-18T01:36:07.865640+00:00"}