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In this paper we obtain an expression for $t(L(S_r(G)))$ in terms of spanning trees of $G$ by a combinatorial approach. This result generalizes some known results on the relation between $t(L(S_r(G)))$ and $t(G)$ and gives an explicit expression $t(L(S_r(G)))=k^{m+s-n-1}(rk+2)^{m-n+1}t(G)$ if $G$ is of order $n+s$ and size $m+s$ in w"},"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":"1507.08022","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-29T05:46:48Z","cross_cats_sorted":[],"title_canon_sha256":"16bd28d921e819dbf1d2fd1be723f820467cfb4d8a6d88b783c5aba7d486884f","abstract_canon_sha256":"fed5d7c636ff3feb91c583bd5bbc920f7e8ca3c53981865b5ae3ed508921d9db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:02.836983Z","signature_b64":"lZ23Q51OKk88HnV2KKz16R5k9D7diP0MIdg+F4+e7FHMVkgWFSR66/4gMlFbtptdgWVf3ENxZeUe1vNq6TXPBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4da72c8a0eff503db73e787f8297f93929a8da8bf9eeba4e3ce3bbc9710bebd","last_reissued_at":"2026-05-18T00:46:02.836572Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:02.836572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fengming Dong, Weigen Yan","submitted_at":"2015-07-29T05:46:48Z","abstract_excerpt":"For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$ be the line graph of $G$ and for any non-negative integer $r$, $S_r(G)$ be the graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$ connecting the two ends of $e$. In this paper we obtain an expression for $t(L(S_r(G)))$ in terms of spanning trees of $G$ by a combinatorial approach. This result generalizes some known results on the relation between $t(L(S_r(G)))$ and $t(G)$ and gives an explicit expression $t(L(S_r(G)))=k^{m+s-n-1}(rk+2)^{m-n+1}t(G)$ if $G$ is of order $n+s$ and size $m+s$ in w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08022","kind":"arxiv","version":3},"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":"1507.08022","created_at":"2026-05-18T00:46:02.836625+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.08022v3","created_at":"2026-05-18T00:46:02.836625+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08022","created_at":"2026-05-18T00:46:02.836625+00:00"},{"alias_kind":"pith_short_12","alias_value":"YTNHFSFA572Q","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"YTNHFSFA572QHW3T","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"YTNHFSFA","created_at":"2026-05-18T12:29:52.810259+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/YTNHFSFA572QHW3T46D7QKL7SO","json":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO.json","graph_json":"https://pith.science/api/pith-number/YTNHFSFA572QHW3T46D7QKL7SO/graph.json","events_json":"https://pith.science/api/pith-number/YTNHFSFA572QHW3T46D7QKL7SO/events.json","paper":"https://pith.science/paper/YTNHFSFA"},"agent_actions":{"view_html":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO","download_json":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO.json","view_paper":"https://pith.science/paper/YTNHFSFA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.08022&json=true","fetch_graph":"https://pith.science/api/pith-number/YTNHFSFA572QHW3T46D7QKL7SO/graph.json","fetch_events":"https://pith.science/api/pith-number/YTNHFSFA572QHW3T46D7QKL7SO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO/action/storage_attestation","attest_author":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO/action/author_attestation","sign_citation":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO/action/citation_signature","submit_replication":"https://pith.science/pith/YTNHFSFA572QHW3T46D7QKL7SO/action/replication_record"}},"created_at":"2026-05-18T00:46:02.836625+00:00","updated_at":"2026-05-18T00:46:02.836625+00:00"}