{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:AITPL5ETS3JQVXEAC7D2RWZKCJ","short_pith_number":"pith:AITPL5ET","canonical_record":{"source":{"id":"1411.5509","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-20T11:43:19Z","cross_cats_sorted":[],"title_canon_sha256":"d9c4e3ecf614250a2e6a8274f7567e1f6d7da13a4829eb2580bc04b2613fb800","abstract_canon_sha256":"e51004d1efb7cfb4a32d83be05a9f775e709da35275d72354b4216e72dba62d4"},"schema_version":"1.0"},"canonical_sha256":"0226f5f49396d30adc8017c7a8db2a126563aa3dac8feff98f9a31e491c81056","source":{"kind":"arxiv","id":"1411.5509","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.5509","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"arxiv_version","alias_value":"1411.5509v1","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.5509","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"pith_short_12","alias_value":"AITPL5ETS3JQ","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"AITPL5ETS3JQVXEA","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"AITPL5ET","created_at":"2026-05-18T12:28:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:AITPL5ETS3JQVXEAC7D2RWZKCJ","target":"record","payload":{"canonical_record":{"source":{"id":"1411.5509","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-20T11:43:19Z","cross_cats_sorted":[],"title_canon_sha256":"d9c4e3ecf614250a2e6a8274f7567e1f6d7da13a4829eb2580bc04b2613fb800","abstract_canon_sha256":"e51004d1efb7cfb4a32d83be05a9f775e709da35275d72354b4216e72dba62d4"},"schema_version":"1.0"},"canonical_sha256":"0226f5f49396d30adc8017c7a8db2a126563aa3dac8feff98f9a31e491c81056","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:33:24.460468Z","signature_b64":"WM0YDP4h+tIKLW3mazTduT0Lzb1HSbuJIr+tmhC2+IK7o5T/JK0W6kHnzBtoyRPGlPp1rhrUgCjkMXnrbUaNCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0226f5f49396d30adc8017c7a8db2a126563aa3dac8feff98f9a31e491c81056","last_reissued_at":"2026-05-18T02:33:24.459897Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:33:24.459897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.5509","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:33:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TLJrNHmaHQbOYuVb8/fs5JoufGCqxz+KqZq9Q+4wuTAH3U6xKu/jt9FY40DbaTvh4QegO89+y55HPnutP7SjDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:02:53.662736Z"},"content_sha256":"980fcbb0ef6593bb9658e0c96a38a61d4fcd8c11a28649f26675783c6e73c94c","schema_version":"1.0","event_id":"sha256:980fcbb0ef6593bb9658e0c96a38a61d4fcd8c11a28649f26675783c6e73c94c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:AITPL5ETS3JQVXEAC7D2RWZKCJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Laplacian polynomial of graphs derived from regular graphs and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fu-Tao Hu, Jia-Bao Liu, Xiang-Feng Pan","submitted_at":"2014-11-20T11:43:19Z","abstract_excerpt":"Let $R(G)$ be the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and by joining each new vertex to the end vertices of the corresponding edge. Let $RT(G)$ be the graph obtained from $R(G)$ by adding a new edge corresponding to every vertex of $G$, and by joining each new edge to every vertex of $G$. In this paper, we determine the Laplacian polynomials of $RT(G)$ of a regular graph $G$. Moreover, we derive formulae and lower bounds of Kirchhoff index of the graphs. Finally we also present the formulae for calculating the Kirchhoff index of some special graphs "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5509","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:33:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i23V60mde27EO2+OoGB/sju8V7RzM2nfGsBmhDUJwfAo84fpSvVCG+nSx0BUmHFKauHwfqRab4uVikvCNgv2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:02:53.663082Z"},"content_sha256":"8ab69503b9cc6613bdf684dcf3d6c78e90feb56f2a5d8100fd5240fb2ca0daad","schema_version":"1.0","event_id":"sha256:8ab69503b9cc6613bdf684dcf3d6c78e90feb56f2a5d8100fd5240fb2ca0daad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AITPL5ETS3JQVXEAC7D2RWZKCJ/bundle.json","state_url":"https://pith.science/pith/AITPL5ETS3JQVXEAC7D2RWZKCJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AITPL5ETS3JQVXEAC7D2RWZKCJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T11:02:53Z","links":{"resolver":"https://pith.science/pith/AITPL5ETS3JQVXEAC7D2RWZKCJ","bundle":"https://pith.science/pith/AITPL5ETS3JQVXEAC7D2RWZKCJ/bundle.json","state":"https://pith.science/pith/AITPL5ETS3JQVXEAC7D2RWZKCJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AITPL5ETS3JQVXEAC7D2RWZKCJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:AITPL5ETS3JQVXEAC7D2RWZKCJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e51004d1efb7cfb4a32d83be05a9f775e709da35275d72354b4216e72dba62d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-20T11:43:19Z","title_canon_sha256":"d9c4e3ecf614250a2e6a8274f7567e1f6d7da13a4829eb2580bc04b2613fb800"},"schema_version":"1.0","source":{"id":"1411.5509","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.5509","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"arxiv_version","alias_value":"1411.5509v1","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.5509","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"pith_short_12","alias_value":"AITPL5ETS3JQ","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"AITPL5ETS3JQVXEA","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"AITPL5ET","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:8ab69503b9cc6613bdf684dcf3d6c78e90feb56f2a5d8100fd5240fb2ca0daad","target":"graph","created_at":"2026-05-18T02:33:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $R(G)$ be the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and by joining each new vertex to the end vertices of the corresponding edge. Let $RT(G)$ be the graph obtained from $R(G)$ by adding a new edge corresponding to every vertex of $G$, and by joining each new edge to every vertex of $G$. In this paper, we determine the Laplacian polynomials of $RT(G)$ of a regular graph $G$. Moreover, we derive formulae and lower bounds of Kirchhoff index of the graphs. Finally we also present the formulae for calculating the Kirchhoff index of some special graphs ","authors_text":"Fu-Tao Hu, Jia-Bao Liu, Xiang-Feng Pan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-20T11:43:19Z","title":"The Laplacian polynomial of graphs derived from regular graphs and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5509","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:980fcbb0ef6593bb9658e0c96a38a61d4fcd8c11a28649f26675783c6e73c94c","target":"record","created_at":"2026-05-18T02:33:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e51004d1efb7cfb4a32d83be05a9f775e709da35275d72354b4216e72dba62d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-20T11:43:19Z","title_canon_sha256":"d9c4e3ecf614250a2e6a8274f7567e1f6d7da13a4829eb2580bc04b2613fb800"},"schema_version":"1.0","source":{"id":"1411.5509","kind":"arxiv","version":1}},"canonical_sha256":"0226f5f49396d30adc8017c7a8db2a126563aa3dac8feff98f9a31e491c81056","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0226f5f49396d30adc8017c7a8db2a126563aa3dac8feff98f9a31e491c81056","first_computed_at":"2026-05-18T02:33:24.459897Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:33:24.459897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WM0YDP4h+tIKLW3mazTduT0Lzb1HSbuJIr+tmhC2+IK7o5T/JK0W6kHnzBtoyRPGlPp1rhrUgCjkMXnrbUaNCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:33:24.460468Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.5509","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:980fcbb0ef6593bb9658e0c96a38a61d4fcd8c11a28649f26675783c6e73c94c","sha256:8ab69503b9cc6613bdf684dcf3d6c78e90feb56f2a5d8100fd5240fb2ca0daad"],"state_sha256":"98a60c0bef66d33d20c4bdde2713028a612fca5c39abbe9f7b20377797da2bff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VJBgOWR4bSB2lYnBhYzCVjPDOlZu1QhnLKWJt/J4ZCtqW4rNi6D0XQR522vRrxmygTtmD4lyQSqDPSiV+wzFCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T11:02:53.665216Z","bundle_sha256":"38a5b9591be88098b4c10dc8fd7a55f51a99defcccc5946e8b2ee525685e5f39"}}