{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JDE4NRK35VC3DKASLVHDLORCJV","short_pith_number":"pith:JDE4NRK3","canonical_record":{"source":{"id":"1511.03080","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-10T12:18:05Z","cross_cats_sorted":[],"title_canon_sha256":"31b72a84d853922d21e440a8483847663bd404b7a0a7aef6eda191fffa77b9a0","abstract_canon_sha256":"60b443ae7631d4d1c6670ca9f12ddce05fa393634fe568f418f20eb3653bbef7"},"schema_version":"1.0"},"canonical_sha256":"48c9c6c55bed45b1a8125d4e35ba224d56a3ffafd8e115b79deff11dc406384f","source":{"kind":"arxiv","id":"1511.03080","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.03080","created_at":"2026-05-18T01:27:17Z"},{"alias_kind":"arxiv_version","alias_value":"1511.03080v1","created_at":"2026-05-18T01:27:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.03080","created_at":"2026-05-18T01:27:17Z"},{"alias_kind":"pith_short_12","alias_value":"JDE4NRK35VC3","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JDE4NRK35VC3DKAS","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JDE4NRK3","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JDE4NRK35VC3DKASLVHDLORCJV","target":"record","payload":{"canonical_record":{"source":{"id":"1511.03080","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-10T12:18:05Z","cross_cats_sorted":[],"title_canon_sha256":"31b72a84d853922d21e440a8483847663bd404b7a0a7aef6eda191fffa77b9a0","abstract_canon_sha256":"60b443ae7631d4d1c6670ca9f12ddce05fa393634fe568f418f20eb3653bbef7"},"schema_version":"1.0"},"canonical_sha256":"48c9c6c55bed45b1a8125d4e35ba224d56a3ffafd8e115b79deff11dc406384f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:17.062529Z","signature_b64":"ajHqeG7hTeNcBhxEEKKkmi18oPWfGEhEbtfj4KbGW8PrQH7o9EDH8ZX6at1NIxtn6YlQG17bpZiLr/VN01boDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48c9c6c55bed45b1a8125d4e35ba224d56a3ffafd8e115b79deff11dc406384f","last_reissued_at":"2026-05-18T01:27:17.062038Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:17.062038Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.03080","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":"2026-05-18T01:27:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vw4oQJWOVdpOvMfEi2JzDdoBGE6VrEmAdSPg5AIF+04QaXEY4EWLdM/ZMgt+y1wNx6NcPzVrteO49TGh4+dkDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:38:55.715062Z"},"content_sha256":"8840005356efe22d3fe09c03b25e013c9513e1502f025ce0e627fd3b42f3a218","schema_version":"1.0","event_id":"sha256:8840005356efe22d3fe09c03b25e013c9513e1502f025ce0e627fd3b42f3a218"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JDE4NRK35VC3DKASLVHDLORCJV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cacti with maximum Kirchhoff index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wen-Rui Wang, Xiang-Feng Pan","submitted_at":"2015-11-10T12:18:05Z","abstract_excerpt":"The concept of resistance distance was first proposed by Klein and Randi\\'c. The Kirchhoff index $Kf(G)$ of a graph $G$ is the sum of resistance distance between all pairs of vertices in $G$. A connected graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. Let $Cat(n;t)$ be the set of connected cacti possessing $n$ vertices and $t$ cycles, where $0\\leq t \\leq \\lfloor\\frac{n-1}{2}\\rfloor$. In this paper, the maximum kirchhoff index of cacti are characterized, as well as the corresponding extremal graph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03080","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":"2026-05-18T01:27:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bc2nfhfRSmL3Kxhhh8yyqc4U4kAc52EXotxPNNe+Btaeshr9WW8+PD0hS9m3aqg5T3zdYTpsQ4KT+pomHx6MBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:38:55.715805Z"},"content_sha256":"1553e86cb3f51fbf7725d95ec97416915f614419eada5b6d3cba78136ed3cf9a","schema_version":"1.0","event_id":"sha256:1553e86cb3f51fbf7725d95ec97416915f614419eada5b6d3cba78136ed3cf9a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JDE4NRK35VC3DKASLVHDLORCJV/bundle.json","state_url":"https://pith.science/pith/JDE4NRK35VC3DKASLVHDLORCJV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JDE4NRK35VC3DKASLVHDLORCJV/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":"2026-06-08T14:38:55Z","links":{"resolver":"https://pith.science/pith/JDE4NRK35VC3DKASLVHDLORCJV","bundle":"https://pith.science/pith/JDE4NRK35VC3DKASLVHDLORCJV/bundle.json","state":"https://pith.science/pith/JDE4NRK35VC3DKASLVHDLORCJV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JDE4NRK35VC3DKASLVHDLORCJV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JDE4NRK35VC3DKASLVHDLORCJV","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":"60b443ae7631d4d1c6670ca9f12ddce05fa393634fe568f418f20eb3653bbef7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-10T12:18:05Z","title_canon_sha256":"31b72a84d853922d21e440a8483847663bd404b7a0a7aef6eda191fffa77b9a0"},"schema_version":"1.0","source":{"id":"1511.03080","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.03080","created_at":"2026-05-18T01:27:17Z"},{"alias_kind":"arxiv_version","alias_value":"1511.03080v1","created_at":"2026-05-18T01:27:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.03080","created_at":"2026-05-18T01:27:17Z"},{"alias_kind":"pith_short_12","alias_value":"JDE4NRK35VC3","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JDE4NRK35VC3DKAS","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JDE4NRK3","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:1553e86cb3f51fbf7725d95ec97416915f614419eada5b6d3cba78136ed3cf9a","target":"graph","created_at":"2026-05-18T01:27:17Z","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":"The concept of resistance distance was first proposed by Klein and Randi\\'c. The Kirchhoff index $Kf(G)$ of a graph $G$ is the sum of resistance distance between all pairs of vertices in $G$. A connected graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. Let $Cat(n;t)$ be the set of connected cacti possessing $n$ vertices and $t$ cycles, where $0\\leq t \\leq \\lfloor\\frac{n-1}{2}\\rfloor$. In this paper, the maximum kirchhoff index of cacti are characterized, as well as the corresponding extremal graph.","authors_text":"Wen-Rui Wang, Xiang-Feng Pan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-10T12:18:05Z","title":"Cacti with maximum Kirchhoff index"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03080","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":"sha256:8840005356efe22d3fe09c03b25e013c9513e1502f025ce0e627fd3b42f3a218","target":"record","created_at":"2026-05-18T01:27:17Z","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":"60b443ae7631d4d1c6670ca9f12ddce05fa393634fe568f418f20eb3653bbef7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-10T12:18:05Z","title_canon_sha256":"31b72a84d853922d21e440a8483847663bd404b7a0a7aef6eda191fffa77b9a0"},"schema_version":"1.0","source":{"id":"1511.03080","kind":"arxiv","version":1}},"canonical_sha256":"48c9c6c55bed45b1a8125d4e35ba224d56a3ffafd8e115b79deff11dc406384f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48c9c6c55bed45b1a8125d4e35ba224d56a3ffafd8e115b79deff11dc406384f","first_computed_at":"2026-05-18T01:27:17.062038Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:27:17.062038Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ajHqeG7hTeNcBhxEEKKkmi18oPWfGEhEbtfj4KbGW8PrQH7o9EDH8ZX6at1NIxtn6YlQG17bpZiLr/VN01boDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:27:17.062529Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.03080","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8840005356efe22d3fe09c03b25e013c9513e1502f025ce0e627fd3b42f3a218","sha256:1553e86cb3f51fbf7725d95ec97416915f614419eada5b6d3cba78136ed3cf9a"],"state_sha256":"565ca69ca7e57ee144fb5f94311d5da91d0565611b1037a3416542805e0425c8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"thoKB1JZHMzindjxZeTy8RAFBPyzLgu58hGHjOMFBzAamKTwRaoeff8eAalmV1cgEmn0/Pr4oslsoxiK98SHDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T14:38:55.719490Z","bundle_sha256":"71505dd0e98f40c907864dbf87e7cb05562c37d77c26f2bb0ccf2e56fc9a90fc"}}