{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XRJAZKXE6CGIPMUCJRJOIEZNB5","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":"753cd01e387330d0bf6a27320943b943df9b2acc544af597a48a53d6b3d1307d","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-21T19:46:01Z","title_canon_sha256":"a626e8c331f1241657a3271f450ad9a99e03cc76c61c44de8c3a509ead78416d"},"schema_version":"1.0","source":{"id":"1109.4622","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4622","created_at":"2026-05-18T03:04:06Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4622v3","created_at":"2026-05-18T03:04:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4622","created_at":"2026-05-18T03:04:06Z"},{"alias_kind":"pith_short_12","alias_value":"XRJAZKXE6CGI","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XRJAZKXE6CGIPMUC","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XRJAZKXE","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:7a55736722461b764fcb8d2c864183b218cb4cf124d4119cd6dfb5a805e106fa","target":"graph","created_at":"2026-05-18T03:04:06Z","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":"In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set (m,n)-graphs (i.e. graphs with n vertices and m edges) and some operations on graphs that are monotone with respect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as well as with some probabilistic characteristics of graphs with randomly deleted edges. Section 2 provides some notions, notation, and simple observations. Section 3 contains some basic facts on the Laplacian polynomial of a graph","authors_text":"Alexander Kelmans","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-21T19:46:01Z","title":"Operations on Graphs Increasing Some Graph Parameters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4622","kind":"arxiv","version":3},"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:e34b5b804c4ac3be7b7b4c3212c712cf8df561a69867d247529f1fdacecedeb9","target":"record","created_at":"2026-05-18T03:04:06Z","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":"753cd01e387330d0bf6a27320943b943df9b2acc544af597a48a53d6b3d1307d","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-21T19:46:01Z","title_canon_sha256":"a626e8c331f1241657a3271f450ad9a99e03cc76c61c44de8c3a509ead78416d"},"schema_version":"1.0","source":{"id":"1109.4622","kind":"arxiv","version":3}},"canonical_sha256":"bc520caae4f08c87b2824c52e4132d0f5145301089509a7203a13cb5fa0ed345","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc520caae4f08c87b2824c52e4132d0f5145301089509a7203a13cb5fa0ed345","first_computed_at":"2026-05-18T03:04:06.970539Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:06.970539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jDt/Xwj89LI2YNEOaWClj2YbNklR3YkFvdPmFQtXiFUDsSPjBfhNAXxCHdRLmO6KPPISYAL8aktdYgPz8mCzBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:06.971120Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.4622","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e34b5b804c4ac3be7b7b4c3212c712cf8df561a69867d247529f1fdacecedeb9","sha256:7a55736722461b764fcb8d2c864183b218cb4cf124d4119cd6dfb5a805e106fa"],"state_sha256":"ac5910e0998fbf4f448be133dcfc894e0fc9709e523b34fcad580d64e97a3a2b"}