{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:NAYAZLJK4HIDNQBYA7AY4IAQFQ","short_pith_number":"pith:NAYAZLJK","canonical_record":{"source":{"id":"1007.2472","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-15T04:15:51Z","cross_cats_sorted":[],"title_canon_sha256":"360815a3fe54dbd80cf6346323d070243f91a5ea0224a330f3b42b1c27a0be33","abstract_canon_sha256":"df6ce1331bf4481ee2554b971986a1610acc5284d1ece61d6741ffae207244ed"},"schema_version":"1.0"},"canonical_sha256":"68300cad2ae1d036c03807c18e20102c24f6b37c2e573ab0e2ceba36bf0b6177","source":{"kind":"arxiv","id":"1007.2472","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2472","created_at":"2026-05-18T00:36:04Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2472v3","created_at":"2026-05-18T00:36:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2472","created_at":"2026-05-18T00:36:04Z"},{"alias_kind":"pith_short_12","alias_value":"NAYAZLJK4HID","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"NAYAZLJK4HIDNQBY","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"NAYAZLJK","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:NAYAZLJK4HIDNQBYA7AY4IAQFQ","target":"record","payload":{"canonical_record":{"source":{"id":"1007.2472","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-15T04:15:51Z","cross_cats_sorted":[],"title_canon_sha256":"360815a3fe54dbd80cf6346323d070243f91a5ea0224a330f3b42b1c27a0be33","abstract_canon_sha256":"df6ce1331bf4481ee2554b971986a1610acc5284d1ece61d6741ffae207244ed"},"schema_version":"1.0"},"canonical_sha256":"68300cad2ae1d036c03807c18e20102c24f6b37c2e573ab0e2ceba36bf0b6177","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:04.371348Z","signature_b64":"CpVWGVlJw/NQhkBjt0j/gJoXsEHttHWhDVwdHN+y66S68FNd55JwkxD5DgHVfrTqYOxo1mKdX9hmsEOeY7ehCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68300cad2ae1d036c03807c18e20102c24f6b37c2e573ab0e2ceba36bf0b6177","last_reissued_at":"2026-05-18T00:36:04.370726Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:04.370726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.2472","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:36:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K4raWf4mSY4MrVRfGI/BLmCsIYl5gfRVrZX1uo4hQZrlhL35PxggfoQIiEQZ6yo/V/hbj45JHdmH/+P5aECYDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T20:04:47.002628Z"},"content_sha256":"031adc4a3919056a4c030565e9933c4e7f1bc25e588272cf5b3d91d636cba443","schema_version":"1.0","event_id":"sha256:031adc4a3919056a4c030565e9933c4e7f1bc25e588272cf5b3d91d636cba443"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:NAYAZLJK4HIDNQBYA7AY4IAQFQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Laplacian spectral characterization of some graph products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Suijie Wang, Xiaogang Liu","submitted_at":"2010-07-15T04:15:51Z","abstract_excerpt":"This paper studies the Laplacian spectral characterization of some graph products. We consider a class of connected graphs: $\\mathscr{G}={G : |EG|\\leq|VG|+1}$, and characterize all graphs $G\\in\\mathscr{G}$ such that the products $G\\times K_m$ are $L$-DS graphs. The main result of this paper states that, if $G\\in\\mathscr{G}$, except for $C_{6}$ and $\\Theta_{3,2,5}$, is $L$-DS graph, so is the product $G\\times K_{m}$. In addition, the $L$-cospectral graphs with $C_{6}\\times K_{m}$ and $\\Theta_{3,2,5}\\times K_{m}$ have been found."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2472","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:36:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9vnDGqDfzT9ReOWuWSRsISlG30pSgIQKIzG8XIyFNwM+C4XnukmiuawD1nDt20mtjlqiOet2MVkTLzdLV7d/Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T20:04:47.003334Z"},"content_sha256":"a55dd158e29922b101232874a4c0c1d136cba61ce5b179db890574d4b00f5835","schema_version":"1.0","event_id":"sha256:a55dd158e29922b101232874a4c0c1d136cba61ce5b179db890574d4b00f5835"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NAYAZLJK4HIDNQBYA7AY4IAQFQ/bundle.json","state_url":"https://pith.science/pith/NAYAZLJK4HIDNQBYA7AY4IAQFQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NAYAZLJK4HIDNQBYA7AY4IAQFQ/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":"2026-06-05T20:04:47Z","links":{"resolver":"https://pith.science/pith/NAYAZLJK4HIDNQBYA7AY4IAQFQ","bundle":"https://pith.science/pith/NAYAZLJK4HIDNQBYA7AY4IAQFQ/bundle.json","state":"https://pith.science/pith/NAYAZLJK4HIDNQBYA7AY4IAQFQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NAYAZLJK4HIDNQBYA7AY4IAQFQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:NAYAZLJK4HIDNQBYA7AY4IAQFQ","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":"df6ce1331bf4481ee2554b971986a1610acc5284d1ece61d6741ffae207244ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-15T04:15:51Z","title_canon_sha256":"360815a3fe54dbd80cf6346323d070243f91a5ea0224a330f3b42b1c27a0be33"},"schema_version":"1.0","source":{"id":"1007.2472","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2472","created_at":"2026-05-18T00:36:04Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2472v3","created_at":"2026-05-18T00:36:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2472","created_at":"2026-05-18T00:36:04Z"},{"alias_kind":"pith_short_12","alias_value":"NAYAZLJK4HID","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"NAYAZLJK4HIDNQBY","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"NAYAZLJK","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:a55dd158e29922b101232874a4c0c1d136cba61ce5b179db890574d4b00f5835","target":"graph","created_at":"2026-05-18T00:36:04Z","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":"This paper studies the Laplacian spectral characterization of some graph products. We consider a class of connected graphs: $\\mathscr{G}={G : |EG|\\leq|VG|+1}$, and characterize all graphs $G\\in\\mathscr{G}$ such that the products $G\\times K_m$ are $L$-DS graphs. The main result of this paper states that, if $G\\in\\mathscr{G}$, except for $C_{6}$ and $\\Theta_{3,2,5}$, is $L$-DS graph, so is the product $G\\times K_{m}$. In addition, the $L$-cospectral graphs with $C_{6}\\times K_{m}$ and $\\Theta_{3,2,5}\\times K_{m}$ have been found.","authors_text":"Suijie Wang, Xiaogang Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-15T04:15:51Z","title":"Laplacian spectral characterization of some graph products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2472","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":"sha256:031adc4a3919056a4c030565e9933c4e7f1bc25e588272cf5b3d91d636cba443","target":"record","created_at":"2026-05-18T00:36:04Z","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":"df6ce1331bf4481ee2554b971986a1610acc5284d1ece61d6741ffae207244ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-15T04:15:51Z","title_canon_sha256":"360815a3fe54dbd80cf6346323d070243f91a5ea0224a330f3b42b1c27a0be33"},"schema_version":"1.0","source":{"id":"1007.2472","kind":"arxiv","version":3}},"canonical_sha256":"68300cad2ae1d036c03807c18e20102c24f6b37c2e573ab0e2ceba36bf0b6177","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68300cad2ae1d036c03807c18e20102c24f6b37c2e573ab0e2ceba36bf0b6177","first_computed_at":"2026-05-18T00:36:04.370726Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:36:04.370726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CpVWGVlJw/NQhkBjt0j/gJoXsEHttHWhDVwdHN+y66S68FNd55JwkxD5DgHVfrTqYOxo1mKdX9hmsEOeY7ehCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:36:04.371348Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.2472","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:031adc4a3919056a4c030565e9933c4e7f1bc25e588272cf5b3d91d636cba443","sha256:a55dd158e29922b101232874a4c0c1d136cba61ce5b179db890574d4b00f5835"],"state_sha256":"596c0390ffd260679851d3d22bfe2b84fb218cc9f9f47e83b96debc4e75bfcf8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fUUlfS8iZdzPSC5CPb1v2iC/FFywfK53LF+yApT7SyXx46Hx86A5dVnqZfvpCX+YDxPIya5rd+aIJLVDW+s5CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T20:04:47.007689Z","bundle_sha256":"592d4b179ef71ef89e23824dce2f9d2a39edcf642b45f01130c00fe56eba0c1e"}}