{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:YHPWOC2ZNVBRO7JH5DC2HA63DP","short_pith_number":"pith:YHPWOC2Z","canonical_record":{"source":{"id":"1712.06308","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-18T09:28:14Z","cross_cats_sorted":[],"title_canon_sha256":"93c350668655fecb8b6513ce9ae7df5236f082937ac3eed6a496c326ad849768","abstract_canon_sha256":"bc1697aecad6a277b24f70b451315b0c88c566c8f6afcc41d49cdae1bf028d0f"},"schema_version":"1.0"},"canonical_sha256":"c1df670b596d43177d27e8c5a383db1bc97254275600ea4c43dfa1f15fd171ce","source":{"kind":"arxiv","id":"1712.06308","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.06308","created_at":"2026-05-18T00:27:50Z"},{"alias_kind":"arxiv_version","alias_value":"1712.06308v1","created_at":"2026-05-18T00:27:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.06308","created_at":"2026-05-18T00:27:50Z"},{"alias_kind":"pith_short_12","alias_value":"YHPWOC2ZNVBR","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YHPWOC2ZNVBRO7JH","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YHPWOC2Z","created_at":"2026-05-18T12:31:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:YHPWOC2ZNVBRO7JH5DC2HA63DP","target":"record","payload":{"canonical_record":{"source":{"id":"1712.06308","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-18T09:28:14Z","cross_cats_sorted":[],"title_canon_sha256":"93c350668655fecb8b6513ce9ae7df5236f082937ac3eed6a496c326ad849768","abstract_canon_sha256":"bc1697aecad6a277b24f70b451315b0c88c566c8f6afcc41d49cdae1bf028d0f"},"schema_version":"1.0"},"canonical_sha256":"c1df670b596d43177d27e8c5a383db1bc97254275600ea4c43dfa1f15fd171ce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:50.682087Z","signature_b64":"bwuN58lThW7ScQ4ZWu/Wp8H8MjYcfsvsGETTAGPiJygaflpVvsL7KQ+H+R7hWj0uLIK2YwoaM6M4wb06PtksDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c1df670b596d43177d27e8c5a383db1bc97254275600ea4c43dfa1f15fd171ce","last_reissued_at":"2026-05-18T00:27:50.681552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:50.681552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.06308","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":"2026-05-18T00:27:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YbYwFVcGknEkRyc1M80yssnty0+34Pj+U+1a5eXc+iLB2a68xnubpe8oM1/+tsOxABbSxBiWDKcu7GtdHCPFDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T00:03:04.253703Z"},"content_sha256":"cb98a808d7811c8014029f0263ecbcee4fc1a9ba3ec457b3002b490b6459b69d","schema_version":"1.0","event_id":"sha256:cb98a808d7811c8014029f0263ecbcee4fc1a9ba3ec457b3002b490b6459b69d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:YHPWOC2ZNVBRO7JH5DC2HA63DP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mixed Moore Cayley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Grahame Erskine","submitted_at":"2017-12-18T09:28:14Z","abstract_excerpt":"The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree $r$ and directed out-degree $z$, a straightforward counting argument yields an upper bound $M(z,r,2)=(z+r)^2+z+1$ for the order of the graph. Apart from the case $r=1$, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06308","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":"2026-05-18T00:27:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3UFANGM8bQIs9Yw4hz68C/gvm69hiS7ufPNSqOeOqaqVdtA7wVgbnehxhV86idl/1sskFYOAVvBkrvcBr1lrBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T00:03:04.254335Z"},"content_sha256":"67ec8400f4dda9cbe802eb0c84a3f88df95c7d8eb3cb641ac1a7418b7b20ff8b","schema_version":"1.0","event_id":"sha256:67ec8400f4dda9cbe802eb0c84a3f88df95c7d8eb3cb641ac1a7418b7b20ff8b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YHPWOC2ZNVBRO7JH5DC2HA63DP/bundle.json","state_url":"https://pith.science/pith/YHPWOC2ZNVBRO7JH5DC2HA63DP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YHPWOC2ZNVBRO7JH5DC2HA63DP/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":"2026-06-22T00:03:04Z","links":{"resolver":"https://pith.science/pith/YHPWOC2ZNVBRO7JH5DC2HA63DP","bundle":"https://pith.science/pith/YHPWOC2ZNVBRO7JH5DC2HA63DP/bundle.json","state":"https://pith.science/pith/YHPWOC2ZNVBRO7JH5DC2HA63DP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YHPWOC2ZNVBRO7JH5DC2HA63DP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:YHPWOC2ZNVBRO7JH5DC2HA63DP","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":"bc1697aecad6a277b24f70b451315b0c88c566c8f6afcc41d49cdae1bf028d0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-18T09:28:14Z","title_canon_sha256":"93c350668655fecb8b6513ce9ae7df5236f082937ac3eed6a496c326ad849768"},"schema_version":"1.0","source":{"id":"1712.06308","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.06308","created_at":"2026-05-18T00:27:50Z"},{"alias_kind":"arxiv_version","alias_value":"1712.06308v1","created_at":"2026-05-18T00:27:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.06308","created_at":"2026-05-18T00:27:50Z"},{"alias_kind":"pith_short_12","alias_value":"YHPWOC2ZNVBR","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YHPWOC2ZNVBRO7JH","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YHPWOC2Z","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:67ec8400f4dda9cbe802eb0c84a3f88df95c7d8eb3cb641ac1a7418b7b20ff8b","target":"graph","created_at":"2026-05-18T00:27:50Z","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 degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree $r$ and directed out-degree $z$, a straightforward counting argument yields an upper bound $M(z,r,2)=(z+r)^2+z+1$ for the order of the graph. Apart from the case $r=1$, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there ar","authors_text":"Grahame Erskine","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-18T09:28:14Z","title":"Mixed Moore Cayley graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06308","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:cb98a808d7811c8014029f0263ecbcee4fc1a9ba3ec457b3002b490b6459b69d","target":"record","created_at":"2026-05-18T00:27:50Z","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":"bc1697aecad6a277b24f70b451315b0c88c566c8f6afcc41d49cdae1bf028d0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-18T09:28:14Z","title_canon_sha256":"93c350668655fecb8b6513ce9ae7df5236f082937ac3eed6a496c326ad849768"},"schema_version":"1.0","source":{"id":"1712.06308","kind":"arxiv","version":1}},"canonical_sha256":"c1df670b596d43177d27e8c5a383db1bc97254275600ea4c43dfa1f15fd171ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c1df670b596d43177d27e8c5a383db1bc97254275600ea4c43dfa1f15fd171ce","first_computed_at":"2026-05-18T00:27:50.681552Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:50.681552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bwuN58lThW7ScQ4ZWu/Wp8H8MjYcfsvsGETTAGPiJygaflpVvsL7KQ+H+R7hWj0uLIK2YwoaM6M4wb06PtksDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:50.682087Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.06308","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb98a808d7811c8014029f0263ecbcee4fc1a9ba3ec457b3002b490b6459b69d","sha256:67ec8400f4dda9cbe802eb0c84a3f88df95c7d8eb3cb641ac1a7418b7b20ff8b"],"state_sha256":"eb2d074bbb6fab54e34eb9c6464ef72b944bfe5577a9431d2e25b5bc03560638"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0Bt7Jm+MCWBvZUdnIYoGkCdIl2drTcva/zMx6NGJICjdWgnsv3r17WODqLoS25nU6qyOi6VUaNaadK5Jhs5eBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T00:03:04.257751Z","bundle_sha256":"fc0aec36016f5f141c6a5a00622fe0ff30810117dabd38618fc5948c60a114cd"}}