{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:A37FCROJTE4QGY4CHSWAGEGRYN","short_pith_number":"pith:A37FCROJ","canonical_record":{"source":{"id":"1809.03297","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-10T13:28:52Z","cross_cats_sorted":[],"title_canon_sha256":"030bd7fdd2b367a27482bd40f5317ffa5971e655711a11e05a9304b54dbc7e36","abstract_canon_sha256":"8821ef96e700aae40b91eceb4587823a25ca34f540216fbc395ad5816d096657"},"schema_version":"1.0"},"canonical_sha256":"06fe5145c999390363823cac0310d1c3756835db0b3f69fbfb972736b32ab8ce","source":{"kind":"arxiv","id":"1809.03297","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.03297","created_at":"2026-05-18T00:06:09Z"},{"alias_kind":"arxiv_version","alias_value":"1809.03297v1","created_at":"2026-05-18T00:06:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.03297","created_at":"2026-05-18T00:06:09Z"},{"alias_kind":"pith_short_12","alias_value":"A37FCROJTE4Q","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"A37FCROJTE4QGY4C","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"A37FCROJ","created_at":"2026-05-18T12:32:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:A37FCROJTE4QGY4CHSWAGEGRYN","target":"record","payload":{"canonical_record":{"source":{"id":"1809.03297","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-10T13:28:52Z","cross_cats_sorted":[],"title_canon_sha256":"030bd7fdd2b367a27482bd40f5317ffa5971e655711a11e05a9304b54dbc7e36","abstract_canon_sha256":"8821ef96e700aae40b91eceb4587823a25ca34f540216fbc395ad5816d096657"},"schema_version":"1.0"},"canonical_sha256":"06fe5145c999390363823cac0310d1c3756835db0b3f69fbfb972736b32ab8ce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:09.315098Z","signature_b64":"BN9iFd+UYgTYwf74uH+OYXkq+/ezZ1MI8Pvl64ypYMfhPp0LhutdTkkBnDSxkK9GJUvUxeQ7BtR5RhA15E80BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06fe5145c999390363823cac0310d1c3756835db0b3f69fbfb972736b32ab8ce","last_reissued_at":"2026-05-18T00:06:09.314444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:09.314444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.03297","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":"2026-05-18T00:06:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BUf0+dcXuV5Pn1K+siGL1FQ6VG3iiPglOF1GayiC69ljdQ1AAHNc706dBx7LOTGROJBjDwcJ67Xs4a6JhIrVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T21:04:48.064881Z"},"content_sha256":"5aa3c697b94976a0b10bc583bc3b8177a9b5bd1ca605eda3ba34837b96937382","schema_version":"1.0","event_id":"sha256:5aa3c697b94976a0b10bc583bc3b8177a9b5bd1ca605eda3ba34837b96937382"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:A37FCROJTE4QGY4CHSWAGEGRYN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Properties of stepwise irregular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Prithwineel Paul, Somnath Bera","submitted_at":"2018-09-10T13:28:52Z","abstract_excerpt":"Stepwise irregular (SI) graphs were introduced by Ivan Gutman recently in 2018 and in these graphs the difference between the degrees of any two adjacent vertices is exactly one. In this work, we show the existence of connected bicyclic SI graphs of order 5 and 9 which has been claimed to be non-existent by Gutman. We also show the existence of SI graphs of different order and cyclomatic numbers when there exist a SI graph with a vertex of degree 1 or 2. We give a characterization on the order of the connected tricyclic SI graphs. At the end, we investigate some properties of the SI graphs und"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.03297","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":"2026-05-18T00:06:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9KTiS1NW0JGzCUeHbq6ZKIeCJDpGz5ehZlHetjqsrmLZ978VGBQ/oJPCqm55KGwmIf0g7S/50r0MlCXUe+7ICg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T21:04:48.065235Z"},"content_sha256":"d38548fafa00041eb104de7dc5ea1211393e69d8e2d1047be3fb5639410765dd","schema_version":"1.0","event_id":"sha256:d38548fafa00041eb104de7dc5ea1211393e69d8e2d1047be3fb5639410765dd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A37FCROJTE4QGY4CHSWAGEGRYN/bundle.json","state_url":"https://pith.science/pith/A37FCROJTE4QGY4CHSWAGEGRYN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A37FCROJTE4QGY4CHSWAGEGRYN/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":"2026-07-01T21:04:48Z","links":{"resolver":"https://pith.science/pith/A37FCROJTE4QGY4CHSWAGEGRYN","bundle":"https://pith.science/pith/A37FCROJTE4QGY4CHSWAGEGRYN/bundle.json","state":"https://pith.science/pith/A37FCROJTE4QGY4CHSWAGEGRYN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A37FCROJTE4QGY4CHSWAGEGRYN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:A37FCROJTE4QGY4CHSWAGEGRYN","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":"8821ef96e700aae40b91eceb4587823a25ca34f540216fbc395ad5816d096657","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-10T13:28:52Z","title_canon_sha256":"030bd7fdd2b367a27482bd40f5317ffa5971e655711a11e05a9304b54dbc7e36"},"schema_version":"1.0","source":{"id":"1809.03297","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.03297","created_at":"2026-05-18T00:06:09Z"},{"alias_kind":"arxiv_version","alias_value":"1809.03297v1","created_at":"2026-05-18T00:06:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.03297","created_at":"2026-05-18T00:06:09Z"},{"alias_kind":"pith_short_12","alias_value":"A37FCROJTE4Q","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"A37FCROJTE4QGY4C","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"A37FCROJ","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:d38548fafa00041eb104de7dc5ea1211393e69d8e2d1047be3fb5639410765dd","target":"graph","created_at":"2026-05-18T00:06:09Z","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":"Stepwise irregular (SI) graphs were introduced by Ivan Gutman recently in 2018 and in these graphs the difference between the degrees of any two adjacent vertices is exactly one. In this work, we show the existence of connected bicyclic SI graphs of order 5 and 9 which has been claimed to be non-existent by Gutman. We also show the existence of SI graphs of different order and cyclomatic numbers when there exist a SI graph with a vertex of degree 1 or 2. We give a characterization on the order of the connected tricyclic SI graphs. At the end, we investigate some properties of the SI graphs und","authors_text":"Prithwineel Paul, Somnath Bera","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-10T13:28:52Z","title":"Properties of stepwise irregular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.03297","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":"sha256:5aa3c697b94976a0b10bc583bc3b8177a9b5bd1ca605eda3ba34837b96937382","target":"record","created_at":"2026-05-18T00:06:09Z","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":"8821ef96e700aae40b91eceb4587823a25ca34f540216fbc395ad5816d096657","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-10T13:28:52Z","title_canon_sha256":"030bd7fdd2b367a27482bd40f5317ffa5971e655711a11e05a9304b54dbc7e36"},"schema_version":"1.0","source":{"id":"1809.03297","kind":"arxiv","version":1}},"canonical_sha256":"06fe5145c999390363823cac0310d1c3756835db0b3f69fbfb972736b32ab8ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06fe5145c999390363823cac0310d1c3756835db0b3f69fbfb972736b32ab8ce","first_computed_at":"2026-05-18T00:06:09.314444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:09.314444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BN9iFd+UYgTYwf74uH+OYXkq+/ezZ1MI8Pvl64ypYMfhPp0LhutdTkkBnDSxkK9GJUvUxeQ7BtR5RhA15E80BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:09.315098Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.03297","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5aa3c697b94976a0b10bc583bc3b8177a9b5bd1ca605eda3ba34837b96937382","sha256:d38548fafa00041eb104de7dc5ea1211393e69d8e2d1047be3fb5639410765dd"],"state_sha256":"c2cb651d4877118435bd19f82712cb0feb19041979c16b9e96c7eaa85e7ca737"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BJPVh6St3N68V0rnH4JWKkdkCl68EVgynOwDyalbq55utA5c2eq2+cJDESz5xHyHySgQlBMwKQd1EB7f066HBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T21:04:48.067218Z","bundle_sha256":"bb65345309837e09b7ad5e0d925325cb1fb0cceac2fff2a834b54b744a3b2654"}}