{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3N5PO3W5T52S6OMVR2UPSTWSR2","short_pith_number":"pith:3N5PO3W5","canonical_record":{"source":{"id":"2605.22160","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.RA","submitted_at":"2026-05-21T08:29:20Z","cross_cats_sorted":[],"title_canon_sha256":"13752f038f990e706050fb2e53d5abed7cef09297606b7487df4947fcf8ff9e5","abstract_canon_sha256":"4913ad465f2b37a2499e948a1b5a0b2ff13473b5099a8252dfaa936f9a026954"},"schema_version":"1.0"},"canonical_sha256":"db7af76edd9f752f39958ea8f94ed28ea0f16ff4cec71a26e99868cbd583e921","source":{"kind":"arxiv","id":"2605.22160","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.22160","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"arxiv_version","alias_value":"2605.22160v1","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22160","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"pith_short_12","alias_value":"3N5PO3W5T52S","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"pith_short_16","alias_value":"3N5PO3W5T52S6OMV","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"pith_short_8","alias_value":"3N5PO3W5","created_at":"2026-05-22T01:04:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3N5PO3W5T52S6OMVR2UPSTWSR2","target":"record","payload":{"canonical_record":{"source":{"id":"2605.22160","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.RA","submitted_at":"2026-05-21T08:29:20Z","cross_cats_sorted":[],"title_canon_sha256":"13752f038f990e706050fb2e53d5abed7cef09297606b7487df4947fcf8ff9e5","abstract_canon_sha256":"4913ad465f2b37a2499e948a1b5a0b2ff13473b5099a8252dfaa936f9a026954"},"schema_version":"1.0"},"canonical_sha256":"db7af76edd9f752f39958ea8f94ed28ea0f16ff4cec71a26e99868cbd583e921","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:29.176337Z","signature_b64":"XTqBINA+8beHVZ3XPXsvj8ZjH2nXBSxItIL67Zz3VhV5Tr2Capooqbqd0lLUe/3GFOEp+HRkmfmMe4qYq3MdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db7af76edd9f752f39958ea8f94ed28ea0f16ff4cec71a26e99868cbd583e921","last_reissued_at":"2026-05-22T01:04:29.175507Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:29.175507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.22160","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":"2026-05-22T01:04:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tP03KwVcW8/WDShlN165YsPZKkLBwWeYelSbZi6DfPeOAqAgVrtQUY64Ury1R+fYwUa94Nv3qW3mLW+rHRYCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:50:06.124303Z"},"content_sha256":"9090525da94485c0c835cedb720fa5a3a1df3a9bd9a4cee31fcc334451b4a64e","schema_version":"1.0","event_id":"sha256:9090525da94485c0c835cedb720fa5a3a1df3a9bd9a4cee31fcc334451b4a64e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3N5PO3W5T52S6OMVR2UPSTWSR2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimum second neighborhood degree energy of commuting graphs of finite rings","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jutirekha Dutta, Payal Tak, Rajat Kanti Nath","submitted_at":"2026-05-21T08:29:20Z","abstract_excerpt":"In this paper, we compute minimum second neighborhood degree spectrum and energy of commuting graphs of certain finite non-commutative rings. In particular, we consider non-commutative rings of order $p^2, p^3, p^4, p^5, p^2q$ and $p^3q$, where $p$ and $q$ are primes. We shall also show that the commuting graphs of these rings are MSN-integral but not MSN-hyperintegral. Finally, employing the techniques used in this paper, we prove Conjecture 3 of [Nath, R. K., Fasfous, W. N. T., Das, K. C. and Shang, Y. Common neighbourhood energy of commuting graphs of finite groups, {\\em Symmetry} {\\bf 13}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22160","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.22160/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-22T01:04:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BqyKV4ysMComyB/mBS9ra/7r2iOc22d9mXQN01t43VUhOPHp+h5nNkIPQVJ6nhFcpcK9eAHaLFZIbFgXe4DgAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:50:06.125032Z"},"content_sha256":"0841afb57d0394adf15c0d7250afa78d091d1756f40dd5ba8cc9a810f3259792","schema_version":"1.0","event_id":"sha256:0841afb57d0394adf15c0d7250afa78d091d1756f40dd5ba8cc9a810f3259792"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3N5PO3W5T52S6OMVR2UPSTWSR2/bundle.json","state_url":"https://pith.science/pith/3N5PO3W5T52S6OMVR2UPSTWSR2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3N5PO3W5T52S6OMVR2UPSTWSR2/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":"2026-05-31T17:50:06Z","links":{"resolver":"https://pith.science/pith/3N5PO3W5T52S6OMVR2UPSTWSR2","bundle":"https://pith.science/pith/3N5PO3W5T52S6OMVR2UPSTWSR2/bundle.json","state":"https://pith.science/pith/3N5PO3W5T52S6OMVR2UPSTWSR2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3N5PO3W5T52S6OMVR2UPSTWSR2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3N5PO3W5T52S6OMVR2UPSTWSR2","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":"4913ad465f2b37a2499e948a1b5a0b2ff13473b5099a8252dfaa936f9a026954","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.RA","submitted_at":"2026-05-21T08:29:20Z","title_canon_sha256":"13752f038f990e706050fb2e53d5abed7cef09297606b7487df4947fcf8ff9e5"},"schema_version":"1.0","source":{"id":"2605.22160","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.22160","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"arxiv_version","alias_value":"2605.22160v1","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22160","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"pith_short_12","alias_value":"3N5PO3W5T52S","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"pith_short_16","alias_value":"3N5PO3W5T52S6OMV","created_at":"2026-05-22T01:04:29Z"},{"alias_kind":"pith_short_8","alias_value":"3N5PO3W5","created_at":"2026-05-22T01:04:29Z"}],"graph_snapshots":[{"event_id":"sha256:0841afb57d0394adf15c0d7250afa78d091d1756f40dd5ba8cc9a810f3259792","target":"graph","created_at":"2026-05-22T01:04:29Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.22160/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we compute minimum second neighborhood degree spectrum and energy of commuting graphs of certain finite non-commutative rings. In particular, we consider non-commutative rings of order $p^2, p^3, p^4, p^5, p^2q$ and $p^3q$, where $p$ and $q$ are primes. We shall also show that the commuting graphs of these rings are MSN-integral but not MSN-hyperintegral. Finally, employing the techniques used in this paper, we prove Conjecture 3 of [Nath, R. K., Fasfous, W. N. T., Das, K. C. and Shang, Y. Common neighbourhood energy of commuting graphs of finite groups, {\\em Symmetry} {\\bf 13}(","authors_text":"Jutirekha Dutta, Payal Tak, Rajat Kanti Nath","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.RA","submitted_at":"2026-05-21T08:29:20Z","title":"Minimum second neighborhood degree energy of commuting graphs of finite rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22160","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":"sha256:9090525da94485c0c835cedb720fa5a3a1df3a9bd9a4cee31fcc334451b4a64e","target":"record","created_at":"2026-05-22T01:04:29Z","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":"4913ad465f2b37a2499e948a1b5a0b2ff13473b5099a8252dfaa936f9a026954","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.RA","submitted_at":"2026-05-21T08:29:20Z","title_canon_sha256":"13752f038f990e706050fb2e53d5abed7cef09297606b7487df4947fcf8ff9e5"},"schema_version":"1.0","source":{"id":"2605.22160","kind":"arxiv","version":1}},"canonical_sha256":"db7af76edd9f752f39958ea8f94ed28ea0f16ff4cec71a26e99868cbd583e921","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db7af76edd9f752f39958ea8f94ed28ea0f16ff4cec71a26e99868cbd583e921","first_computed_at":"2026-05-22T01:04:29.175507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:04:29.175507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XTqBINA+8beHVZ3XPXsvj8ZjH2nXBSxItIL67Zz3VhV5Tr2Capooqbqd0lLUe/3GFOEp+HRkmfmMe4qYq3MdCQ==","signature_status":"signed_v1","signed_at":"2026-05-22T01:04:29.176337Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.22160","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9090525da94485c0c835cedb720fa5a3a1df3a9bd9a4cee31fcc334451b4a64e","sha256:0841afb57d0394adf15c0d7250afa78d091d1756f40dd5ba8cc9a810f3259792"],"state_sha256":"6f1f24c925272c86aa88bf87db29263a5baebc400405aeda7dd56ae13b3e7470"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0nWNsuLG5FKKobuJVCVKTGvuS0Q2vgbHjAaGpPK88tF/8T6Dqs27e21uszf6Z3RLsVtIDs6P4EexEXyvMkrXDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T17:50:06.128701Z","bundle_sha256":"32db41013226629fe51e955fd6d49425a44c5d36e630e0257f0e84612e8b44ad"}}