{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GMFC4F7BCLZ5HNWV6MSWMUHI3G","short_pith_number":"pith:GMFC4F7B","canonical_record":{"source":{"id":"2602.08684","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-09T14:06:32Z","cross_cats_sorted":[],"title_canon_sha256":"31f8cef7e68bf03d00b04ce9c69cf5b1e4c5d61a5baa0e14ff521dcc159bf404","abstract_canon_sha256":"f9a5c9ff7f1c01612587333b21fdc5791c5f95321b727c92992ad97db5bece64"},"schema_version":"1.0"},"canonical_sha256":"330a2e17e112f3d3b6d5f3256650e8d98d0c9be71a0646aae65390269478f52b","source":{"kind":"arxiv","id":"2602.08684","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.08684","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"arxiv_version","alias_value":"2602.08684v2","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.08684","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"pith_short_12","alias_value":"GMFC4F7BCLZ5","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"pith_short_16","alias_value":"GMFC4F7BCLZ5HNWV","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"pith_short_8","alias_value":"GMFC4F7B","created_at":"2026-05-26T02:04:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GMFC4F7BCLZ5HNWV6MSWMUHI3G","target":"record","payload":{"canonical_record":{"source":{"id":"2602.08684","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-09T14:06:32Z","cross_cats_sorted":[],"title_canon_sha256":"31f8cef7e68bf03d00b04ce9c69cf5b1e4c5d61a5baa0e14ff521dcc159bf404","abstract_canon_sha256":"f9a5c9ff7f1c01612587333b21fdc5791c5f95321b727c92992ad97db5bece64"},"schema_version":"1.0"},"canonical_sha256":"330a2e17e112f3d3b6d5f3256650e8d98d0c9be71a0646aae65390269478f52b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:05.874013Z","signature_b64":"tyfeL+nDFtGj/Uo+coGgDv6bxbIPJxNk27M5q5A8lIgWfmHO8YKtELI2WJet0B6Fic5o1I83HJVBUMkl9fO3Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"330a2e17e112f3d3b6d5f3256650e8d98d0c9be71a0646aae65390269478f52b","last_reissued_at":"2026-05-26T02:04:05.873020Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:05.873020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.08684","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-26T02:04:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Na1Qitwo0Lia4JqGCqKDdlIvMa8LnPSlv4gCZul7jrtvf9FvN4KbyddtZKB0XnuLsCy2LG6VfErAHWDcq8YyCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:19:46.276580Z"},"content_sha256":"3f08a4647e391d46ccf22ec2f54309a868a882891e57de0ca55f47a7a23dac05","schema_version":"1.0","event_id":"sha256:3f08a4647e391d46ccf22ec2f54309a868a882891e57de0ca55f47a7a23dac05"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GMFC4F7BCLZ5HNWV6MSWMUHI3G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Laplacian Pair State Transfer on Total Graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akash Kalita, Bikash Bhattacharjya","submitted_at":"2026-02-09T14:06:32Z","abstract_excerpt":"The total graph of a graph $G$, denoted $\\mathcal{T}(G)$, is defined as the graph whose vertex set is the union of the vertex set of $G$ and the edge set of $G$, such that two vertices of $\\mathcal{T}(G)$ are adjacent if the corresponding elements of $G$ are either adjacent or incident. In this paper, we investigate the existence of Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on $\\mathcal{T}(G)$, where $G$ is an $r$-regular graph. We prove that if $G$ is Laplacian integral, $r \\geq 3$, and $r+1$ is not a Laplacian eigenvalue of $G$, then $\\mathcal{T}(G)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.08684","kind":"arxiv","version":2},"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/2602.08684/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":"2026-05-26T02:04:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lSpu15/i+w8j+4EPdGUi77V71rGF0EzPeko3NzuIOUXvla9L7JergKIIFViZ2KG6h5fXCpdU35WbrnJ4o8ouAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:19:46.277287Z"},"content_sha256":"709cdc7d4e465a2969e7afa8e8f68cece08d088c642bdad8b9fc222dd5491a16","schema_version":"1.0","event_id":"sha256:709cdc7d4e465a2969e7afa8e8f68cece08d088c642bdad8b9fc222dd5491a16"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GMFC4F7BCLZ5HNWV6MSWMUHI3G/bundle.json","state_url":"https://pith.science/pith/GMFC4F7BCLZ5HNWV6MSWMUHI3G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GMFC4F7BCLZ5HNWV6MSWMUHI3G/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":"2026-06-06T21:19:46Z","links":{"resolver":"https://pith.science/pith/GMFC4F7BCLZ5HNWV6MSWMUHI3G","bundle":"https://pith.science/pith/GMFC4F7BCLZ5HNWV6MSWMUHI3G/bundle.json","state":"https://pith.science/pith/GMFC4F7BCLZ5HNWV6MSWMUHI3G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GMFC4F7BCLZ5HNWV6MSWMUHI3G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GMFC4F7BCLZ5HNWV6MSWMUHI3G","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":"f9a5c9ff7f1c01612587333b21fdc5791c5f95321b727c92992ad97db5bece64","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-09T14:06:32Z","title_canon_sha256":"31f8cef7e68bf03d00b04ce9c69cf5b1e4c5d61a5baa0e14ff521dcc159bf404"},"schema_version":"1.0","source":{"id":"2602.08684","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.08684","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"arxiv_version","alias_value":"2602.08684v2","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.08684","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"pith_short_12","alias_value":"GMFC4F7BCLZ5","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"pith_short_16","alias_value":"GMFC4F7BCLZ5HNWV","created_at":"2026-05-26T02:04:05Z"},{"alias_kind":"pith_short_8","alias_value":"GMFC4F7B","created_at":"2026-05-26T02:04:05Z"}],"graph_snapshots":[{"event_id":"sha256:709cdc7d4e465a2969e7afa8e8f68cece08d088c642bdad8b9fc222dd5491a16","target":"graph","created_at":"2026-05-26T02:04:05Z","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/2602.08684/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The total graph of a graph $G$, denoted $\\mathcal{T}(G)$, is defined as the graph whose vertex set is the union of the vertex set of $G$ and the edge set of $G$, such that two vertices of $\\mathcal{T}(G)$ are adjacent if the corresponding elements of $G$ are either adjacent or incident. In this paper, we investigate the existence of Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on $\\mathcal{T}(G)$, where $G$ is an $r$-regular graph. We prove that if $G$ is Laplacian integral, $r \\geq 3$, and $r+1$ is not a Laplacian eigenvalue of $G$, then $\\mathcal{T}(G)$","authors_text":"Akash Kalita, Bikash Bhattacharjya","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-09T14:06:32Z","title":"Laplacian Pair State Transfer on Total Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.08684","kind":"arxiv","version":2},"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":"sha256:3f08a4647e391d46ccf22ec2f54309a868a882891e57de0ca55f47a7a23dac05","target":"record","created_at":"2026-05-26T02:04:05Z","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":"f9a5c9ff7f1c01612587333b21fdc5791c5f95321b727c92992ad97db5bece64","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-09T14:06:32Z","title_canon_sha256":"31f8cef7e68bf03d00b04ce9c69cf5b1e4c5d61a5baa0e14ff521dcc159bf404"},"schema_version":"1.0","source":{"id":"2602.08684","kind":"arxiv","version":2}},"canonical_sha256":"330a2e17e112f3d3b6d5f3256650e8d98d0c9be71a0646aae65390269478f52b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"330a2e17e112f3d3b6d5f3256650e8d98d0c9be71a0646aae65390269478f52b","first_computed_at":"2026-05-26T02:04:05.873020Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T02:04:05.873020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tyfeL+nDFtGj/Uo+coGgDv6bxbIPJxNk27M5q5A8lIgWfmHO8YKtELI2WJet0B6Fic5o1I83HJVBUMkl9fO3Cg==","signature_status":"signed_v1","signed_at":"2026-05-26T02:04:05.874013Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.08684","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3f08a4647e391d46ccf22ec2f54309a868a882891e57de0ca55f47a7a23dac05","sha256:709cdc7d4e465a2969e7afa8e8f68cece08d088c642bdad8b9fc222dd5491a16"],"state_sha256":"da171262c5ba6baaf81f6494a3b163f06767449a59877ef479181d4df6cba204"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ANhZylu7qyuQcGFnxQGsxdQlFlyEwrPdwiDqagqPmvFQWXWEJc01ePH7CwVzjsBGtutoWrNYz4Ustq3Asm78Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T21:19:46.280902Z","bundle_sha256":"891c0a3c127dc72402e15e8387016a596ac5a2390d5ee3ed525f6cbb8c176ddd"}}