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We determine $S(\\Gamma)$ of the generalized de Bruijn graphs $\\Gamma=\\mathrm{DB}(n,d)$ with vertices $0,\\dots,n-1$ and arcs $(i,di+k)$ for $0\\leq i\\leq n-1$ and $0\\leq k\\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs.\n  Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\\in\\math"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1405.0113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-01T08:23:05Z","cross_cats_sorted":[],"title_canon_sha256":"aba73b32a891248073a9b8789d8cbd783a8cb49ab97994033d93157b62e564e5","abstract_canon_sha256":"4100932307cd781a7cdaba0502c9838bbca5352c3103a3023be0a0fe8fa19565"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:52.976328Z","signature_b64":"zhOs/DfVdav7kAR5fKXg8wmisn1ZmYX6O+jkwgfeTmil1UoYuQbDfsqb/YZApB34Ih9ADjpmTlQ3EjsICW7CBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cf574d6b9d6be1202321c5d9a9a68e6e64ac7a26c9cac8f66a5b956514a5abc","last_reissued_at":"2026-05-18T02:38:52.976001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:52.976001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dmitrii V. Pasechnik, Henk D.L. Hollmann, Swee Hong Chan","submitted_at":"2014-05-01T08:23:05Z","abstract_excerpt":"A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $\\Gamma$ gives rise to a finite group $\\mathbb{Z}^{n-1}/\\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(\\Gamma)$ of $\\Gamma$. We determine $S(\\Gamma)$ of the generalized de Bruijn graphs $\\Gamma=\\mathrm{DB}(n,d)$ with vertices $0,\\dots,n-1$ and arcs $(i,di+k)$ for $0\\leq i\\leq n-1$ and $0\\leq k\\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs.\n  Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\\in\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0113","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1405.0113","created_at":"2026-05-18T02:38:52.976048+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.0113v1","created_at":"2026-05-18T02:38:52.976048+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.0113","created_at":"2026-05-18T02:38:52.976048+00:00"},{"alias_kind":"pith_short_12","alias_value":"TT2XJVVZ227B","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"TT2XJVVZ227BEARS","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"TT2XJVVZ","created_at":"2026-05-18T12:28:52.271510+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43","json":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43.json","graph_json":"https://pith.science/api/pith-number/TT2XJVVZ227BEARSDROZVGTI43/graph.json","events_json":"https://pith.science/api/pith-number/TT2XJVVZ227BEARSDROZVGTI43/events.json","paper":"https://pith.science/paper/TT2XJVVZ"},"agent_actions":{"view_html":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43","download_json":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43.json","view_paper":"https://pith.science/paper/TT2XJVVZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.0113&json=true","fetch_graph":"https://pith.science/api/pith-number/TT2XJVVZ227BEARSDROZVGTI43/graph.json","fetch_events":"https://pith.science/api/pith-number/TT2XJVVZ227BEARSDROZVGTI43/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43/action/storage_attestation","attest_author":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43/action/author_attestation","sign_citation":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43/action/citation_signature","submit_replication":"https://pith.science/pith/TT2XJVVZ227BEARSDROZVGTI43/action/replication_record"}},"created_at":"2026-05-18T02:38:52.976048+00:00","updated_at":"2026-05-18T02:38:52.976048+00:00"}