{"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"}