{"paper":{"title":"Walk-Sums, Continued Fractions and Unique Factorisation on Digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"cs.DM","authors_text":"D. Jaksch, P.-L. Giscard, S. J. Thwaite","submitted_at":"2012-02-24T18:38:23Z","abstract_excerpt":"We show that the series of all walks between any two vertices of any (possibly weighted) directed graph $\\mathcal{G}$ is given by a universal continued fraction of finite depth and breadth involving the simple paths and simple cycles of $\\mathcal{G}$. A simple path is a walk forbidden to visit any vertex more than once. We obtain an explicit formula giving this continued fraction. Our results are based on an equivalent to the fundamental theorem of arithmetic: we demonstrate that arbitrary walks on $\\mathcal{G}$ factorize uniquely into nesting products of simple paths and simple cycles, where "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5523","kind":"arxiv","version":5},"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"}