{"paper":{"title":"On Cycles in Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Madhav P. Desai","submitted_at":"2010-09-30T06:29:36Z","abstract_excerpt":"We consider the geometric random (GR) graph on the $d-$dimensional torus with the $L_\\sigma$ distance measure ($1 \\leq \\sigma \\leq \\infty$). Our main result is an exact characterization of the probability that a particular labeled cycle exists in this random graph. For $\\sigma = 2$ and $\\sigma = \\infty$, we use this characterization to derive a series which evaluates to the cycle probability. We thus obtain an exact formula for the expected number of Hamilton cycles in the random graph (when $\\sigma = \\infty$ and $\\sigma = 2$). We also consider the adjacency matrix of the random graph and deri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.6046","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"}