{"paper":{"title":"Reliability Polynomials and their Asymptotic Limits for Families of Graphs","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Robert Shrock, Shu-Chiuan Chang","submitted_at":"2002-08-27T23:40:26Z","abstract_excerpt":"We present exact calculations of reliability polynomials $R(G,p)$ for lattice strips $G$ of fixed widths $L_y \\le 4$ and arbitrarily great length $L_x$ with various boundary conditions. We introduce the notion of a reliability per vertex, $r(\\{G\\},p) = \\lim_{|V| \\to \\infty} R(G,p)^{1/|V|}$ where $|V|$ denotes the number of vertices in $G$ and $\\{G\\}$ denotes the formal limit $\\lim_{|V| \\to \\infty} G$. We calculate this exactly for various families of graphs. We also study the zeros of $R(G,p)$ in the complex $p$ plane and determine exactly the asymptotic accumulation set of these zeros ${\\cal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cond-mat/0208538","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":""},"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"}