{"paper":{"title":"On the spectral distributions of distance-k graph of free product graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.CO","authors_text":"Octavio Arizmendi, Tulio Gaxiola","submitted_at":"2016-01-18T23:02:30Z","abstract_excerpt":"We calculate the distribution with respect to the vacuum state of the distance-$k$ graph of a $d$-regular tree. From this result we show that the distance-$k$ graph of a $d$-regular graphs converges to the distribution of the distance-$k$ graph of a regular tree.\n  Finally, we prove that, properly normalized, the asymptotic distributions of distance-$k$ graphs of the $d$-fold free product graph, as $d$ tends to infinity, is given by the distribution of $P_k(s)$, where $s$ is a semicircle random variable and $P_k$ is the $k$-th Chebychev polynomial."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04752","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"}