{"paper":{"title":"The Commuting Graph of the Symmetric Inverse Semigroup","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Janusz Konieczny, Jo\\~ao Ara\\'ujo, Wolfram Bentz","submitted_at":"2012-05-08T11:47:28Z","abstract_excerpt":"The commuting graph of a finite non-commutative semigroup $S$, denoted $\\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\\mi(X)$ has the symmetric group $\\sym(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\\sym(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of $\\cg(\\sym(X))$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.1664","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"}