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• W2950593950 abstract "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 $cg(sym(X))$, and in 2011, Dolu{z}an and Oblak claimed (but their proof has a GAP) that this upper bound is in fact the exact value. The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup $mi(X)$. We calculate the clique number of $cg(mi(X))$, the diameters of the commuting graphs of the proper ideals of $mi(X)$, and the diameter of $cg(mi(X))$ when $|X|$ is even or a power of an odd prime. We show that when $|X|$ is odd and divisible by at least two primes, then the diameter of $cg(mi(X))$ is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of $mi(X)$ of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of $mi(X)$. The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory." @default.
• W2950593950 created "2019-06-27" @default.
• W2950593950 creator A5090238147 @default.
• W2950593950 date "2015-01-01" @default.
• W2950593950 modified "2023-09-27" @default.
• W2950593950 title "THE COMMUTING GRAPH OF THE SYMMETRIC INVERSE SEMIGROUP" @default.
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