Summary: The commuting graph of a finite non-commutative semigroup \(S\), denoted \(\mathcal{G}(S)\), 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 \(\mathcal{I}(X)\) be the symmetric inverse semigroup of partial injective transformations on a finite set \(X\). The semigroup \(\mathcal{I}(X)\) has the symmetric group \(\mathrm{Sym}(X)\) of permutations on \(X\) as its group of units. J. M. Burns and B. Goldsmith [Bull. Lond. Math. Soc. 21, No. 1, 70–72 (1989; Zbl 0632.20003)] determined the clique number of the commuting graph of \(\mathrm{Sym}(X)\). A. Iranmanesh and A. Jafarzadeh [J. Algebra Appl. 7, No. 1, 129–146 (2008; Zbl 1151.20013)] found an upper bound of the diameter of \(G(\mathrm{Sym}(X))\), and D. Dolžan and P. Oblak [Linear Algebra Appl. 435, No. 7, 1657–1665 (2011; Zbl 1231.16037)] claimed 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 \(\mathcal{I}(X)\).
We calculate the clique number of \(\mathcal{G}(\mathcal{I}(X))\), the diameters of the commuting graphs of the proper ideals of \(\mathcal{I}(X)\), and the diameter of \(\mathcal{G}(\mathcal{I}(X))\) 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 \(\mathcal{G}(\mathcal{I}(X))\) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of \(\mathcal{I}(X)\) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of \(\mathcal{I}(X)\). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.