Minimal paths in the commuting graphs of semigroups

Let SS be a finite non-commutative semigroup. The commuting graph of SS, denoted G(S)G(S), is the graph whose vertices are the non-central elements of SS and whose edges are the sets {a,b}{a,b} of vertices such that a≠ba≠b and ab=baab=ba. Denote by T(X)T(X) the semigroup of full transformations on a finite set XX. Let JJ be any ideal of T(X)T(X) such that JJ is different from the ideal of constant transformations on XX. We prove that if |X|≥4|X|≥4, then, with a few exceptions, the diameter of G(J)G(J) is 55. On the other hand, we prove that for every positive integer nn, there exists a semigroup SS such that the diameter of G(S)G(S) is nn.We also study the left paths in G(S)G(S), that is, paths a1−a2−⋯−ama1−a2−⋯−am such that a1≠ama1≠am and a1ai=amaia1ai=amai for all i∈{1,…,m}i∈{1,…,m}. We prove that for every positive integer n≥2n≥2, except n=3n=3, there exists a semigroup whose shortest left path has length nn. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.