Minimal paths in the commuting graphs of semigroups

Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non-central elements of S and whose edges are the sets {a, b} of vertices such that a 6= b and ab = ba. Denote by T (X) the semigroup of full transformations on a finite set X. Let J be any ideal of T (X) such that J is different from the ideal of constant transformations on X. We prove that if |X| ≥ 4, then, with a few exceptions, the diameter of G(J) is 5. On the other hand, we prove that for every positive integer n, there exists a semigroup S such that the diameter of G(S) is n. We also study the left paths in G(S), that is, paths a1− a2− · · ·− am such that a1 6= am and a1ai = amai for all i ∈ {1, . . . ,m}. We prove that for every positive integer n ≥ 2, except n = 3, there exists a semigroup whose shortest left path has length n. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein. 2010 Mathematics Subject Classification. 05C25, 05C12, 20M20.