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 ≠ b and a b = b a . 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 . o study the left paths in G ( S ) , that is, paths a 1 − a 2 − ⋯ − a m such that a 1 ≠ a m and a 1 a i = a m a i 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.