11 research outputs found
Minimal paths in the commuting graphs of semigroups
Let be a finite non-commutative semigroup. The commuting graph of ,
denoted \cg(S), is the graph whose vertices are the non-central elements of
and whose edges are the sets of vertices such that and
. Denote by the semigroup of full transformations on a finite set
. Let be any ideal of such that is different from the ideal
of constant transformations on . We prove that if , then, with a
few exceptions, the diameter of \cg(J) is 5. On the other hand, we prove that
for every positive integer , there exists a semigroup such that the
diameter of \cg(S) is . We also study the left paths in \cg(S), that is,
paths such that and for all
i\in \{1,\ldot, m\}. We prove that for every positive integer ,
except , there exists a semigroup whose shortest left path has length .
As a corollary, we use the previous results to solve a purely algebraic old
problem posed by B.M. Schein.Comment: 23 pages; v.2: Lemma 2.1 corrected; v.3: final version to appear in
European J. of Combinatoric
On the diameter of the commuting graph of the full matrix ring over the real numbers
In a recent paper C. Miguel proved that the diameter of the commuting graph of the matrix ring Mn(R) is equal to 4 if either n = 3 or n = 5. But the case n = 4 remained open, since the diameter could be 4 or 5. In this work we close the problem showing that also in this case the diameter is 4
The commuting graph of the symmetric inverse semigroup
The commuting graph of a finite non-commutative semigroup S, denoted 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 I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(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 G(Sym(X)), and in 2011, Dol˘zan and Oblak 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 I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(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 G(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 I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory