A note on commuting graphs for symmetric groups

The commuting graph C(G;X) , where G is a group and X a subset of G, has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C(G;X) is investigated when G is the symmetric group and X a conjugacy class of G

A novel method to construct NSSD molecular graphs

A graph is said to be NSSD (=non-singular with a singular deck) if it has no eigenvalue equal to zero, whereas all its vertex-deleted subgraphs have eigenvalues equal to zero. NSSD graphs are of importance in the theory of conductance of organic compounds. In this paper, a novel method is described for constructing NSSD molecular graphs from the commuting graphs of the Hv-group. An algorithm is presented to construct the NSSD graphs from these commuting graphsThis research is partially funded through Quaid-i-Azam University grant URF-201

Minimal paths in the commuting graphs of semigroups

Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted \cg(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 $a\ne b$ 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 $|X|\geq4$, then, with a few exceptions, the diameter of \cg(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 \cg(S) is $n$. We also study the left paths in \cg(S), that is, paths $a_1-a_2-...-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all i\in \{1,\ldot, m\}. We prove that for every positive integer $n\geq2$, 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.Comment: 23 pages; v.2: Lemma 2.1 corrected; v.3: final version to appear in European J. of Combinatoric

Maximum and minimum degree energy of commuting graph for dihedral groups

If G is a finite group and Z(G) is the centre of G, then the commuting graph for G, denoted by ΓG, has G\Z(G) as its vertices set with two distinct vertices vp and vq are adjacent if vp vq = vq vp. The degree of the vertex vp of ΓG, denoted b

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