Given a non-abelian finite group G, let π(G) denote the set of prime divisors of the order of G and denote by Z(G) the center of G. The prime graph of G is the graph with vertex set π(G) where two distinct primes p and q are joined by an edge if and only if G contains an element of order pq and the non-commuting graph of G is the graph with the vertex set G−Z(G) where two non-central elements x...

In this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid G(n). Also we show that G(n) contains commuting regular subsemigroup and give a necessary and sufficient condition for the groupoid G(n) to be commuting regular.

Let G be a non-abelian group. The non-commuting graph AG of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ the maximum size of a complete subgraph of Γ is called the clique number of Γ and it is denoted by ω(Γ). In this paper we characterize all non-solvable groups G with ω(AG)...

Let G be a abelian finite group. The non-commuting graph Δ(G) of G is defined as follows: The vertex set is G− Z(G), two vertex x and y are joined by an edge whenever xy = yx. Note that if G is abelian, then Δ(G) has no vertices. So, throughout this article let G be a nonabelian finite group. There are many papers on algebraic structure, using the properties of graphs, for instance see [4, 2, 3...

in this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid g(n) . also we show that g(n) contains commuting regular subsemigroup and give a necessary and sucient condition for the groupoid g(n) to be commuting regular.

let $g = (v, e)$ be a simple graph. denote by $d(g)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $a(g)$ the adjacency matrix of $g$. the  signless laplacianmatrix of $g$ is $q(g) = d(g) + a(g)$ and the $k-$th signless laplacian spectral moment of  graph $g$ is defined as $t_k(g)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...

let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

In this paper we introduce and study a family An(q) of abelian subgroups of GLn(q) covering every element of GLn(q). We show that An(q) contains all the centralizers of cyclic matrices and equality holds if q > n. For q > 2, we obtain an infinite product expression for a probabilistic generating function for |An(q)|. This leads to upper and lower bounds which show in particular that c1q −n ≤ |A...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in X. If |X| ≥ |Y | for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset of pairwise non-commuting elements. In this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...