For a finite group G, let Z(G) be the centre of G. Then non-commuting graph on denoted by ΓG, has G\Z(G) as its vertex set with two distinct vertices vp and vq joined an edge whenever vpvq ≠ vqvp. The degree sum matrix is square whose (p,q)-th entry dvp + dvq p different from q, otherwise, it zero, where dvi vi. This study presents general formula for energy, EDS (ΓG), dihedral groups order 2n,...

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...

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N -qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized q...

We present a two-parameter family (Gm,k)m,k∈N≥2 , of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős– Rényi random graph. As well as being of independent interest, our groups...

Suppose n is a fixed positive integer. We introduce the relative n-th non-commuting graph ΓH,G, associated to the nonabelian subgroup H of group G. The vertex set is G \ C H,G in which C H,G = {x ∈ G : [x, y] = 1 and [x, y] = 1 for all y ∈ H}. Moreover, {x, y} is an edge if x or y belong to H and xy 6= yx or xy 6= yx. In fact, the relative n-th commutativity degree, Pn(H,G) the probability that...

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N -qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized q...

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...

Let G be a finite non-abelian group and denote by Z(G) its center. The non-commuting graph of on transversal the center is whose vertices are non-central elements in two x y adjacent whenever xy=yx. In this work, we classify groups centeris double-toroidal or 1-planar.