a. r. ashrafi

The commuting graph of a finite group GG, C(G)C(G), is a simple graph with vertex set GG in which two vertices xx and yy are adjacent if and only if xy=yxxy=yx. The aim of this paper is to compute the distance Laplacian spectrum and the distance Laplacian energy of the commuting graph of CACA-groups.

The number of subgroups, normal subgroups and characteristic subgroups of a finite group G are denoted by Sub(G), NSub(G) and CSub(G), respectively. The main goal of this paper is to present a matrix model for computing these positive integers for dicyclic groups, semi-dihedral groups, and three sequences U6n, V8n and H(n) of groups that can be presented as follows: U6n = ⟨a, b | a 2n = b 3 = e, bab = a⟩, V8n = ⟨a, b | a 2n = b 4 = e, aba = b −1 , ab−1a = b⟩, H(n) = ⟨a, b, c | a 2 n−2 = b 2 = c 2 = e, [a, b] = [b, c] = e, ac = ab⟩. For each group, a matrix model containing all information is given.

Suppose that G is a finite non-abelian group. Define the graph Γ(G) with the non-central conjugacy classes of G as vertex set and two distinct vertices A and B are adjacent if and only if there are x∈A and y∈B such that xy=yx. The graph Γ(G) is called the commuting conjugacy class graph of G and introduced by Mohammadian et al. in [A. Mohammadian, A. Erfanian, M. Farrokhi D. G. and B. Wilkens, Triangle-free commuting conjugacy class graphs, {J. Group Theory} {19} (3) (2016) 1049--1061]. In this paper, the graph structure of the commuting conjugacy class graph of nilpotent groups of order n are obtained in which n is not divisible by p5, for every prime factor p of n.