On the clique number of non-commuting graphs of certain groups

完全简单半群的交换图

Commuting graphs of completely simple semigroups

本研究刻画群上Rees矩阵半群的交换图及其多项性质，推导完全简单半群交换图的性质与对应图类，还限定了群交换图部分性质的取值范围。

逆半群与完全正则半群的交换图

Commuting graphs of inverse semigroups and completely regular semigroups

本文针对Clifford半群、逆半群与完全正则半群类，研究能否找到对应半群使其交换图的围长、团数、色数、联结度等数值属性等于给定自然数的问题。

单位上三角群UT(4; p)的交换图

Commuting Graph of Unitriangular Group UT(4; p)

本文研究有限域Fp上的4阶单位上三角群UT(4; p)的交换图，明确其交换关系，证明约化交换图连通且直径为3，还确定了其最大团大小、色数等图论性质。

某些循环-交换群的交换图

The commuting graphs of certain cyclic-by-abelian groups

针对形如$G=AN$的有限非交换群（$A$交换、$N$循环正规），证明其交换图要么是直径≤4的连通图，要么是$|G'|+1$个完全图的不交并，结果适用于所有有限亚循环群及无平方因子阶群。

Commuting graph of $A$-orbits

Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $Γ(G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on $Γ(G,A)$.

Abelian covers of alternating groups

Let $G=A_n$, a finite alternating group. We study the commuting graph of $G$ and establish, for all possible values of $n$ barring $13, 14, 17$ and $19$, whether or not the independence number is equal to the clique-covering number.

Spectrum and energy of non-commuting graphs of finite groups

Let $G$ be a finite non-abelian group and $Γ_{nc}(G)$ be its non-commuting graph. In this paper, we compute spectrum and energy of $Γ_{nc}(G)$ for certain classes of finite groups. As a consequence of our results we construct infinite families of integral complete $r$-partite graphs. We compare energy and Laplacian energy (denoted by $E(Γ_{nc}(G))$ and $LE(Γ_{nc}(G))$ respectively) of $Γ_{nc}(G)$ and conclude that $E(Γ_{nc}(G)) \leq LE(Γ_{nc}(G))$ for those groups except for some non-abelian groups of order $pq$. This shows that the conjecture posed in [Gutman, I., Abreu, N. M. M., Vinagre, C.…

Explicit constructions of loops with commuting inner mappings

In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. Until now, no other examples were known. We construct many such loops from groups of nilpotency class two by replacing the product $xy$ with $xyh$ in certain positions, where $h$ is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.

奇偶群的Gill-Guillot交换图

The Gill-Guillot commuting graph for sporadic and related groups

本文研究了奇点简单群和某些例外Schur乘子简单群的Gill-Guillot交换图的连通性。

Tidy subgroups for commuting automorphisms of totally disconnected groups: an analogue of simultaneous triangularisation of matrices

Let αbe an automorphism of the totally disconnected group G. The compact open subgroup, V, if G is tidy for αif [α(V') : α(V')\cap V'] is minimised at V, where V' ranges over all compact open subgroups of G. Identifying a subgroup tidy for αis analogous to identifying a basis which puts a linear transformation into Jordan canonical form. This analogy is developed here by showing that commuting automorphisms have a common tidy subgroup of G and, conversely, that a group H of automorphisms having a common tidy subgroup V is abelian modulo the automorphisms which leave V invariant. Certain subgro…

具有同构幂图和交换图的无限群

Infinite groups with isomorphic power graph and commuting graph

本文研究了无限群上的幂图、增强幂图和交换图，证明了任意两图相等的充要条件，扩展了有限群的结果，并回答了P. J. Cameron提出的关于不同群间幂图与交换图同构存在性的问题。

On the structure of the power graph and the enhanced power graph of a group

Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the \emph{commuting graph} by introducing a new graph which …

有限群的超交换图及其Zagreb指数

Super commuting graphs of finite groups and their Zagreb indices

本文研究有限群上由相等、共轭和同阶关系定义的超交换图，刻画多类非交换群族的图结构，计算其Zagreb指数并验证Hansen-Vukičević猜想。

Commuting involution graphs of linear groups

In this paper, we determine the diameter of the commuting involution graphs of special and general linear groups over an arbitrary field. It turns out that our results also determine the diameter for certain projective special linear groups over finite fields. Moreover, we find the diameter of the commuting graphs of general linear groups on the set of all involutions over a field of characteristic 2, which completes the diameter of general linear groups on the set of all involutions. As an application, we classify the structure of the four-dimensional linear groups over finite fields accordin…

Commuting probabilities of infinite groups

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a…