半环上矩阵交换图的直径

部分变换半群的交换图直径

Diameters of commuting graphs of partial transformation semigroups

本文针对有限集上的部分变换半群，确定其交换图直径，证明其与已知的对应变换半群交换图直径相等，证实存在半群与其真子半群交换图直径相等的结论。

A note on commuting graphs of matrix rings over fields

We will give a short proof of the fact that if the algebraic closure of a field $\mathbb F$ is a finite extension, then for $n\geq 3$ the commuting graph $Γ(M_n(\mathbb F))$ is connected and its diameter is four.

A Note on a Conjecture about Commuting Graphs

We prove that the diameter of the commuting graph of the full ma- trix ring over the real numbers is at most five. This answers, in the affir- mative, a conjecture proposed by Akbari-Mohammadian-radjavi-Raja, for the special case of the field of real numbers.

Okubo代数的交换图

Commuting graphs of Okubo algebras

本文研究Okubo代数的交换图连通性，证明特征非3且含本原三次单位根的域上伪八元数代数的交换图与M₃(𝔽)同构，得出代数闭域上Okubo代数及实可除Okubo代数的交换图直径均为4且后者连通，证明基于幂等元中心化子交集非零的性质。

Relative non-commuting graph of a finite ring

Let $S$ be a subring of a finite ring $R$ and $C_R(S) = \{r \in R : rs = sr \;\forall\; s \in S\}$. The relative non-commuting graph of the subring $S$ in $R$, denoted by $Γ_{S, R}$, is a simple undirected graph whose vertex set is $R \setminus C_R(S)$ and two distinct vertices $a, b$ are adjacent if and only if $a$ or $b \in S$ and $ab \neq ba$. In this paper, we discuss some properties of $Γ_{S, R}$, determine diameter, girth, some dominating sets and chromatic index for $Γ_{S, R}$. Also, we derive some connections between $Γ_{S, R}$ and the relative commuting probability of $S$ in $R$. Fina…

完全0-单半群的交换图

Commuting graphs of completely 0-simple semigroups

本文研究完全0-单半群的交换图，利用0-Rees矩阵半群特征，给出连通性判定、直径、团数、围长、色数上界及编织度的计算方法。

On generalized non-commuting graph of a finite ring

Let $S, K$ be two subrings of a finite ring $R$. Then the generalized non-commuting graph of subrings $S, K$ of $R$, denoted by $Γ_{S, K}$, is a simple graph whose vertex set is $(S \cup K) \setminus (C_K(S) \cup C_S(K))$ and two distinct vertices $a, b$ are adjacent if and only if $a \in S$ or $b \in S$ and $ab \neq ba$. We determine the diameter, girth and some dominating sets for $Γ_{S, K}$. Some connections between the $Γ_{S, K}$ and $\Pr(S, K)$ are also obtained. Further, $\Z$-isoclinism between two pairs of finite rings is defined and showed that the generalized non-commuting graphs of t…

$p$-进矩阵的交换图

Commuting graphs of $p$-adic matrices

本文研究$p$-进矩阵交换图的连通性与直径，证明连通条件为$n\geq 3$且$n$非素数或$p$的幂，并首次给出不依赖选择公理的直径$6$例子及一般界。

交换图与半群构造

Commuting graphs and semigroup constructions

本文研究了交换图与零并、直积两种半群构造的相互作用，探讨了新构造半群交换图的直径、团数等图论性质，并揭示了其与原始半群交换图性质之间的关系。

Commuting Graphs of Boundedly Generated Semigroups

Araújo, Kinyon and Konieczny (2011) pose several problems concerning the construction of arbitrary commuting graphs of semigroups. We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems. We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs o…

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。

关于三角环的交换图

On Commuting graphs of triangular rings

本文研究了非交换环上$2\times 2$上三角矩阵环的交换图，计算了$R$为有限域时的边数、团数及独立数，并证明了当$R=\mathbb{Z}_{n}$非域时该图连通且直径为3。

交换更新下线性循环的终止性

Termination of linear loops under commutative updates

本文研究交换可逆矩阵更新下多路径线性循环的终止性判定问题，通过将其转化为多项式环与多项式半环上模中是否存在正元素的代数问题，证明了该问题是可判定的。