Commuting graph of $A$-orbits

The diameter of the commuting graph of a finite group with trivial centre

The commuting graph of a finite group with trivial centre is examined. It is shown that the connected components of the commuting graph have diameter at most 10.

交换图及一类与中心化子相关联的图

The commuting graph and a graph associated with centralizers

针对$p$-群场景，本文研究交换图结构与$|G:Z(G)|$的关系，构造了满足特定交换图性质的群族，并引入一类关联中心化子、与交换图相关的新图。

Laplacian Spectrum of non-commuting graphs of finite groups

In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups considered in this paper are L-integral. We also obtain some conditions on a group $G$ so that its non-commuting graph is L-integral.

On the genus of the commuting graphs of finite non-abelian groups

The commuting graph of a non-abelian group is a simple graph in which the vertices are the non-central elements of the group, and two distinct vertices are adjacent if and only if they commute. In this paper, we classify (up to isomorphism) all finite non-abelian groups whose commuting graphs are acyclic, planar or toroidal. We also derive explicit formulas for the genus of the commuting graphs of some well-known class of finite non-abelian groups, and show that, every collection of finite non-abelian groups whose commuting graphs have the same genus is finite.

Commuting involution graphs for $\tilde C_n$

In this article we consider commuting graphs of involution conjugacy classes in the affine Weyl group of type $\tilde C_n$. We show that where the graph is connected the diameter is at most $n+2$.

Finite groups whose commuting graphs are integral

A finite non-abelian group $G$ is called commuting integral if the commuting graph of $G$ is integral. In this paper, we show that a finite group is commuting integral if its central factor is isomorphic to ${\mathbb{Z}}_p \times {\mathbb{Z}}_p$ or $D_{2m}$, where $p$ is any prime integer and $D_{2m}$ is the dihedral group of order $2m$.

On Laplacian energy of non-commuting graphs of finite groups

In this paper, we compute Laplacian energy of the non-commuting graphs of some classes of finite non-abelian groups.

Non-commuting graphs of nilpotent groups

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-commuting graph $Γ_G$ associated to $G$ is the graph whose vertex set is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only if $xy\neq yx$. We prove that if $G$ and $H$ are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then $|G|=|H|$.

The energy and spectrum of non commuting graph

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ(G) of G is a graph with vertex set is non central elements of G and two vertices x, y are adjacent if and only if they are commute. In this paper we calculate the energy, Laplacian energy and spectrum of non-commuting graph of dihedral group D2n. Also we will obtain the energy of non-commuting graph of D2n \times D2n and G \times H, where G is a non-abelian finite group and H is an abelian finite group

Groups with maximum vertex degree commuting graphs

Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph $Γ(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices are adjacent if they commute. We say that $Γ(G)$ is strong $k$ star free if the $k$ star graph is not a subgraph of $Γ(G)$. In this paper, we characterize all strong $5$ star free commuting graphs. As a byproduct, we classify all strong claw-free graphs. Also, we prove that the set of all non-abelian groups whose commuting graph is strong $k$ star free is finite.

A Note on Commuting Involution Graphs in Affine Coxeter Groups

Commuting involution graphs have been studied for finite Coxeter groups and for affine groups of classical type. The purpose of this short note is to establish some general results for commuting involution graphs in affine Coxeter groups, and to deal with types $\tilde F_4$ and $\tilde G_2$. Types $\tilde E_6$ $\tilde E_7$ and $\tilde E_8$ are more substantial and we will address these in a forthcoming paper.

可解 A-群的交换图

The Commuting Graph of a Solvable A-Group

本文研究了可解 A-群连通交换图直径的上界，证明当导出长度为 2 时直径至多为 4，一般情形下至多为 6，并提供示例说明这些上界无法改进。

Spectrum and energies of commuting conjugacy class graph of a finite group

In this paper we compute spectrum, Laplacian spectrum, signless Laplacian spectrum and their corresponding energies of commuting conjugacy class graph of the group $G(p, m, n) = \langle x, y : x^{p^m} = y^{p^n} = [x, y]^p = 1, [x, [x, y]] = [y, [x, y]] = 1\rangle$, where $p$ is any prime, $m \geq 1$ and $n \geq 1$. We derive some consequences along with the fact that commuting conjugacy class graph of $G(p, m, n)$ is super integral. We also compare various energies and determine whether commuting conjugacy class graph of $G(p, m, n)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.

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。

Extending Morgan and Parker's results about commuting graphs

Morgan and Parker have proved that if $G$ is a group satisfying the condition that $Z(G) = 1$, then the connected components of the commuting graph of $G$ have diameter at most $10$. Parker has proved that if in addition $G$ is solvable, then the commuting graph of $G$ is disconnected if and only if $G$ is a Frobenius group or a $2$-Frobenius group, and if the commuting graph of $G$ is connected, then its diameter is at most $8$. We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap Z(G) = 1$. We also prove that if $G$ is solvable and $G/Z(G)$ is either a Frob…

Common neighbourhood spectrum and energy of commuting conjugacy class graph

In this paper we compute common neighbourhood (abbreviated as CN) spectrum and energy of commuting conjugacy class graph of several families of finite non-abelian groups. As a consequence of our results we show that the commuting conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose central quotient is isomorphic to $D_{2n}$ or $Z_p \times Z_p$, for some prime $p$, are CN-integral but not CN-hyperenergetic.

有限群非交换图的无符号拉普拉斯能量及相关结果

Signless Laplacian energies of non-commuting graphs of finite groups and related results

本文研究了有限群非交换图的无符号拉普拉斯谱和能量，给出了使其成为Q-integral的条件，并探讨了能量关系及hyper-和hypo-energetic性质。

Commuting Involution Graphs in Classical Affine Weyl Groups

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let $W$ be a classical Weyl group of rank $n$, with $\tilde W$ its corresponding affine Weyl group. Our main result is that if $X$ is a conjugacy class of involutions in $\tilde W$, then the commuting involution graph $\mathcal{C}(\tilde W, X)$ is either disconnected or has diameter at most $n+2$. This bound is known to hold for types $\tilde A_n$ and $\tilde C_n$, so the main work of this paper is to prove the theorem for types $\tilde B_n$ and $\tilde D_n$.

交换图的公共邻域Laplacian谱与无符号Laplacian谱及能量

Common neighborhood Laplacian and signless Laplacian spectra and energies of commuting graphs

针对若干有限非阿贝尔群的交换图，本文计算其公共邻域Laplacian与无符号Laplacian谱及能量，证明整谱性并刻画超能量情形。

AC群的非交换图：作为拟阵

Non-commuting graph of AC-groups: as matroids

本文证明有限群G为AC群等价于其非交换图是拟阵，借助拟阵性质探究AC群特征，还给出其非交换图团数计算公式，为这类群结构研究提供新视角。

$PSL_2(q)$交换交换involution图的自同构群

Automorphism Groups of the $PSL_2(q)$ Commuting Involution Graphs

本文研究了PSL_2(q)交换involution图的自同构群，通过分析共轭类X的性质，确定了该图的自同构群结构。

On the commuting graphs of Brandt semigroups

The commuting graph of a finite non-commutative semigroup S, denoted by Δ(S), is the simple graph whose vertices are the non-central elements of S and two distinct vertices x; y are adjacent if xy = yx. In the present paper, we study various graph-theoretic properties of the commuting graph Δ(B_n) of Brandt semigroup B_n including its diameter, clique number, chromatic number, independence number, strong metric dimension and dominance number. Moreover, we obtain the automorphism group Aut(Δ(Bn)) and the endomorphism monoid End(Δ(Bn)) of Δ(Bn). We show that Aut(Δ(Bn)) = S_n \times Z_2, where S_…

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

Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ 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 $ω(\mathcal{A}_G)\leq 57$, where the number 57 is the clique number of the non-commuting graph of the projective special linear group $\mathrm{PSL}(2,7)$. We also complete the det…

有限群的交换共轭类图与Hansen-Vukičević猜想

Commuting Conjugacy Class Graphs of Finite Groups and the Hansen-Vukičević Conjecture

本文计算了有限群交换共轭类图的Zagreb指数，并证明了多种有限群及其商群的交换共轭类图满足Hansen-Vukičević猜想。

中心有 $p^{n-2}$ 个元素的 $p^n$ 阶非交换群的交换图

Commuting graph of non-abelian groups of order $p^n$ with center having $p^{n-2}$ elements

研究中心大小为 $p^{n-2}$ 的 $p^n$ 阶非交换 $p$-群的交换图，分析其绕路距离、度量维数、分辨多项式和谱性质。

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 group action with few edges

Let $A$ be a group acting by automorphisms on the group $G.$ \textit{The commuting graph $Γ(G,A)$ of $A$-orbits} of this action is the simple graph with vertex set $\{x^{A} : 1\ne x \in G \}$, the set of all $A$-orbits on $G\setminus \{1\}$, where two distinct vertices $x^{A}$ and $y^{A}$ are joined by an edge if and only if there exist $x_{1}\in x^{A}$ and $y_{1}\in y^{A}$ such that $[x_{1},y_{1}]=1$. The present paper characterizes the groups $G$ for which $Γ(G,A)$ is an $\mathcal{F}$-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.

Counting Centralizers of a Finite Group with an Application in Constructing the Commuting Conjugacy Class Graph

The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property that $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a consequence, it is proved that such a group has exactly $[(p+1)^2+1]$ element centralizers and the structure of the commuting conjugacy class graph of $G$ is completely determined.

Between the enhanced power graph and the commuting graph

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that the automorphism group of $G$ acts as automorphisms of the deep commuting graph.

Abelian coverings of finite general linear groups and an application to their non-commuting graph

In this paper we introduce and study a family $\mathcal{A}_n(q)$ of abelian subgroups of $\GL_n(q)$ covering every element of $\GL_n(q)$. We show that $\mathcal{A}_n(q)$ contains all the centralisers of cyclic matrices and equality holds if $q>n$. Also, for $q>2$, we prove a simple closed formula for the size of $\mathcal{A}_n(q)$ and give an upper bound if $q=2$. A subset $X$ of a finite group $G$ is said to be pairwise non-commuting if $xy\not=yx$, for distinct elements $x, y$ in $X$. As an application of our results on $\mathcal{A}_n(q)$, we prove lower and upper bounds for the maximum size…

On commuting elements and embeddings of graph groups and monoids

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G(Gamma) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Gamma) does not have a subgroup isomorphic to a direct product of non-abelian free groups. We also obtain corresponding and more general results for monoids.

Commuting graph of a group on a transversal

Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\neq y$ and $xy=yx$. Let $T$ be a transversal of the center $Z(G)$ of $G$. When $G$ is a finite non-abelian group and $X=T\setminus Z(G)$, we denote the graph ${\cal C}(G,X)$ by ${\cal T}(G)$. In this paper, we show that ${\cal T}(G)$ is a connected strongly regular graph if and only if $G$ is isoclinic to an extraspecial $2$-group of order at least $32$. We also characterize the finite non-abel…

Genus of commuting conjugacy class graph of groups

For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute the genus of $\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, tor…

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.…

There is no upper bound for the diameter of the commuting graph of a finite group

We construct a family of finite special 2-groups which have commuting graph of increasing diameter

Spectral aspects of commuting conjugacy class graph of finite groups

The commuting conjugacy class graph of a non-abelian group $G$, denoted by $\mathcal{CCC}(G)$, is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups $D_{2n}, Q_{4m}, U_{(n, m)}, V_{8n}$ and $SD_{8n}$. Our computation shows that $\mathcal{CCC}(G)$ is super integral for these groups. We compare vari…

On the Non-Commuting Graph of Dihedral Group

For a nonabelian group G, the non-commuting graph $Γ_G$ of $G$ is defined as the graph with vertex set $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices of $Γ_G$ are adjacent if they do not commute in $G$. In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of non-commuting graph on $D_{2n}$. We also find the mean distance of non-commuting graph on $D_{2n}$.

Recognizing the Commuting Graph of a Finite Group

In this paper we study the realizability question for commuting graphs of finite groups: Given an undirected graph $X$ is it the commuting graph of a group $G$? And if so, to determine such a group. We seek efficient algorithms for this problem. We make some general observations on this problem, and obtain a polynomial-time algorithm for the case of extraspecial groups.

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。

Commuting graphs of odd prime order elements in simple groups

We study the commuting graph on elements of odd prime order in finite simple groups. The results are used in a forthcoming paper describing the structure of Bruck loops and Bol loops of exponent 2.

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。

具有小亏格的有限群交换图的刻画

Characterization of commuting graphs of finite groups having small genus

本文刻画了小亏格有限群的交换图，明确可实现为双环面、三环面交换图的图类，验证了相关图满足两类猜想，还发现新的超能量非完全图，为Gutman猜想提供反例。

On Bounding the Diameter of the Commuting Graph of a Group

The commuting graph of a group $G$ is the simple undirected graph whose vertices are the non-central elements of $G$ and two distinct vertices are adjacent if and only if they commute. It is conjectured by Jafarzadeh and Iranmanesh that there is a universal upper bound on the diameter of the commuting graphs of finite groups when the commuting graph is connected. In this paper we determine upper bounds on the diameter of the commuting graph for some classes of groups to rule them out as possible counterexamples to this conjecture. We also give an example of an infinite family of groups with tr…

The non-commuting, non-generating graph of a nilpotent group

For a nilpotent group $G$, let $Ξ(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $Ξ(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $Ξ^+(G)$ be the subgraph of $Ξ(G)$ induced by its non-isolated vertices. We show that if $Ξ(G)$ has an edge, then $Ξ^+(G)$ is connected with diameter $2$ or $3$, with $Ξ(G) = Ξ^+(G)$ in the diameter $3$ case. In the infinite case, our res…

On The Commuting Graph of Semidihedral Group

The commuting graph $Δ(G)$ of a finite non-abelian group $G$ is a simple graph with vertex set $G$ and two distinct vertices $x, y$ are adjacent if $xy = yx$. In this paper, among some properties of $Δ(G)$, we investigate $Δ(SD_{8n})$ the commuting graph of the semidihedral group $SD_{8n}$. In this connection, we discuss various graph invariants of $Δ(SD_{8n})$ including minimum degree, vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $Δ(SD_{8n})$.

有限单群的非交换、非生成图

The non-commuting, non-generating graph of a finite simple group

研究有限单群的非交换、非生成图的连通性和直径上界，证明其补图直径上限为4。

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。

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。

有限群的深度交换图

On the deep commuting graph of a finite group

本文研究有限群的深度交换图，证明其为完全图当且仅当群是循环群，分类了完美图情况，探究了欧拉图性等性质，还分类了与增强幂图重合的阿贝尔群并刻画支配顶点。

Commuting graphs on Coxeter groups, Dynkin diagrams and finite subgroups of $SL(2,\mathbb{C})$

For a group $H$ and a non empty subset $Γ\subseteq H$, the commuting graph $G=\mathcal{C}(H,Γ)$ is the graph with $Γ$ as the node set and where any $x,y \in Γ$ are joined by an edge if $x$ and $y$ commute in $H$. We prove that any simple graph can be obtained as a commuting graph of a Coxeter group, solving the realizability problem in this setup. In particular we can recover every Dynkin diagram of ADE type as a commuting graph. Thanks to the relation between the ADE classification and finite subgroups of $\SL(2,\C)$, we are able to rephrase results from the {\em McKay correspondence} in term…

A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2). We pose a new conjecture (Conjecture 3) that, together with the assumption $|Z(G)|\geq|Z(H)|$, implies Conjecture 2 and we prove it for groups in which all centralizers of non-central elements are abelian.

Can connected commuting graphs of finite groups have arbitrarily large diameter ?

We present a family of finite, non-abelian groups and propose that there are members of this family whose commuting graphs are connected and of arbitrarily large diameter. If true, this would disprove a conjecture of Iranmanesh and Jafarzadeh. While unable to prove our claim, we present a heuristic argument in favour of it. We also present the results of simulations which yielded explicit examples of groups whose commuting graphs have all possible diameters up to and including 10. Previously, no finite group whose commuting graph had diameter greater than 6 was known.

若干有限群的共轭超交换图与阶超交换图的谱

Spectrum of conjugacy and order super commuting graphs of some finite groups

该研究定义了等价关系下的超图结构，证明其为完全图的广义并，进而推导三类有限群的共轭与阶超交换图的邻接谱和拉普拉斯谱。

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。