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Register a new userThe Complexity of Power Graphs Associated With Finite Groups
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The power graph $mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity of a clique--replaced graph and study some applications. Second, we derive some explicit formulas concerning the complexity $kappa(mathcal{P}(G))$ for various groups $G$ such as the cyclic group of order $n$, the simple groups $L_2(q)$, the extra--special $p$--groups of order $p^3$, the Frobenius groups, etc.
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In this paper we have investigated some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it has been shown that the simple group $L_2(7)$ can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, we have obtained an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.
Given a group $G$, we define the power graph $mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $langle xranglesubseteq langle yrangle$ or $langle yranglesubseteq langle xrangle$. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. In the present paper, among other results, we will find the number of spanning trees of the power graph associated with specific finite groups. We also determine, up to isomorphism, the structure of a finite group $G$ whose power graph has exactly $n$ spanning trees, for $n<5^3$. Finally, we show that the alternating group $mathbb{A}_5$ is uniquely determined by tree-number of its power graph among all finite simple groups.
Let $G$ be a finite group and $sigma$ a partition of the set of all? primes $Bbb{P}$, that is, $sigma ={sigma_i mid iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_i$ and $sigma_icap sigma_j= emptyset $ for all $i e j$. If $n$ is an integer, we write $sigma(n)={sigma_i mid sigma_{i}cap pi (n) e emptyset }$ and $sigma (G)=sigma (|G|)$. We call a graph $Gamma$ with the set of all vertices $V(Gamma)=sigma (G)$ ($G e 1$) a $sigma$-arithmetic graph of $G$, and we associate with $G e 1$ the following three directed $sigma$-arithmetic graphs: (1) the $sigma$-Hawkes graph $Gamma_{Hsigma }(G)$ of $G$ is a $sigma$-arithmetic graph of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{Hsigma }(G))$ if $sigma_jin sigma (G/F_{{sigma_i}}(G))$; (2) the $sigma$-Hall graph $Gamma_{sigma Hal}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{sigma Hal}(G))$ if for some Hall $sigma_i$-subgroup $H$ of $G$ we have $sigma_jin sigma (N_{G}(H)/HC_{G}(H))$; (3) the $sigma$-Vasilev-Murashko graph $Gamma_{{mathfrak{N}_sigma }}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{{mathfrak{N}_sigma}}(G))$ if for some ${mathfrak{N}_{sigma }}$-critical subgroup $H$ of $G$ we have $sigma_i in sigma (H)$ and $sigma_jin sigma (H/F_{{sigma_i}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.
In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of order $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any $nin mathbb{N}$, solvable groups of $n^text{th}$-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or $A_5$ and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
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