On the Connectivity of Proper Power Graphs of Finite Groups

Doostabadi, Alireza; Ghouchan, Mohammad Farrokhi Derakhshandeh

doi:10.1080/00927872.2014.945093

Cited by 34 publications

(40 citation statements)

References 22 publications

(21 reference statements)

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“…In the last section of the paper we consider G = S n , finding c 0 (S n ), c 0 (T (S n )) and c 0 (O(S n )). In particular we find, with a different approach, the values of c 0 (S n ) in [9,Theorem 4.2]. Throughout the paper we denote by P the set of prime numbers and put P + 1 = {x ∈ N : x = p + 1 for some p ∈ P }.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In a forthcoming paper [3] we will treat the alternating group A n computing c 0 (A n ), c 0 (T (A n )) and c 0 (O(A n )). We will also correct some mistakes about c 0 (A n ) found in [9]. We believe that our algorithmic method [2, Theorem B] may help, more generally, to obtain c 0 (G) where G is simple and almost simple.…”

Section: In All the Above Cases The Main Component Is Never Completementioning

confidence: 91%

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Quotient graphs for power graphs

Bubboloni

Iranmanesh

Shaker

2017

Rend. Sem. Mat. Univ. Padova

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In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph P 0 (G) of a finite group G, finding a formula for the number c(P 0 (G)) of its components which is particularly illuminative when G ≤ Sn is a fusion controlled permutation group. We make use of the proper quotient power graph P 0 (G), the proper order graph O 0 (G) and the proper type graph T 0 (G). We show that all those graphs are quotient of P 0 (G) and demonstrate a strong link between them dealing with G = Sn. We find simultaneously c(P 0 (Sn)) as well as the number of components of P 0 (Sn), O 0 (Sn) and T 0 (Sn). MSC(2010): Primary: 05C25; Secondary: 20B30.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: In All the Above Cases The Main Component Is Never Completementioning

confidence: 91%

Quotient graphs for power graphs

Bubboloni

Iranmanesh

Shaker

2017

Rend. Sem. Mat. Univ. Padova

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“…In [30], the authors asked the structure of all non-abelian simple groups with 2-connected power graphs. Doostabadi et al [16] computed the number of component in the proper power graphs of the alternating groups which shows that the power graph of these simple groups can be 2−connected. This result recently corrected by Bubboloni et al [4] which again shows the existence of simple group with 2−connected power graph.…”

Section: Power Graph Of Finite Groups: a Literature Reviewmentioning

confidence: 99%

Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Hamzeh

Reza

2017

Filomat

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Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x, y ∈ G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

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“…One of the major graph representation amongst them is the power graphs of finite groups. We found several papers in this context [2,5,6,9,10,13,14,23,26,27,29]. Example 1.…”

Section: Introductionmentioning

confidence: 99%

Forbidden Subgraphs of Power Graphs

Manna

Cameron

Mehatari

2021

Electron. J. Combin.

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The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

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On the Connectivity of Proper Power Graphs of Finite Groups

Cited by 34 publications

References 22 publications

Quotient graphs for power graphs

Quotient graphs for power graphs

Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Forbidden Subgraphs of Power Graphs

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