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Minimal cut-sets in the power graph of certain finite non-cyclic groups
TL;DR: The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other.
Abstract: The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. We study (minimal) cut-sets of the power graph of a (finite) non-cyclic (nilpotent) group which are associated with its maximal cyclic subgroups. Let $G$ be a finite non-cyclic nilpotent group whose order is divisible by at least two distinct primes. If $G$ has a Sylow subgroup which is neither cyclic nor a generalized quaternion $2$-group and all other Sylow subgroups of $G$ are cyclic, then under some conditions we prove that there is only one minimum cut-set of the power graph of $G$. We apply this result to find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes.
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TL;DR: In this article , the Laplacian spectral radius and domination number of the proper enhanced power graphs of nilpotent groups are derived. But they do not explicitly calculate the domination number.
Abstract: Abstract For a group šŗ, the enhanced power graph of šŗ is a graph with vertex set šŗ in which two distinct vertices x , y x,y are adjacent if and only if there exists an element š¤ in šŗ such that both š„ and š¦ are powers of š¤. The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set G ā S G\setminus S , where š is the set of dominating vertices of the enhanced power graph. In this paper, we at first classify all nilpotent groups šŗ such that the proper enhanced power graphs are connected and calculate their diameter. We also explicitly calculate the domination number of the proper enhanced power graphs of finite nilpotent groups. Finally, we determine the multiplicity of the Laplacian spectral radius of the enhanced power graphs of nilpotent groups.
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Posted Contentā¢
TL;DR: In this article, it was shown that for any integer n ā„ 3, under certain conditions involving the prime divisors of the cyclic group of order n, one can identify at most n-1 vertices such that the degree of at least one of these vertices is equal to that of the prime degree of the other vertices.
Abstract: The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum degree $\delta(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\in\{1,2\}$, see [18]. For $r\geq 3$, under certain conditions involving the prime divisors of $n$, we identify at most $r-1$ vertices such that $\delta(\mathcal{P}(C_n))$ is equal to the degree of at least one of these vertices. If $r=3$ or if $n$ is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of $n$.
2Ā citations
TL;DR: In this article , the minimum degree and the vertex connectivity of the enhanced power graph of a finite group were determined and the upper and lower bounds of the Wiener index of the vertices were obtained.
Abstract: The enhanced power graph of a finite group $$G$$ , denoted by $$P_E(G)$$ , is a simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if $$x, y \in \langle z \rangle$$ for some $$z \in G$$ . In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of $$P_E(G)$$ are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain the upper and lower bounds of the Wiener index of $$P_E(G)$$ , where G is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.
2Ā citations
04 Jul 2023
TL;DR: In this paper , the authors extend the results of Bera et al. to arbitrary finite groups and show that the proper power graphs of dihedral groups are the complement of line graphs.
Abstract: S. Bera (Line graph characterization of power graphs of finite nilpotent groups, \textit{Communication in Algebra}, 50(11), 4652-4668, 2022) characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian $2$-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose power graphs, proper power graphs, enhanced power graphs and proper enhanced power graphs are the complement of line graphs, respectively.
1Ā citations
15 Dec 2022
TL;DR: For non-nilpotent groups, the dominating vertices of a graph D (G) of a symmetric group S n (or alternating group A n ) are a star graph, dominatable, split graph, chordal graph, threshold graph and cograph as mentioned in this paper .
Abstract: . The power graph of a ļ¬nite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The enhanced power graph of a ļ¬nite group G is a simple undirected graph whose vertex set is the group G and two vertices a and b are adjacent if there exists c ā G such that both a and b are powers of c . In this paper, we study the diļ¬erence graph D ( G ) of a ļ¬nite group G which is the diļ¬erence of the enhanced power graph and the power graph of G with all isolated vertices removed. For an arbitrary ļ¬nite group G , ļ¬rst we study the dominating vertices of D ( G ). We give a necessary and suļ¬cient condition on G such that D ( G ) is a star graph, dominatable, split graph, chordal graph, threshold graph and cograph, respectively. Using this, we classify all the nilpotent groups G for which D ( G ) is a star graph, dominatable, split graph, chordal graph, threshold graph and cograph, respectively. Moreover, we characterise all the nilpotent groups whose diļ¬erence graphs are bipartite, planar, outerplanar and Eulerian, respectively. In order to study, the diļ¬erence graph for non-nilpotent groups, in this paper, we classify all the values of n for which the diļ¬erence graph of the symmetric group S n (or alternating group A n ) is cograph and chordal, respectively.
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References
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Bookā¢
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14 Aug 2008
TL;DR: This book provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal, and is suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science.
Abstract: Graph theory is a flourishing discipline containing a body of beautiful and powerful theorems of wide applicability. Its explosive growth in recent years is mainly due to its role as an essential structure underpinning modern applied mathematics computer science, combinatorial optimization, and operations research in particular but also to its increasing application in the more applied sciences. The versatility of graphs makes them indispensable tools in the design and analysis of communication networks, for instance. The primary aim of this book is to present a coherent introduction to the subject, suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science. It provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal. Commonly used proof techniques are described and illustrated, and a wealth of exercises - of varying levels of difficulty - are provided to help the reader master the techniques and reinforce their grasp of the material. A second objective is to serve as an introduction to research in graph theory. To this end, sections on more advanced topics are included, and a number of interesting and challenging open problems are highlighted and discussed in some detail. Despite this more advanced material, the book has been organized in such a way that an introductory course on graph theory can be based on the first few sections of selected chapters. Visit the graph theory book blog at: http://blogs.springer.com/bondyandmurty/.
3,296Ā citations
"Minimal cut-sets in the power graph..." refers methods in this paper
...We refer to [2] for the unexplained terminology from graph theory used in this paper....
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TL;DR: In this article, it was shown that the multiplicative semigroup ā¤n and its subgroup Un is complete if and only if n = 1,2,4,p or 2p, where p is a Fermat prime.
Abstract: The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,bāS are adjacent if and only if aā b and am=b or bm=a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or pm. Particular attention is given to the multiplicative semigroup ā¤n and its subgroup Un, where G(Un) is a major component of G(ā¤n). It is proved that G(Un) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(Un) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(Un) has no Hamiltonian cycle.
266Ā citations
"Minimal cut-sets in the power graph..." refers background in this paper
...Then the undirected power graph of a semigroup, in particular, of a group was defined in [3]....
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TL;DR: In this paper, a complete description of all commutative semigroups satisfying three other combinatorial properties defined in terms of directed graphs is given, by a graph we mean a directed graph without loops or multiple edges.
198Ā citations
13 Nov 2013
TL;DR: This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature.
Abstract: This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.
187Ā citations
"Minimal cut-sets in the power graph..." refers background in this paper
...More on these graphs can be found in the survey paper [1] and the references therein....
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01 Jan 2000
161Ā citations
"Minimal cut-sets in the power graph..." refers background in this paper
...The notion of directed power graph of a group was introduced in [10], which was further extended to semigroups in [11, 12]....
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