Open accessJournal Article

# Minimal cut-sets in the power graphs of certain finite non-cyclic groups

04 Mar 2021-Communications in Algebra (Informa UK Limited)-Vol. 49, Iss: 3, pp 1195-1211
Abstract: We study minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups. As a result, we find the vertex connectivity of the power...

Topics: Cyclic group (62%), Nilpotent group (61%)
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Journal Article
Abstract: The enhanced power graph Pe(G) of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, ind...

Topics: Vertex (geometry) (55%), Finite group (55%)

10 Citations

Open accessJournal Article
Abstract: Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated t...

7 Citations

Open accessPosted Content
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 $n$.

Topics: Cyclic group (54%), Vertex (geometry) (52%), Finite group (52%)

2 Citations

Open accessPosted Content
Abstract: The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. We characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree.

Topics: , Group (mathematics) (56%), Nilpotent (55%)

1 Citations

Open accessJournal Article
Abstract: The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. In this paper, we characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree.

Topics: , Group (mathematics) (56%), Nilpotent (55%)

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18 results found

Open accessBook
19 Oct 2011-
Abstract: This is a detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the different branches of group theory and its principal accomplishments.

Topics: Abstract algebra (57.99%), Group theory (56.99%), Ideal theory (56.99%) ... show more

3,224 Citations

Open accessBook
14 Aug 2008-
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/.

Topics: Graph theory (56.99%),

3,061 Citations

Journal Article
Ivy Chakrabarty1, Shamik Ghosh1, M. K. Sen2Institutions (2)
27 Jan 2009-Semigroup Forum
Abstract: The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠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 ℤn and its subgroup Un, where G(Un) is a major component of G(ℤn). 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.

Topics: Bound graph (63%), Graph power (62%), Graph toughness (59%) ... show more

210 Citations

Open accessJournal Article
Andrei V. Kelarev1, Stephen Quinn1Institutions (1)
01 May 2002-Journal of Algebra
Abstract: Combinatorial properties of words in groups and semigroups have been investigated by many authors. For recent results on this topic we refer to the book [4] and survey [11] (see also [10]). In particular, Justin [9] described all repetitive commutative semigroups (a section of [4] and a chapter of [12] are devoted to this combinatorial property, see also [3, 4]). In this paper we obtain complete descriptions of all commutative semigroups satisfying three other combinatorial properties defined in terms of directed graphs. Throughout, by a graph we mean a directed graph without loops or multiple edges. The power graph Pow S of a semigroup S has all elements of S as vertices and has edges u v for all u v ∈ G such that u = v and v is a power of u. The divisibility graph Div S has vertex set S and edges u v , where u = v and u divides v; i.e., u belongs to the ideal generated by v. The annihilator graph Ann S of a semigroup S with 0 has vertex set S and the set of edges u v ∈ S × S uv = 0 u = v .

Topics: Semigroup (56.99%), Directed graph (56.99%), Multiple edges (55%) ... show more

180 Citations

Open accessJournal Article
13 Nov 2013-
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.

148 Citations

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###### Related Papers (5)
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