Minimal cut-sets in the power graphs of certain finite non-cyclic groups
TL;DR: In this article, minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups are studied.
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...
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TL;DR: This paper aims to demonstrate the efforts towards in-situ applicability of EMMR-II, which aims to provide real-time information about the response of the immune system to EMTs.
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TL;DR: In this article, the minimum degree, indescrete 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.
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...
22 citations
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 𝐺, the enhanced power graph of 𝐺 is a graph with vertex set 𝐺 in which two distinct vertices x , y x,y are adjacent if and only if there exists an element 𝑤 in 𝐺 such that both 𝑥 and 𝑦 are powers of 𝑤. The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set G ∖ S G\setminus S , where 𝑆 is the set of dominating vertices of the enhanced power graph. In this paper, we at first classify all nilpotent groups 𝐺 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.
4 citations
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TL;DR: In this article, it was shown that for any integer n ≥ 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 $n$.
2 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.
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Book•
19 Oct 2011
TL;DR: A detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative is given in this article, where the reader is provided with only a basic knowledge of modern algebra.
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.
3,406 citations
Book•
[...]
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/.
3,296 citations
TL;DR: In this article, it was shown that the multiplicative semigroup ℤn 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,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.
266 citations
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.
198 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.
187 citations