Journal ArticleDOI
On the enhanced power graph of a finite group
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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...read more
Citations
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On the Connectivity of Enhanced Power Graphs of Finite Groups
TL;DR: In this article, the vertex connectivity of enhanced power graphs of finite groups was studied and an upper bound for vertex connectivity was derived for any general abelian group G. The vertex connectivity for the enhanced power graph of any general group G is given.
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On the connectivity of enhanced power graph of finite group
TL;DR: An upper bound of vertex connectivity is derived for the enhanced powergraph of any general abelian group G, such that the proper enhanced power graph is connected.
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On the Commuting Graph of Semidihedral Group
TL;DR: In this article, the edge connectivity and minimum degree of the commuting graph of a semidihedral group were investigated and the necessary and sufficient condition on the group G such that the interior and center of the graph are equal.
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Strong metric dimensions for power graphs of finite groups
TL;DR: In this article, the authors characterized the strong metric dimension of the order supergraph, the enhanced power graph and the reduced power graph of a finite group, and showed that for any finite group there exists a strong metric of order supergraphs with vertices whose vertices are adjacent if they generate cyclic subgroups.
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A survey on enhanced power graphs of finite groups
TL;DR: In this paper , the authors survey known results on enhanced power graphs of finite groups and open problems, questions and suggestions for future work are also included, as well as some open problems.
References
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Book
A Course in the Theory of Groups
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.
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The Strong Perfect Graph Theorem
TL;DR: The strong perfect graph conjecture as discussed by the authors states that a graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced sub graph of G is an odd cycle of length at least five or the complement of one.
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Elementary number theory
TL;DR: The book as mentioned in this paper is designed for a first course in number theory with minimal prerequisites and is designed to stimulate curiosity about numbers and their properties, including almost a thousand imaginative exercises and problems.
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The strong perfect graph theorem
TL;DR: In this article, the authors prove that every Berge graph either falls into one of a few basic classes, or it has a kind of separation that cannot occur in a minimal imperfect graph.
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Undirected power graphs of semigroups
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.