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A study of enhanced power graphs of finite groups
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In this paper, the enhanced power graph is defined as the graph with vertex set G such that two vertices x and y are adjacent if they are contained in the same cyclic subgroup.Abstract:
The enhanced power graph 𝒢e(G) of a group G is the graph with vertex set G such that two vertices x and y are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups...read more
Citations
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On the enhanced power graph of a finite group
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.
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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.
Journal ArticleDOI
On enhanced power graphs of certain groups
Sandeep Dalal,Jitender Kumar +1 more
TL;DR: This paper obtains various graph invariants viz. independence number, minimum degree and matching number of [Formula] is the dicyclic group or a class of groups of order [formula: see text].
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Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph
TL;DR: The Gruenberg-Kegel graph has been the subject of much recent interest; see as mentioned in this paper for a survey of some of this material, relating to groups with the same Gruenberg Kegel graphs.
References
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Journal ArticleDOI
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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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.