Characterizing finite nilpotent groups associated with a graph theoretic equality
Ramesh Prasad Panda,Kamal Lochan Patra,Kamal Lochan Patra,Binod Kumar Sahoo,Binod Kumar Sahoo +4 more
TLDR
The power graph of a group is defined as the simple graph whose vertices are the group elements and two nodes are adjacent whenever one of them is a positive power of the other.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.read more
Citations
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On the minimum degree of power graphs of finite nilpotent groups
TL;DR: In this article , the minimum degree δ(P(G)) of a power graph P(G) of a non-cyclic nilpotent group G was studied under some conditions involving the prime divisors of |G| and the Sylow subgroups of G.
References
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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.
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Directed Graphs and Combinatorial Properties of Semigroups
Andrei V. Kelarev,Stephen Quinn +1 more
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.
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The power graph of a finite group, II
TL;DR: In this article, it was shown that the undirected power graph determines the directed power graph up to isomorphism, and that two finite groups which have isomorphic undirectED power graphs have the same number of elements of each order.
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Some applications of graph theory to finite groups
TL;DR: Results on vertex coloring and the vertex independence number of a finite graph are used to prove the theorem that G cannot be covered by the union of fewer than [|G|^1^3] abelian subgroups.