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On the minimum degree of the power graph of a finite cyclic group
Ramesh Prasad Panda,Ramesh Prasad Panda,Kamal Lochan Patra,Kamal Lochan Patra,Binod Kumar Sahoo,Binod Kumar Sahoo +5 more
TLDR
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 $n$.read more
Citations
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Recent developments on the power graph of finite groups - a survey
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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Co-Prime Order graph of a finite abelian Group and Dihedral Group
Amit Sehgal,Manjeet,Dalip Singh +2 more
TL;DR: In this paper, the degree of a vertex in the Co-Prime Order graph of a finite abelian group or a dihedral group was investigated. And the Laplacian spectrum of the co-prime order graph was also investigated.
References
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Book
Introduction to Graph Theory
TL;DR: In this article, the authors introduce the concept of graph coloring and propose a graph coloring algorithm based on the Eulers formula for k-chromatic graphs, which can be seen as a special case of the graph coloring problem.
Journal ArticleDOI
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.
Journal ArticleDOI
The power graph of a finite group
Peter J. Cameron,Shamik Ghosh +1 more
TL;DR: It is conjectured that two finite groups with isomorphic power graphs have the same number of elements of each order, and that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.
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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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Power graphs: A survey
TL;DR: This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature.
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