Journal ArticleDOI
Undirected power graphs of semigroups
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TLDR
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.read more
Citations
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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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.
Journal ArticleDOI
The structure and metric dimension of the power graph of a finite group
TL;DR: This work first shows that P G has a transitive orientation, so it is a perfect graph and its core is a complete graph, and uses the poset on all cyclic subgroups of G (under usual inclusion) to characterize the structure of P G .
Journal ArticleDOI
Certain properties of the power graph associated with a finite group
TL;DR: The power graph of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other as mentioned in this paper.
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.
Book
Contemporary Abstract Algebra
TL;DR: Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists.
Journal ArticleDOI
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.
Book
Elementary number theory
Gareth Jones,J. Mary Jones +1 more
TL;DR: An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples, is given in this article, where elementary ideas about groups and rings are used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography.