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Undirected power graphs of semigroups. (English) Zbl 1207.05075
Summary: The undirected power graph 𝒢(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,bS are adjacent if and only if ab and a m =b or b m =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 p m . Particular attention is given to the multiplicative semigroup n and its subgroup U n , where 𝒢(U n ) is a major component of 𝒢( n ). It is proved that 𝒢(U n ) 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(U n ) 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(U n ) has no Hamiltonian cycle.
MSC:
05C25Graphs and abstract algebra
20M10General structure theory of semigroups
05C40Connectivity
05C45Eulerian and Hamiltonian graphs
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