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Undirected power graphs of semigroups. (English) Zbl 1207.05075

Summary: The undirected power graph \(\mathcal G(S)\) of a semigroup \(S\) is an undirected graph whose vertex set is \(S\) and two vertices \(a,b\in S\) are adjacent if and only if \(a\neq b\) 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 \(\mathcal G(S)\) is connected or complete. As a consequence we prove that \(\mathcal G(G)\) is connected for any finite group \(G\) and \(\mathcal G(G)\) 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 \(\mathbb Z_{n }\) and its subgroup \(U _{n }\), where \(\mathcal G(U _{n })\) is a major component of \(\mathcal G(\mathbb Z_{n })\). It is proved that \(\mathcal G(U _{n })\) 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 \(\mathcal G(G)\) 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, \(\mathcal G(G)\) is Hamiltonian and list some values of \(n\) for which \(\mathcal G(U _{n })\) has no Hamiltonian cycle.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20M10 General structure theory for semigroups
05C40 Connectivity
05C45 Eulerian and Hamiltonian graphs
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