Chakrabarty, Ivy; Ghosh, Shamik; Sen, M. K. Undirected power graphs of semigroups. (English) Zbl 1207.05075 Semigroup Forum 78, No. 3, 410-426 (2009). 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. Cited in 1 ReviewCited in 153 Documents MSC: 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20M10 General structure theory for semigroups 05C40 Connectivity 05C45 Eulerian and Hamiltonian graphs Keywords:semigroup; group; divisibility graph; power graph; connected graph; complete graph; Fermat prime; planar graph; Eulerian graph; Hamiltonian graph PDFBibTeX XMLCite \textit{I. Chakrabarty} et al., Semigroup Forum 78, No. 3, 410--426 (2009; Zbl 1207.05075) Full Text: DOI References: [1] Bosak, J.: The graphs of semigroups. In: Theory of Graphs and Application, pp. 119–125. Academic Press, New York (1964) · Zbl 0161.20901 [2] Budden, F.: Cayley graphs for some well-known groups. Math. Gaz. 69, 271–278 (1985) · Zbl 0584.05040 [3] Gallian, J.A.: Contemporary Abstract Algebra. Narosa Publishing House, London (1999) · Zbl 0972.00001 [4] Howie, J.M.: An Introduction to Semigroup Theory. Academic Press, London (1976) · Zbl 0355.20056 [5] Hungerford, T.W.: Algebra. Springer, Berlin (1974) [6] Jones, G.A., Jones, J.M.: Elementary Number Theory. Springer, New Delhi (2005) · JFM 42.0088.02 [7] Kelarev, A.V., Quinn, S.J.: Directed graph and combinatorial properties of semigroups. J. Algebra 251, 16–26 (2002) · Zbl 1005.20043 [8] Keller, W.: Prime factors of Fermat numbers. http://www.prothsearch.net/fermat.html · Zbl 0813.11069 [9] West, D.B.: Introduction to Graph Theory. Prentice-Hall, New Delhi (2003) [10] Witte, D., Letzter, G., Gallian, J.A.: On Hamiltonian circuits in Cartesian products of Cayley digraphs. Discrete Math. 43, 297–307 (1983) · Zbl 0503.05041 [11] Zelinka, B.: Intersection graphs of finite Abelian groups. Czech. Math. J. 25(100), 171–174 (1975) · Zbl 0311.05119 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.