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Undirected power graphs of semigroups. (English) Zbl 1207.05075
Summary: The undirected power graph $𝒢\left(S\right)$ 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 $a\ne b$ 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 $𝒢\left(S\right)$ is connected or complete. As a consequence we prove that $𝒢\left(G\right)$ is connected for any finite group $G$ and $𝒢\left(G\right)$ 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 ${ℤ}_{n}$ and its subgroup ${U}_{n}$, where $𝒢\left({U}_{n}\right)$ is a major component of $𝒢\left({ℤ}_{n}\right)$. It is proved that $𝒢\left({U}_{n}\right)$ 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 $𝒢\left(G\right)$ for a finite group $G$ and apply this result to determine the values of $n$ for which $G\left({U}_{n}\right)$ is planar. Finally we show that for any cyclic group of order greater than or equal to 3, $𝒢\left(G\right)$ is Hamiltonian and list some values of $n$ for which $𝒢\left({U}_{n}\right)$ has no Hamiltonian cycle.
##### MSC:
 05C25 Graphs and abstract algebra 20M10 General structure theory of semigroups 05C40 Connectivity 05C45 Eulerian and Hamiltonian graphs
##### References:
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