Undirected power graphs of semigroups
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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 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 ℤ n and its subgroup U n , where G(U n ) is a major component of G(ℤ 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.
KeywordsSemigroup Group Divisibility graph Power graph Connected graph Complete graph Fermat prime Planar graph Eulerian graph Hamiltonian graph
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