Semigroup Forum

, Volume 78, Issue 3, pp 410–426

Undirected power graphs of semigroups

Research Article

DOI: 10.1007/s00233-008-9132-y

Cite this article as:
Chakrabarty, I., Ghosh, S. & Sen, M.K. Semigroup Forum (2009) 78: 410. doi:10.1007/s00233-008-9132-y


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 am=b or bm=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 pm. Particular attention is given to the multiplicative semigroup ℤn and its subgroup Un, where G(Un) is a major component of G(ℤn). It is proved that G(Un) 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(Un) 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(Un) has no Hamiltonian cycle.


SemigroupGroupDivisibility graphPower graphConnected graphComplete graphFermat primePlanar graphEulerian graphHamiltonian graph

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsJadavpur UniversityKolkataIndia
  2. 2.Department of Pure MathematicsUniversity of CalcuttaKolkataIndia