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REALIZABILITY PROBLEM FOR COMMUTING GRAPHS

Part of:Representation theory of groupsGraph theorySemigroups

Published online by Cambridge University Press:  13 May 2016

MICHAEL GIUDICI and
BOJAN KUZMA
MICHAEL GIUDICI
Affiliation:
Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia email michael.giudici@uwa.edu.au
BOJAN KUZMA*
Affiliation:
University of Primorska, Glagoljaška 8, SI-6000 Koper, Slovenia email bojan.kuzma@famnit.upr.siIMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia
*
✉ bojan.kuzma@famnit.upr.si
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Abstract

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We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain complete classifications of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.

Keywords

finite groupsfinite semigroupscommuting graphclassification problem

MSC classification

Primary:05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary:20M99: None of the above, but in this section20D99: None of the above, but in this section

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 3 , December 2016, pp. 335 - 355
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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