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REALIZABILITY PROBLEM FOR COMMUTING GRAPHS

Part of: Semigroups Graph theory Representation theory of groups

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.si IMFM, 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 section 20D99: None of the above, but in this section
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. 

References

Afkhami, M., Farrokhi, M. and Khashyarmanesh, K., ‘Planar, toroidal, and projective commuting and noncommuting graphs’, Comm. Algebra 43(7) (2015), 2964–2970.CrossRefGoogle Scholar
Ambrozie, C., Bračič, J., Kuzma, B. and Müller, V., ‘The commuting graph of bounded linear operators on a Hilbert space’, J. Funct. Anal. 264(4) (2013), 1068–1087.CrossRefGoogle Scholar
Arad, Z. and Herfort, W., ‘Classification of finite groups with a CC-subgroup’, Comm. Algebra 32(6) (2004), 2087–2098.CrossRefGoogle Scholar
Araújo, J., Kinyon, M. and Konieczny, J., ‘Minimal paths in the commuting graphs of semigroups’, European J. Combin. 32 (2011), 178–197.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I. The user language’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
Brauer, R. and Fowler, K. A., ‘On groups of even order’, Ann. of Math. (2) 62 (1955), 565–583.CrossRefGoogle Scholar
Das, A. K. and Nongsiang, D., On the genus of the commuting graphs of finite non-abelian groups. arXiv:1311.6342.Google Scholar
Feit, W. and Thompson, J. G., ‘Finite groups which contain a self-centralizing subgroup of order 3’, Nagoya Math. J. 21 (1962), 185–197.CrossRefGoogle Scholar
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.2; 2013, (http://www.gap-system.org).Google Scholar
Giudici, M. and Parker, C., ‘There is no upper bound for the diameter of the commuting graph of a finite group’, J. Combin. Theory Ser. A 120 (2013), 1600–1603.CrossRefGoogle Scholar
Gorenstein, D., Finite Groups (AMS Chelsea Publishing, 1968).Google Scholar
Itô, N., ‘On finite groups with given conjugate types. I’, Nagoya Math. J. 6 (1953), 17–28.CrossRefGoogle Scholar
Mazurov, V. D., ‘On groups that contain a self-centralizing subgroup of order 3’, Algebra Logika 42(1) (2003), 51–64; translation in Algebra Logic 42(1) (2003), 29–36).CrossRefGoogle Scholar
Morgan, G. L. and Parker, C. W., ‘The diameter of the commuting graph of a finite group with trivial centre’, J. Algebra 393 (2013), 41–59.CrossRefGoogle Scholar
Pisanski, T., ‘Universal commutator graphs’, Discrete Math. 78(1–2) (1989), 155–156.CrossRefGoogle Scholar
Solomon, R. and Woldar, A., ‘Simple groups are characterized by their non-commuting graphs’, J. Group Theory 16 (2013), 793–824.CrossRefGoogle Scholar
Vahidi, J. and Talebi, A. A., ‘The commuting graphs on groups D 2n and Q n ’, J. Math. Comput. Sci. 1 (2010), 123–127.CrossRefGoogle Scholar
Wong, W. J., ‘On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2’, J. Aust. Math. Soc. 4 (1964), 90–112.CrossRefGoogle Scholar
Wong, W. J., ‘Finite groups with a self-centralizing subgroup of order 4’, J. Aust. Math. Soc. 7 (1967), 570–576.CrossRefGoogle Scholar