Noncommuting graphs of matrices over semirings |
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Authors: | David Dol?an |
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Institution: | a Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia b Faculty of Computer and Information Science, Tr?aška 25, SI-1000 Ljubljana, Slovenia |
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Abstract: | We study diameters and girths of noncommuting graphs of semirings. For a noncommutative semiring that is either multiplicatively or additively cancellative, we find the diameter and the girth of its noncommuting graph and prove that it is Hamiltonian. Moreover, we find diameters and girths of noncommuting graphs of all nilpotent matrices over a semiring, all invertible matrices over a semiring, all noninvertible matrices over a semiring, and the full matrix semiring. In nearly all cases we prove that diameters are less than or equal to 2 and girths are less than or equal to 3, except in the case of 2×2 nilpotent matrices. |
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Keywords: | 16Y60 15A27 05C50 |
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