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Original languageEnglish
Number of pages24
Pages (from-to)1-24
JournalInternational Journal of Algebra and Computation
Journal publication date26 Sep 2017
Early online date26 Sep 2017
DOIs
StateE-pub ahead of print - 26 Sep 2017

Abstract

The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. We refine and prove Rhodes's conjecture on the structure of the maximal groups in the flow semigroup for finite, antisymmetric, strongly connected graphs. Building on this result, we investigate and fully describe the structure and actions of the maximal subgroups of the flow semigroup acting on all but k points for all finite digraphs and graphs for all k >=1. A linear algorithm is presented to determine these so-called 'defect k groups' for any finite (di)graph. Finally, we prove that the complexity of the flow semigroup of a 2-vertex connected (and strongly connected di)graph with n vertices is n- 2, completely confirming Rhodes's conjecture for such (di)graphs.

Notes

Preprint of an article first published online in International Journal of Algebra and Computation, September 2017, doi: https://doi.org/10.1142/S0218196717500412. © 2017 Copyright World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijac. Accepted Manuscript version is under embargo. Embargo end date: 26 September 2018.

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