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On the Diameters of Commuting Graphs Arising from Random Skew-Symmetric Matrices

Published online by Cambridge University Press:  04 February 2014

PETER HEGARTY  and
DMITRII ZHELEZOV
PETER HEGARTY
Affiliation:
Mathematical Sciences, Chalmers, 41296 Gothenburg, Sweden (e-mail: hegarty@chalmers.se, zhelezov@chalmers.se) Mathematical Sciences, University of Gothenburg, 41296 Gothenburg, Sweden
DMITRII ZHELEZOV
Affiliation:
Mathematical Sciences, Chalmers, 41296 Gothenburg, Sweden (e-mail: hegarty@chalmers.se, zhelezov@chalmers.se) Mathematical Sciences, University of Gothenburg, 41296 Gothenburg, Sweden
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Abstract

We present a two-parameter family $(G_{m,k})_{m, k \in \mathbb{N}_{\geq 2}}$, of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Rényi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

Keywords

Primary 20P05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 3 , May 2014 , pp. 449 - 459
Copyright
Copyright © Cambridge University Press 2014 

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