Skip to main content Accessibility help
×
Home

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

  • PETER HEGARTY (a1) (a2) and DMITRII ZHELEZOV (a1) (a2)

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.

Copyright

References

Hide All
[1]Alon, N. and Spencer, J. (2000) The Probabilistic Method, second edition, Wiley.
[2]Bollobás, B. (1981) The diameter of random graphs. Trans. Amer. Math. Soc. 267 4152.
[3]Brauer, R. and Fowler, K. A. (1955) On groups of even order. Ann. of Math. (2) 62 565583.
[4]Giudici, M. and Parker, C. W. (2013) There is no upper bound for the diameter of the commuting graph of a finite group. J. Combin. Theory Ser. A 120 16001603.
[5]Guidici, M. and Pope, A. (2013) On bounding the diameter of the commuting graph of a group. J. Group Theory (1) 17 131149.
[6]Hegarty, P. and Zhelezov, D. Can commuting graphs of finite groups have arbitrarily large diameter? Preprint available at arXiv.org/abs/1204.5456.
[7]Iranmanesh, A. and Jafarzadeh, A. (2008) On the commuting graph associated with the symmetric and alternating groups. J. Algebra Appl. 7 129146.
[8]Klee, V. and Larman, D. (1981) Diameters of random graphs. Canad. J. Math. 33 618640.
[9]Morgan, G. L. and Parker, C. W. (2013) The diameter of the commuting graph of a finite group with trivial centre. J. Algebra 393 4159.
[10]Neumann, B. H. (1976) A problem of Paul Erdős on groups. J. Austral. Math. Soc. Ser. A 21 467472.
[11]Riordan, O. and Wormald, N. (2010) The diameter of sparse random graphs. Combin. Probab. Comput. 19 835926.
[12]Segev, Y. and Seitz, G. M. (2002) Anisotropic groups of type An and the commuting graph of finite simple groups. Pacific J. Math. 202 125225.

Keywords

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

  • PETER HEGARTY (a1) (a2) and DMITRII ZHELEZOV (a1) (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed