Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year

IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2017: 0.32

Open Access
See all formats and pricing
More options …
Volume 13, Issue 1


Volume 13 (2015)

Two bounds on the noncommuting graph

Stefano Nardulli
  • Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia, Bloco C, Cidade Universitária, Ilha do Fundão, Caixa Postal 68530, 21941-909, Rio de Janeiro, Brasil
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Francesco G. Russo
  • Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch 7701, Cape Town, South Africa
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-04-15 | DOI: https://doi.org/10.1515/math-2015-0027


Erdős introduced the noncommuting graph in order to study the number of commuting elements in a finite group. Despite the use of combinatorial ideas, his methods involved several techniques of classical analysis. The interest for this graph has become relevant during the last years for various reasons. Here we deal with a numerical aspect, showing for the first time an isoperimetric inequality and an analytic condition in terms of Sobolev inequalities. This last result holds in the more general context of weighted locally finite graphs.

Keywords: Noncommuting graph; Sobolev–Poincaré inequality; Laplacian operator; Isoperimetric inequality


  • [1] Abdollahi A., Akbari S., Maimani H.R., Non-commuting graph of a group, J. Algebra, 2006, 298, 468–492 Google Scholar

  • [2] Ambrosio L., Gigli N., Mondino A., Rajala T., Riemannian Ricci curvature lower bounds in metric measure spaces with σ–finite measure, Trans. Amer. Math. Soc. (in press), preprint available at http://arxiv.org/pdf/1207.4924v2.pdf Google Scholar

  • [3] Ambrosio L., Mondino A., Savaré G., On the Bakry–Émery condition, the gradient estimates and the Local–to–Global property of RCD*(k, n) metric measure spaces, J. Geom. Anal. (in press), preprint available at http://arxiv.org/pdf/1309.4664v1.pdf Google Scholar

  • [4] Aubin T., Nonlinear analysis on manifolds: Monge–Ampére equations, Grundlehren der Mathematischen Wissenschaften, 252, Springer, Berlin, 1982 Google Scholar

  • [5] Bakry D., Coulhon T., Ledoux M., Saloff–Coste L., Sobolev inequalities in disguise, Indiana Univ. Math. J., 1995, 44, 1033–1074 Google Scholar

  • [6] Chung F.R.K., Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, AMS publications, New York, 1996 Google Scholar

  • [7] Chung F.R.K., Grigor’yan A., Yau S.T., Higher eigenvalues and isoperimetric inequalities on riemannian manifolds and graphs, Comm. Anal. Geom., 2000, 8, 969–1026 Google Scholar

  • [8] Chung F.R.K., Discrete isoperimetric inequalities, In: Surveys in differential geometry, Vol. IX, Int. Press, Somerville, MA, 2004, 53–82 Google Scholar

  • [9] Darafsheh M.R., Groups with the same non-commuting graph, Discrete Appl. Math., 2009, 157, 833–837 Web of ScienceGoogle Scholar

  • [10] Hebey E., Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, Vol.5, New York University Courant Institute of Mathematical Sciences, New York, 1999 Google Scholar

  • [11] Hofmann K.H., Russo F.G., The probability that x and y commute in a compact group, Math. Proc. Cambridge Phil. Soc., 2012, 153, 557–571 Web of ScienceGoogle Scholar

  • [12] Hofmann K.H., Russo F.G., The probability that xm and yn commute in a compact group, Bull. Aust. Math. Soc., 2013, 87, 503– 513 Web of ScienceGoogle Scholar

  • [13] Moghaddamfar A.R., About noncommuting graphs, Siberian Math. J., 2005, 47, 1112–1116 Google Scholar

  • [14] Mondino A., Nardulli S., Existence of isoperimetric regions in noncompact riemannian manifolds under Ricci or scalar curvature conditions, Comm. Anal. Geom., preprint available at http://arxiv.org/pdf/1210.0567v1.pdf Google Scholar

  • [15] Nardulli S., The isoperimetric profile of a noncompact Riemannian manifold for small volumes, Calc. Var. PDE, 2014, 49, 173–195 Google Scholar

  • [16] Neumann B.H., A problem of Paul Erd˝os on groups, J. Aust. Math. Soc., 1976, 21, 467–472 Google Scholar

  • [17] Russo F.G., Problems of connectivity between the Sylow graph, the prime graph and the non-commuting graph of a group, Adv. Pure Math., 2012, 2, 373–378 Google Scholar

About the article

Received: 2014-08-11

Accepted: 2014-03-07

Published Online: 2015-04-15

Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0027.

Export Citation

©2015 Stefano Nardulli and Francesco G. Russo. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in