Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access April 15, 2015

Two bounds on the noncommuting graph

  • Stefano Nardulli and Francesco G. Russo
From the journal Open Mathematics

Abstract

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.

References

[1] Abdollahi A., Akbari S., Maimani H.R., Non-commuting graph of a group, J. Algebra, 2006, 298, 468–492 10.1016/j.jalgebra.2006.02.015Search in 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 Search in 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 Search in Google Scholar

[4] Aubin T., Nonlinear analysis on manifolds: Monge–Ampére equations, Grundlehren der Mathematischen Wissenschaften, 252, Springer, Berlin, 1982 10.1007/978-1-4612-5734-9Search in Google Scholar

[5] Bakry D., Coulhon T., Ledoux M., Saloff–Coste L., Sobolev inequalities in disguise, Indiana Univ. Math. J., 1995, 44, 1033–1074 10.1512/iumj.1995.44.2019Search in Google Scholar

[6] Chung F.R.K., Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, AMS publications, New York, 1996 10.1090/cbms/092Search in 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 10.4310/CAG.2000.v8.n5.a2Search in Google Scholar

[8] Chung F.R.K., Discrete isoperimetric inequalities, In: Surveys in differential geometry, Vol. IX, Int. Press, Somerville, MA, 2004, 53–82 10.4310/SDG.2004.v9.n1.a3Search in Google Scholar

[9] Darafsheh M.R., Groups with the same non-commuting graph, Discrete Appl. Math., 2009, 157, 833–837 10.1016/j.dam.2008.06.010Search in Google 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 10.1090/cln/005Search in 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 10.1017/S0305004112000308Search in Google 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 10.1017/S0004972712000573Search in Google Scholar

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

[15] Nardulli S., The isoperimetric profile of a noncompact Riemannian manifold for small volumes, Calc. Var. PDE, 2014, 49, 173–195 10.1007/s00526-012-0577-1Search in Google Scholar

[16] Neumann B.H., A problem of Paul Erd˝os on groups, J. Aust. Math. Soc., 1976, 21, 467–472 10.1017/S1446788700019303Search in 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 Search in Google Scholar

Received: 2014-8-11
Accepted: 2014-3-7
Published Online: 2015-4-15

©2015 Stefano Nardulli and Francesco G. Russo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/math-2015-0027/html
Scroll to top button