Two bounds on the noncommuting graph

TL;DR

This paper establishes new bounds on the noncommuting graph, including an isoperimetric inequality and Sobolev inequalities, advancing the understanding of its analytical properties in finite groups and weighted graphs.

Contribution

It introduces the first isoperimetric inequality and Sobolev inequalities for the noncommuting graph, extending analysis to weighted locally finite graphs.

Findings

Proves an isoperimetric inequality for the noncommuting graph.

Derives Sobolev inequalities applicable to weighted locally finite graphs.

Provides new analytical tools for studying the structure of noncommuting graphs.

Abstract

Erd\H{o}s 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 is becoming relevant in 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.