  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.
