Parabolicity and Cheeger’s constant on graphs

Herein we study p-parabolicity on graphs. We prove that if a uniform graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every \(1<p<\infty \) $1<p<\infty$ . Moreover, we give sufficient conditions for a uniform graph to be non-parabolic and to be p-parabolic for every \(1<p<\infty \) $1<p<\infty$ .