  We prove Cheeger inequalities for p-Laplacians on finite and infinite
weighted graphs. Unlike in previous works, we do not impose boundedness of the
vertex degree, nor do we restrict ourselves to the normalized Laplacian and,
more generally, we do not impose any boundedness assumption on the geometry.
This is achieved by a novel definition of the measure of the boundary which is
using the idea of intrinsic metrics. For the non-normalized case, our bounds on
the spectral gap of p-Laplacians are already significantly better for finite
graphs and for infinite graphs they yield non-trivial bounds even in the case
of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger
constant and by the exponential volume growth of distance balls.
