  We study the volume growth of metric balls as a function of the radius in
discrete spaces, and focus on the relationship between volume growth and
discrete curvature. We improve volume growth bounds under a lower bound on the
so-called Ollivier curvature, and discuss similar results under other types of
discrete Ricci curvature. Following recent work in the continuous setting of
Riemannian manifolds (by the first author), we then bound the eigenvalues of
the Laplacian of a graph under bounds on the volume growth. In particular, the
spectral gap of the graph can be bounded using a weighted discrete Hardy
inequality and the higher eigenvalues of the graph can be bounded by the
eigenvalues of a tridiagonal matrix times a multiplicative factor, both of
which only depend on the volume growth of the graph. As a direct application,
we relate the eigenvalues to the Cheeger isoperimetric constant. Using these
methods, we describe classes of graphs for which the Cheeger inequality is
tight on the second eigenvalue. We also describe a method for proving Buser's
inequality in graphs, particularly under a lower bound assumption on curvature.

Combinatorics