  The Cheeger constant of a graph, or equivalently its coboundary expansion,
quantifies the expansion of the graph. This notion assumes an implicit choice
of a coefficient group, namely, $\mathbb{F}_2$. In this paper, we study
Cheeger-type inequalities for graphs endowed with a generalized coefficient
group, called a sheaf; this is motivated by applications to locally testable
codes. We prove that a graph is a good spectral expander if and only if it has
good coboundary expansion relative to any (resp. some) constant sheaf,
equivalently, relative to any `ordinary' coefficient group. We moreover show
that sheaves that are close to being constant in a well-defined sense are also
good coboundary expanders, provided that their underlying graph is an expander,
thus giving the first example of good coboundary expansion in non-cosntant
sheaves. By contrast, for general sheaves on graphs, it is impossible to relate
the expansion of the graph and the coboundary expansion of the sheaf.
  In addition, we show that the normalized second eigenvalue of the (weighted)
graph underlying a $q$-thick $d$-dimensional spherical building is
$O(\frac{1}{\sqrt{q}-3d})$ if $q>9d^2$, and plug this into our results about
coboundary expansion of sheaves to get explicit bounds on the coboundary
expansion in terms of $q$ and $d$. It approaches a constant as $q$ grows.
  Along the way, we prove a new version of the Expander Mixing Lemma applying
to $r$-partite weighted graphs.
