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.

翻译：图的Cheeger常数（等价于其上同调边界扩张）量化了图的扩张性质。该概念隐式假定了系数群的选择，即$\mathbb{F}_2$。本文研究配备广义系数群（称为层）的图的Cheeger型不等式，其动机源于局部可测试码的应用。我们证明：图是好的谱扩张器当且仅当其相对于任意（或某个）常值层（等价于任意“普通”系数群）具有好的上同调边界扩张。进一步表明：在精确定义下，接近常值层的层在底层图为扩张器时也是好的上同调边界扩张器，从而首次给出非恒定层中良好上同调边界扩张的实例。与之相反，对于一般图上的层，无法关联图的扩张性质与层的上同调边界扩张。此外，我们证明$d$维$q$重球面建筑底层（加权）图的归一化第二特征值为$O(\frac{1}{\sqrt{q}-3d})$（当$q>9d^2$时），并将其代入层上同调边界扩张的结果，得到关于$q$和$d$的显式界——该界随$q$增大趋近于常数。研究过程中，我们证明了适用于$r$部加权图的扩张器混合引理的新形式。