  We study the eigenvalues of the connection Laplacian on a graph with an
orthogonal group or unitary group signature. We establish higher order Buser
type inequalities, i.e., we provide upper bounds for eigenvalues in terms of
Cheeger constants in the case of nonnegative Ricci curvature. In this process,
we discuss the concepts of Cheeger type constants and a discrete Ricci
curvature for connection Laplacians and study their properties systematically.
The Cheeger constants are defined as mixtures of the expansion rate of the
underlying graph and the frustration index of the signature. The discrete
curvature, which can be computed efficiently via solving semidefinite
programming problems, has a characterization by the heat semigroup for
functions combined with a heat semigroup for vector fields on the graph.
