We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on finite graphs and magnetic Laplacians on closed Riemannian manifolds. In this process, we develop spectral clustering algorithms for partially oriented graphs and multi-way spectral clustering algorithms via metrics in lens spaces and complex projective spaces. As a byproduct, we give a unified viewpoint of Harary’s structural balance theory of signed graphs and the gauge invariance of magnetic potentials.

Let [ ] be the Dirichlet Laplacian in the domain [ ]. Here [ ] and [ ] is a family of tiny identical holes (“ice pieces”) distributed periodically in [ ] with period ε. We denote by [ ] the capacity of a single hole. It was known for a long time that [ ] converges to the operator [ ] in strong resolvent sense provided the limit [ ] exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded Ω) an estimate for the difference of the kth eigenvalue of [ ] and [ ]. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.