Cheeger Inequalities For The Discrete Magnetic Laplacian

In this paper,firstly,we introduce the classical Cheeger constant on compact Riemannian manifolds,and we state that the proof procedure of Cheeger inequalities with the Drichlet boundary condition.Secondly,we introduce Cheeger constants on signed graphs for the magnetic Laplacian,which is related to the frustration index.And likewise we state the the proof procedure of Cheeger inequalities on signed graphs.At last,we apply the theory of stochastic decomposition for general metric spaces due to Lee and Noar and geometric properties of complex projective spaces as positive Ricci curvature spaces to derive the existence of bounded Lipschitz random partition of complex projective spaces,by which we improve the high order Cheeger inequalities for the discrete magnetic Laplacian.