Invariants for Laplacians on periodic graphs

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number $\cI$ of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians (matrices) have the minimal number $2\cI$ of coefficients depending on the quasimomentum and show that the number $\cI$ is an invariant of the periodic graph. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the Laplacian spectrum and the effective masses at the bottom of the spectrum in terms of the invariant $\cI$ and the minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber Laplacians. Moreover, similar results for Schr\"odinger operators with periodic potentials are obtained.