Effective masses for Laplacians on periodic graphs

We consider Laplacians on both periodic discrete and periodic metric equilateral graphs. Their spectrum consists of an absolutely continuous part (which is a union of non-degenerate spectral bands) and flat bands, i.e., eigenvalues of infinite multiplicity. We estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs. Moreover, in the case of the bottom of the spectrum we determine two-sided estimates on the effective mass in terms of geometric parameters of the graphs. The proof is based on Floquet theory, factorization of fiber operators, perturbation theory and the relation between effective masses for Laplacians on discrete and metric graphs obtained in our paper.