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Eigenfunctions of Laplacians on periodic metric graphs

We consider the Kirchhoff Laplacians on equilateral periodic metric graphs. We present results about spectral properties of these operators: 1) the decomposition of the metric Laplacians into a constant fiber direct integral (with an exact form of fiber operators); 2) the relation between eigenfunctions of fiber operators for the metric and discrete Laplacians; 3) a uniform bound for the eigenfunctions of the fiber metric Laplacians.