  The aim of this article is to give a simple geometric condition that
guarantees the existence of spectral gaps of the discrete Laplacian on periodic
graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on
the finite quotient and interpret the vector potential as a Floquet parameter.
We develop a procedure of virtualising edges and vertices that produces
matrices whose eigenvalues (written in ascending order and counting
multiplicities) specify the bracketing intervals where the spectrum of the
Laplacian is localised. We prove Higuchi-Shirai's conjecture for Z-periodic
trees and apply our technique in several examples like the polypropylene or the
polyacetylene to show the existence spectral gaps.
