Let $Gamma$ be an arbitrary $mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $mathcal{H}_varepsilon$ on $Gamma$ with the action $-varepsilon^{-1}{mathrm{d}^2/mathrm{d} x^2}$ on its edges; here $varepsilon>0$ is a small parameter. Let $minmathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $sigma(mathcal{H}^varepsilon)$ of $mathcal{H}^varepsilon$ has at least $m$ gaps as $varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $sigma(mathcal{H}^varepsilon)$ as $varepsilonto 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.
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