  We obtain upper bounds for the first eigenvalue of the magnetic Laplacian
associated to a closed potential $1$-form (hence, with zero magnetic field)
acting on complex functions of a planar domain $\Omega$, with magnetic Neumann
boundary conditions. It is well-known that the first eigenvalue is positive
whenever the potential admits at least one non-integral flux. By gauge
invariance the lowest eigenvalue is simply zero if the domain is simply
connected; then, we obtain an upper bound of the ground state energy depending
only on the ratio between the number of holes and the area; modulo a numerical
constant the upper bound is sharp and we show that in fact equality is attained
(modulo a constant) for Aharonov-Bohm-type operators acting on domains
punctured at a maximal $\epsilon$-net. In the last part we show that the upper
bound can be refined, provided that one can transform the given domain in a
simply connected one by performing a number of cuts with sufficiently small
total length; we thus obtain an upper bound of the lowest eigenvalue by the
ratio between the number of holes and the area, multiplied by a Cheeger-type
constant, which tends to zero when the domain is metrically close to a simply
connected one.
