  We consider a family of periodic tight-binding models (combinatorial graphs)
that have the minimal number of links between copies of the fundamental domain.
For this family we establish a local condition of second derivative type under
which the critical points of the dispersion relation can be recognized as
global maxima or minima. Under the additional assumption of time-reversal
symmetry, we show that any local extremum of a dispersion band is in fact its
global extremum if the dimension of the periodicity group is three or less, or
(in any dimension) if the critical point in question is a symmetry point of the
Floquet--Bloch family with respect to complex conjugation. We demonstrate that
our results are nearly optimal with a number of examples.
