  Periodic $2$nd order ordinary differential operators on $\R$ are known to
have the edges of their spectra to occur only at the spectra of periodic and
antiperiodic boundary value problems. The multi-dimensional analog of this
property is false, as was shown in a 2007 paper by some of the authors of this
article. However, one sometimes encounters the claims that in the case of a
single periodicity (i.e., with respect to the lattice $\mathbb{Z}$), the $1D$
property still holds, and spectral edges occur at the periodic and
anti-periodic spectra only. In this work we show that even in the simplest case
of quantum graphs this is not true. It is shown that this is true if the graph
consists of a $1D$ chain of finite graphs connected by single edges, while if
the connections are formed by at least two edges, the spectral edges can
already occur away from the periodic and anti-periodic spectra.
