  We consider the bifurcations of standing wave solutions to the nonlinear
Schr\"odinger equation (NLS) posed on a quantum graph consisting of two loops
connected by a single edge, the so-called dumbbell, recently studied by
Marzuola and Pelinovsky. The authors of that study found the ground state
undergoes two bifurcations, first a symmetry-breaking, and the second which
they call a symmetry-preserving bifurcation. We clarify the type of the
symmetry-preserving bifurcation, showing it to be transcritical. We then reduce
the question, and show that the phenomena described in that paper can be
reproduced in a simple discrete self-trapping equation on a combinatorial graph
of bowtie shape. This allows for complete analysis both by geometric methods
and by parameterizing the full solution space. We then expand the question, and
describe the bifurcations of all the standing waves of this system, which can
be classified into three families, and of which there exists a countably
infinite set.
