  The existence of non-isomorphic graphs which share the same Laplace spectrum
(to be referred to as isospectral graphs) leads naturally to the following
question: What additional information is required in order to resolve
isospectral graphs? It was suggested by Band, Shapira and Smilansky that this
might be achieved by either counting the number of nodal domains or the number
of times the eigenfunctions change sign (the so-called flip count). Recently
examples of (discrete) isospectral graphs with the same flip count and nodal
count have been constructed by K. Ammann by utilising Godsil-McKay switching.
Here we provide a simple alternative mechanism that produces systematic
examples of both discrete and quantum isospectral graphs with the same flip and
nodal counts.
