Resolving the isospectrality of the dihedral graphs by counting nodal domains

We discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in R² which were introduced recently in [1–5] all based on Sunada's construction of isospectral domains [6]. After discussing some of the properties of these graphs, we present an example which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.