Isospectral drums in R2, involution graphs and Euclidean TI-domains

The widely investigated question 'Can one hear the shape of a drum?' which Kac posed in his published lecture (Kac 1966 Am. Math. Mon. 73 1-23) was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1-22) by construction of isospectral pairs in R-2. Up to present, all known planar counter examples are constructed by a certain tiling method, and in this communication, we call such examples isospectral Euclidean TI-domains. From counter examples of this type, one can construct a pair of (finite) involution graphs. In this communication, we address the question as to how the isospectrality of the domains for the Laplacian influences the cospectrality of the involution graphs.