Abstract. We present and discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in R 2 which were introduced recently in [1-5] all based on Sunada's construction of isospectral domains[6]. After presenting some of the properties of these graphs, we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.

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