A Note on the Diamond Fractal

We prove the uniqueness of a “Laplacian” on the compact diamond fractal. More precisely, there exists a unique (up to positive multiples) self-similar, irreducible, local, regular and symmetric Dirichlet form with the normalized Hausdorff measure as reference measure. The main tool is the “eigenvalue test” which allows to use numerical results to set up theorems. The approach is of interest in its own because it also applies to other finitely ramified, graph-directed fractals.