  We consider high-temperature expansions for the free energy of zero-field
Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal
Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order.
As a by-product, we obtain exact vertex-averaged numbers of self-avoiding
polygons on these quasiperiodic graphs. In addition, we analyze periodic
approximants by computing the partition function via the Kac-Ward determinant.
For the critical properties, we find complete agreement with the commonly
accepted conjecture that the models under consideration belong to the same
universality class as those on periodic two-dimensional lattices.
