On the unit interval (I) and the Sierpinski gasket ($SG$), the spectral decimation function of the Laplacian has similar properties that result in a positive minimum spacing of eigenvalues. Other fractals, for example, the level-3 $SG$, $SG_3$, may not enjoy these properties. Our goal is to obtain a simple sufficient criterion for positive infimum spacing between eigenvalues in the spectrum, based on the properties of the corresponding spectral decimation function of the fractal. We also give a sufficient condition for zero infimum spacing between eigenvalues.

We consider the spectra of the Laplacians of two sequences of fractal graphs in the context of the general theory introduced by Sabot in 2003. For the sequence of graphs associated with the pentagasket, we give a description of the eigenvalues in terms of the iteration of a map from (ℂ2)3 to itself. For the sequence of graphs introduced in a previous paper by the author, we show that the results found therein can be related to Sabot's theory.

We consider a simple self-similar sequence of graphs that does not satisfy the symmetry conditions that imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and we obtain a complete description of the spectra of the graphs in the sequence.