There is an ample body of recent research on indexing for structural graph queries. However, as verified by our experiments with a large number of random and scale-free graphs, there may be a great variation in the performances of indexes of graph queries. Unfortunately, the structures of graph indexes are often complex and ad-hoc, so deriving an accurate performance model is a daunting task. As a result, database practitioners may encounter difficulties in choosing the optimal index for their data graphs. In this paper, we address this problem by proposing a spectral decomposition method for predicting the relative performances of graph indexes. Specifically, given a graph, we compute its spectrum. We then propose a similarity function to compare the spectrums of graphs. We adopt a classification algorithm to build a model and a voting algorithm for predicting the optimal index. Our empirical studies on a large number of random and scale-free graphs, using four structurally distinguishable indexes, demonstrate that our spectral decomposition method is robust and almost always exhibits an accuracy of 70% or above.