Title: Spectral Subgraph Localization. Abstract: Several graph mining problems are based on some variant of the subgraph isomorphism problem: Given two graphs, G and Q, does G contain a subgraph isomorphic to Q? As this problem is NP-hard, many methods avoid addressing it explicitly. In this paper, we propose a method that solves the problem by localizing, i.e., finding the position of, Q in G, by means of an alignment among graph spectra. Finding a node correspondence from Q to G thereafter is relegated to a separate task, as an instance of the graph alignment problem. We demonstrate that our spectral approach outperforms a baseline based on the state-of-the-art method for graph alignment in terms of accuracy on real graphs and scales to hundreds of nodes as no other method does.