Spectral Graph Isomorphism

In this paper, we initiate the study of spectral generalizations of Graph Isomorphism. We define two main problems. 1. The spectral graph isomorphism (SGI) problem: given two graphs G and H as input, does there exist a permutation π such that G and π(H) are spectrally close? This is a mathematical formulation of certain problems in shape and image matching, and is also a natural extension of the fundamental problem of graph isomorphism and its variations. 2. The graph dominance (GD) problem: given two graphs G and H as input, does there exist a permutation π such that π(H) is spectrally dominated by G? In this paper, we present the first (to our knowledge) algorithmic and hardness results on SGI and GD respectively. Our two main results are: a polynomial time algorithm that finds a nearly-optimal spectral isomorphism when the input graphs are bounded degree trees and an NP-hardness result for the GD problem and APX-hardness for optimization version of GD problem.