# Spectrally Robust Graph Isomorphism

We initiate the study of spectral generalizations of the graph isomorphism problem. (a)The Spectral Graph Dominance (SGD) problem: On input of two graphs G and H does there exist a permutation π such that G≼π(H)? (b) The Spectrally Robust Graph Isomorphism (SRGI) problem: On input of two graphs G and H, find the smallest number κ over all permutations π such that π(H) ≼ G≼κ c π(H) for some c. SRGI is a natural formulation of the network alignment problem that has various applications, most notably in computational biology. Here G≼ c H means that for all vectors x we have x^T L_G x ≤ c x^T L_H x, where L_G is the Laplacian G. We prove NP-hardness for SGD. We also present a κ-approximation algorithm for SRGI for the case when both G and H are bounded-degree trees. The algorithm runs in polynomial time when κ is a constant.

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