We discuss two generalizations of the graph isomorphism problem:

(a) Spectral Dominance: On input of two PSD matrices A and B, does there exist a permutation matrix P such that P'*B*P is spectrally dominated by A?

(b) Spectrally Robust Isomorphism: On input of two PSD matrices A and B, and a constant k, does there exists a permutation matrix P such that the condition number of the pair (A,P'*B*P) is bounded above by k?

We show that Spectral Dominance is NP-hard, even when A and B are graph Laplacians. On the other hand, we show that Spectrally Robust Isomorphism is tractable when A and B are Laplacians of trees that have bounded degree. We also discuss several related open questions.

[The talk is based on joint work with A. Kolla, V. Madan and A. K. Sinop that appeared in ICALP '18.]