Distinguishing finite metric spaces via similarity spectra

We study spectra and characteristic polynomials of similarity matrices associated with finite metric spaces, where the similarity matrix of a finite metric space $X=\{x_1,\dots,x_n\}$ is given by $\displaystyle Z(q)=(q^{d(x_i,x_j)})_{i,j}$. % We introduce two spectral invariants of finite metric spa