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 ${\displaystyle Z(q)=(q^{d(x_i,x_j)}{)}_{i,j},}$ where $d(x_i,x_j)$ denotes the distance between $x_i$ and $x_j$. % We introduce two spectral invariants of finite metric spaces, the $q$-spectrum and the normalized $q$-spectrum, defined respectively from $Z(q)$ and its normalized transition matrix. In the case of graphs, these invariants recover the adjacency spectrum and the Laplacian spectrum in the limit $q\to 0$. Our main result shows that the $q$-spectrum determines a large class of finite metric spaces under a natural nondegeneracy condition. We also prove that all four-point metric spaces are determined by their $q$-spectra. % The key observation is that the coefficients of the characteristic polynomial of $Z(q)$ encode cycle structures of the underlying metric space. We further investigate the normalized $q$-spectrum and present computational examples comparing these invariants with classical graph spectra.