Spectral invariants of finite metric spaces

TL;DR

This paper introduces two spectral invariants, the $q$-spectrum and transition $q$-spectrum, for finite metric spaces, extending graph spectra and demonstrating their distinguishing capabilities.

Contribution

The paper defines new spectral invariants for finite metric spaces that generalize graph spectra and analyzes their effectiveness in distinguishing different spaces.

Findings

The $q$-spectrum fully distinguishes many finite metric spaces and all spaces with up to 4 points.

The transition $q$-spectrum distinguishes spaces with rationally independent pairwise distances and all spaces with up to 3 points.

Computational experiments show the transition $q$-spectrum often has stronger practical distinguishing power.

Abstract

We introduce two spectral invariants of finite metric spaces, the $q$-spectrum and the transition $q$-spectrum, defined from similarity matrices. These invariants extend the adjacency and Laplacian spectra of graphs to general finite metric spaces, as graph spectra can be obtained as the limit $q \to 0$. We study the problem of distinguishing finite metric spaces by means of these invariants. The $q$-spectrum completely distinguishes a large class of finite metric spaces and all metric spaces on at most 4 points. We also show that the transition $q$-spectrum distinguishes spaces for which the multiset of pairwise distances is independent over the rational numbers, along with all spaces on at most 3 points. Computational experiments indicate that the transition $q$-spectrum often has stronger distinguishing power in practice, despite weaker theoretical guarantees.