Graph Reconstruction and Quantum Statistical Mechanics

TL;DR

This paper explores the extent to which graphs can be reconstructed from associated Banach algebras and quantum statistical systems, showing that while boundary operator algebras reveal only topological info, quantum systems enable full reconstruction of certain multigraphs.

Contribution

It introduces a method to reconstruct multigraphs with minimal degree three from quantum statistical mechanical systems, extending previous algebraic approaches.

Findings

Boundary operator algebras recover topological information only.

Quantum statistical systems enable complete multigraph reconstruction.

Reconstruction is possible for multigraphs with minimal degree three.

Abstract

We study in how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.