Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs

TL;DR

This paper develops a framework using Dirac operators on arbitrary graphs to explicitly compute the Connes distance, connecting non-commutative geometry with graph theory and spectral analysis.

Contribution

It introduces a spectral triplet framework for infinite graphs and derives explicit formulas for the Connes distance, expanding the tools for non-commutative geometric analysis of graphs.

Findings

Explicit Connes-distance formulas for general graphs

Derived a series of a priori estimates for the Connes distance

Compared different graph geometries and Dirac operators in spectral analysis

Abstract

As an outgrowth of our investigation of non-regular spaces within the context of quantum gravity and non-commutative geometry, we develop a graph Hilbert space framework on arbitrary (infinite) graphs and use it to study spectral properties of graph-Laplacians and graph-Dirac-operators. We define a spectral triplet sharing most of the properties of what Connes calls a spectral triple. With the help of this scheme we derive an explicit expression for the Connes-distance function on general directed or undirected graphs. We derive a series of apriori estimates and calculate it for a variety of examples of graphs. As a possibly interesting aside, we show that the natural setting of approaching such problems may be the framework of (non-)linear programming or optimization. We compare our results (arrived at within our particular framework) with the results of other authors and show that the seeming differences depend on the use of different graph-geometries and/or Dirac operators.