Graph-Laplacians and Dirac Operators on (Infinite) Graphs and the Calculation of the Connes-Distance-Functional

TL;DR

This paper develops a non-commutative geometric framework for infinite graphs to analyze spectral properties of graph Laplacians and Dirac operators, leading to explicit formulas for the Connes-distance function and comparisons with other approaches.

Contribution

It introduces a spectral triplet framework for graphs inspired by non-commutative geometry, enabling explicit calculation of the Connes-distance and comparison with existing methods.

Findings

Derived explicit Connes-distance formulas for graphs.

Established estimates and calculations for specific graph examples.

Compared different graph geometries and Dirac operators in the framework.

Abstract

We develop a graph-Hilbert-space framework, inspired by non-commutative geometry, on (infinite) graphs and use it to study spectral properies of \tit{graph-Laplacians} and so-called \tit{graph-Dirac-operators}. Putting the various pieces together we define a {\it spectral triplet} sharing most (if not all, depending on the particular graph model) of the properties of what Connes calls a \tit{spectral triple}. With the help of this scheme we derive an explicit expression for the {\it Connes-distance function} on general graphs and prove both a variety of apriori estimates for it and calculate it for certain examples of graphs. As a possibly interesting aside, we show that the natural setting of approaching such problems may be the framework of \tit{(non-)linear programming} or \tit{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.