Graphs, spectral triples and Dirac zeta functions

TL;DR

This paper constructs a spectral triple for certain finite graphs, using non-commutative geometry to encode graph properties through zeta functions, aiding classification.

Contribution

It introduces a new spectral triple framework for finite graphs, linking graph invariants with non-commutative geometric tools.

Findings

Spectral triples encode graph structure via zeta functions

Provides a classification framework for graphs using non-commutative geometry

Establishes a connection between graph invariants and spectral data

Abstract

To a finite, connected, unoriented graph of Betti-number g>=2 and valencies >=3 we associate a finitely summable, commutative spectral triple (in the sense of Connes), whose induced zeta functions encode the graph. This gives another example where non-commutative geometry provides a rigid framework for classification.