Noncommutative Riemannian geometry on graphs

TL;DR

This paper develops a noncommutative geometric framework for graphs, introducing edge Laplacians that extend traditional graph Laplacians, and explores their spectral properties and geometric structures.

Contribution

It introduces a novel family of edge Laplacians derived from noncommutative geometry, extending graph Laplacians and providing a functorial differential calculus on graphs.

Findings

Eigenvalues of the edge Laplacian are positive except for one zero mode.

Constructs a canonical Euclidean metric and bimodule connection on graphs.

For Cayley graphs, a metric-compatible connection is explicitly constructed.

Abstract

We show that arising out of noncmmutatve geometry is a natural family of {\em edge Laplacians} on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a `finite manifold structure' on the set of vertices. We equip any graph with a canonical `Euclidean metric' and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.