Noncommutative Riemannian geometry on graphs

Details Noncommutative Riemannian geometry on graphs Noncommutative Riemannian geometry on graphs

2010-11-25

arxiv.org/abs/1011.5898v4

Mathematics

Quantum Algebra

28 pages amslatex, with pdf inline figures; cleaned up the presentation of the paper by some mild restructuring and added Sect

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Noncommutative Riemannian geometry on graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Noncommutative Riemannian geometry on graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Noncommutative Riemannian geometry on graphs will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-463691