  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.
