  As an outgrowth of our investigation of non-regular spaces within the context
of quantum gravity and non-commutative geometry, we develop a graph Hilbert
space framework on arbitrary (infinite) graphs and use it to study spectral
properties of graph-Laplacians and graph-Dirac-operators. We define a spectral
triplet sharing most of the properties of what Connes calls a spectral triple.
With the help of this scheme we derive an explicit expression for the
Connes-distance function on general directed or undirected graphs. We derive a
series of apriori estimates and calculate it for a variety of examples of
graphs. As a possibly interesting aside, we show that the natural setting of
approaching such problems may be the framework of (non-)linear programming or
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.
