  We develop a graph-Hilbert-space framework, inspired by non-commutative
geometry, on (infinite) graphs and use it to study spectral properies of
\tit{graph-Laplacians} and so-called \tit{graph-Dirac-operators}. Putting the
various pieces together we define a {\it spectral triplet} sharing most (if not
all, depending on the particular graph model) of the properties of what Connes
calls a \tit{spectral triple}. With the help of this scheme we derive an
explicit expression for the {\it Connes-distance function} on general graphs
and prove both a variety of apriori estimates for it and calculate it for
certain examples of graphs. As a possibly interesting aside, we show that the
natural setting of approaching such problems may be the framework of
\tit{(non-)linear programming} or \tit{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.
