Graph-Laplacians and Dirac Operators on (Infinite) Graphs and the Calculation of the Connes-Distance-Functional

We develop a graph-Hilbert-space framework, inspired by non-commutative geometry, on (infinite) graphs and use it to study spectral properies of graph-Laplacians and so-called graph-Dirac-operators. Putting the various pieces together we define a spectral triplet sharing most (if not all, depending on the particular graph model) 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 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 (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.