Graph-Laplacians and Dirac Operators and the Connes-Distance-Functional

We develop (within a possibly new) framework spectral analysis and operator theory on (almost) general graphs and use it to study spectral properies of the graph-Laplacian and so-called graph-Dirac-operators. That is, we introduce a Hilbert space structure, being in our framework the direct sum of a node-Hilbert-space and a bond-Hilbert-space, a Dirac operator intertwining these components, and which is shown to be closely related to the graph-Laplacian, and, putting these pieces together, 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 we derive an explicit expression for the Connes-distance function on general graphs and prove both a variety of rigorous estimates and calculate it for certain examples of graphs. 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.