2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56474We 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.27 pages, Latex, some more references and comments referring to existing work, some proofs improved and several new results included, the relation of distance calculations to non-linear optimization clarified, under consideration by JPAMathematical PhysicsHigh Energy Physics - Theorytext