Spectral Analysis and Operator Theory on (Infinite) Graphs, Graph-Laplacians and Dirac Operators and the Connes-Distance-Functional

نویسنده

  • Manfred Requardt
چکیده

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 triple. With the help of this we derive an explicit expression for the Connesdistance 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.

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تاریخ انتشار 2000