Graph-Laplacians and Dirac Operators on (Infinite) Graphs and the Calculation of the Connes-Distance-Functional
Manfred Requardt
备注：
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 JPA
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.
Dirac operators on periodic planar graphs: the right definition advocated by flat bands
The Open University of Japan（日本开放大学）；Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío（比奥比奥大学理学院数学系）；Institute for Liberal Arts and Sciences & Graduate School of Mathematics, Nagoya University（名古屋大学人文学部与研究生院数学研究院）
Pythagoras' Theorem on a 2D-Lattice from a "Natural" Dirac Operator and Connes' Distance Formula
A. Connes的非交换几何是一个广义的Dirac算子，它在状态空间上引入了一个度量（Connes距离）。我们推广了A. Dimakis等人的Connes距离将一维格点上的线性距离恢复到二维格点上。这个狄拉克算子被“自然”定义，具有所谓的“局部特征值属性”，并在这个二维格上诱导欧几里得距离。这种Dirac算子可以推广到任意高维格上。
