Manfred Requardt 2002 J. Phys. A: Math. Gen. 35 759 doi:10.1088/0305-4470/35/3/319
Manfred Requardt
Show affiliationsAs an outgrowth of our investigation of non-regular spaces within the context of quantum gravity and non-commutative geometry, we develop a graph Hilbert space framework on arbitrary (infinite) graphs and use it to study spectral properties of graph Laplacians and graph Dirac operators. We define a spectral triplet sharing most 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 directed or undirected graphs. We derive a series of a priori estimates and calculate it for a variety of examples of graphs. As a possibly interesting side, we show that the natural setting for approaching such problems may be the framework of (non)linear programming or optimization. We compare our results (arrived at within our particular framework) with those of other authors and show that the seeming differences depend on the use of different graph geometries and/or Dirac operators.
34L40 Particular operators (Dirac, one-dimensional Schrödinger, etc.)
Issue 3 (25 January 2002)
Received 7 April 2000, in final form 20 August 2001
Published 11 January 2002
Manfred Requardt 2002 J. Phys. A: Math. Gen. 35 759
S Abdullin et al 2002 J. Phys. G: Nucl. Part. Phys. 28 469
P Tavella and A Premoli 1994 Metrologia 30 479
A V Parisi 2005 Eur. J. Phys. 26 313
Jing Cai et al 2007 Phys. Med. Biol. 52 365
Mark Alford and Qinghai Wang 2005 J. Phys. G: Nucl. Part. Phys. 31 719
A Bauch et al 2006 Metrologia 43 109
Jürgen Becker and Günther Grün 2005 J. Phys.: Condens. Matter 17 S291
P Enders and O Schmidtmann 1992 J. Phys. A: Math. Gen. 25 1981
Garry Ludwig and S Brian Edgar 2000 Class. Quantum Grav. 17 1683