Connes' distance function for commutative and noncommutative graphs

摘要

By formulating the concept of a graph algebraically, i.e., as properties of the algebra of functions over the set of vertices and the set of edges, we arrive at a purely algebraic concept of distance related to the one proposed by Connes for manifolds which easily extends also to the noncommutative case. The Dirac operator used in Connes' approach is replaced by a generalized difference operator which can be defined on arbitrary graphs. We speculate on the question of how this operator might be related to the concept of a Dirac operator on graphs. [References: 10]