Measures on Graphs and Groupoid Measures

The main purpose of this paper is to introduce several measures determined by a given finite directed graph. To construct σ-algebras for those measures, we consider several algebraic structures induced by G; (i) the free semigroupoid of the shadowed graph (ii) the graph groupoid of G, (iii) the disgram set and (iv) the reduced diagram set . The graph measures determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures determined by (iii) is the diagram measure measuring how long we moved consequently from the starting positions (which are vertices) of some movements on G. The graph measures and determined by (ii) and (iv) are the (graph) groupoid measure and the (quotient-)groupoid measure, respectively. We show that above graph measurings are invariants on shadowed graphs of finite directed graphs. Also, we will consider the reduced diagram measure theory on graphs. In the final chapter, we will show that if two finite directed graphs G ₁ and G ₂ are graph-isomorphic, then the von Neumann algebras L ^∞(μ ₁) and L ^∞(μ ₂) are *-isomorphic, where μ ₁ and μ ₂ are the same kind of our graph measures of G ₁ and G ₂, respectively. Received: December 7, 2006. Revised: August 3, 2007. Accepted: August 18, 2007.