# Measures on Graphs and Certain Groupoid Measures

#### Abstract

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 F(G) of the shadowed graph G = G ∪ G (ii) the graph groupoid G of G, (iii) the disgram set D(G) and (iv) the reduced diagram set Dr(G). The graph measures μGˆ determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures μδ 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 μG and μδr 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 finite directed graphs, when we consider shadowed graphs are certain two-colored 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 G1 and G2 are graph-isomorphic, then the von Neumann algebras L∞(μ1) and L ∞(μ2) are ∗-isomorphic, where μ1 and μ2 are the same kind of our graph measures of G1 and G2, respectively. The main purpose of this paper is to introduce certain measures induced by a finite directed graph and to introduce certain measures induced by a groupoid, in particular, a groupoid generated by a finite directed graph. Measure Theory of such measures is fundamental and easily understood but it is interesting because of that the theory is depending on combinatorial objects (i.e., graphs). In fact, the operator algebra depending on such measures looks much interesting than the measure theory. In the final chapter of this paper, we briefly consider a von Neumann algebra L (μG), where μG is either one of our measures induced by a finite directed graph or one of our graph groupoid measures. This paper would be the first step of such combinatorial measure theoretic operator algebra. In this paper, we will concentrate on constructing such measures and considering their properties. We will show that our graph measuring is quiet stable object since it is an invariant on finite directed graphs under certain additional assumption. From this invariance, we can see that if two finite directed graphs G1 and G2 are graph-isomorphic, then the corresponding our graph measures μG1 and μG2 are equivalent and hence the von Neumann algebras L (μG1) and L ∞ (μG2) are ∗-isomorphic. Date: June, 2006.

### Cite this paper

@inproceedings{Cho2006MeasuresOG, title={Measures on Graphs and Certain Groupoid Measures}, author={Ilwoo Cho}, year={2006} }