On the (ir)rationality of Kontsevich Weights

2.1. Graphs and weights. A Kontsevich graph [2] is a directed graph with two types of vertices, type I and type II. The type I vertices are usually denoted 1, . . . ,m (m ≥ 0) and the type II vertices 1̄, . . . , n̄ (n ≥ 0). The directed edges are required to start at type I vertices. In fact, we will only consider the case m = 7, n = 2. A Lie graph is a Kontsevich graph with exactly two type II vertices, and with at most one edge ending and exactly two edges starting at every type I vertex. An example is shown in Figure 1. To each Kontsevich graph, one can associate a weight, i.e., a real, conjecturally rational number. The weight is given by an integral of the form