Findings on the Structure of Balanced Measures

Given a finite, complete graph $G=(V,E)$ with some probability measures supported on it's vertices, we can define a subset of these probability measures that have nice properties. We call such measures \textit{balanced}. The structure of such measures has some interesting properties that we're currently exploring. One such property is that, given the graph join of $n$ graphs, we're able to define a unique balanced measure for each subset of the graph join. Furthermore, given a set of graphs $G_i$ that each meet certain conditions, we've been able to find an upper bound on the number of balanced measures that can be supported on the graph join of $n$ such graphs.