Internal graphs of graph products of hyperfinite II${}_1$-factors

In this paper, we show that for a graph $\mathrm{\Gamma }$ from a class named H-rigid graphs, its subgraph $\mathrm{I}\mathrm{n}\mathrm{t}(\mathrm{\Gamma })$, named the internal graph of $\mathrm{\Gamma }$, is an isomorphism invariant of the graph product of hyperfinite II${}_1$-factors $R_{\mathrm{\Gamma }}$. In particular, we can classify $R_{\mathrm{\Gamma }}$ for some typical types of graphs, such as lines, cyclic graphs and infinite regular trees. As an application, we also show that for two isomorphic graph products of hyperfinite II${}_1$-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than 1. Our proof is based on the recent resolution of the Peterson-Thom conjecture.