I will discuss the random matrix theory behind two-component random resistor networks on general graphs. This involves random submatrices of the graph’s natural Gamma projection operator, which is related to the gradient of the Gaussian Free Field on the graph. The particular realization of the submatrix is determined by the disorder of the conductances. Certain combinations of graph symmetries together with different models for the random conductances lead to exactly computable spectral statistics. Our recent results lead to exact spectral statistics for the uniform spanning tree model on a diamond hierarchical lattice. Joint work with Ken Golden, Elena Cherkaev, Ben Murphy, Han Le.