Minimum rank and zero forcing number for butterfly networks

The minimum rank of a simple graph \(G\) is the smallest possible rank over all symmetric real matrices \(A\) whose nonzero off-diagonal entries correspond to the edges of \(G\). Using the zero forcing number, we prove that the minimum rank of the butterfly network is \(\frac19\left[(3r+1)2^{r+1}-2(-1)^r\right]\) and that this is equal to the rank of its adjacency matrix.