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. 
Daniela Ferrero, Cyriac Grigorious, Thomas Kalinowski, Joe Ryan,   Sudeep Stephen
math.CO cs.DM