Minimum rank and zero forcing number for butterfly networks

TL;DR

This paper determines the minimum rank of butterfly networks using the zero forcing number, providing an exact formula and showing it equals the adjacency matrix's rank.

Contribution

The paper introduces a precise formula for the minimum rank of butterfly networks and establishes its equality with the adjacency matrix's rank using zero forcing techniques.

Findings

Exact formula for minimum rank of butterfly networks

Minimum rank equals the adjacency matrix's rank

Zero forcing number effectively bounds the minimum rank

Abstract

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.