Minimum rank, maximum nullity and zero forcing number for selected graph families

Abstract

The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$-th entry (for $i\ne j$) is nonzero whenever $\left\{i,j\right\}$ is an edge in $G$ and is zero otherwise. Maximum nullity is taken over the same set of matrices, and the sum of maximum nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. This paper defines the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide examples showing that the maximum nullity of a graph and its dual may differ, and similarly for the zero forcing number.