Zero forcing parameters and minimum rank problems

TL;DR

This paper explores zero forcing parameters, introduces the positive semidefinite zero forcing number, and applies these concepts to analyze minimum rank problems in graphs, revealing differences between real and complex cases.

Contribution

It introduces the positive semidefinite zero forcing number and relates it to the ordered set number, advancing the study of minimum rank problems in graph theory.

Findings

No vertex is in every zero forcing set for connected graphs of order at least two.

The positive semidefinite zero forcing number equals |G|-OS(G).

An example shows the real positive semidefinite minimum rank can exceed the complex Hermitian minimum rank.

Abstract

The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity / minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z_+(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hermitian positive semidefinite minimum rank is presented.