Critical ideals, minimum rank and zero forcing number

TL;DR

This paper explores the relationships between zero forcing number, minimum rank, and algebraic co-rank in graphs, providing new insights for bounding and computing these parameters.

Contribution

It introduces the algebraic co-rank as a third parameter linking zero forcing number and minimum rank, offering a novel perspective for their analysis.

Findings

Establishes connections between zero forcing number, minimum rank, and algebraic co-rank.

Provides bounds and methods for computing these graph parameters.

Offers new theoretical insights into the interplay of these parameters.

Abstract

There are profound relations between the zero forcing number and minimum rank of a graph. We study the relation of both parameters with a third one, the algebraic co-rank; that is defined as the largest $i$ such that the $i$-th critical ideal is trivial. This gives a new perspective for bounding and computing these three graph parameters.