Graph classes for critical ideals, minimum rank and zero forcing number

TL;DR

This paper characterizes graphs with low real algebraic co-rank, establishing bounds on minimum rank based on algebraic co-rank, and explores their relations with zero forcing number.

Contribution

It provides a new characterization of graphs with algebraic co-rank at most 2, linking minimum rank and algebraic co-rank in a novel way.

Findings

Graphs with real algebraic co-rank ≤ 2 are characterized.

Minimum rank ≤ 3 graphs have bounds related to their algebraic co-rank.

New relations between zero forcing number, minimum rank, and algebraic co-rank are established.

Abstract

Recently, there have been found new relations between the zero forcing number and the minimum rank of a graph with the algebraic co-rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank at most 2. This implies that for any graph with at most minimum rank at most 3, its minimum rank is bounded from above by its real algebraic co-rank.