Using a new zero forcing process to guarantee the Strong Arnold Property

TL;DR

This paper introduces new zero forcing parameters to ensure matrices satisfy the Strong Arnold Property, linking these parameters to maximum nullity and the Colin de Verdière type parameter, and computes these values for small graphs.

Contribution

It defines zero forcing parameters $Z_{SAP}(G)$ and $Z_{vc}(G)$, proves their implications for the Strong Arnold Property, and establishes a relationship between $\xi(G)$ and these parameters for small graphs.

Findings

$Z_{SAP}(G)=0$ guarantees the Strong Arnold Property for all matrices in $\mathcal{S}(G)$.

The inequality $M(G)-Z_{vc}(G)\leq \xi(G)$ holds for all graphs.

Computed $\xi(G)$ for all graphs up to 7 vertices, matching $loor{Z}(G)$.

Abstract

The maximum nullity $M(G)$ and the Colin de Verdi\`ere type parameter $\xi(G)$ both consider the largest possible nullity over matrices in $\mathcal{S}(G)$, which is the family of real symmetric matrices whose $i,j$-entry, $i\neq j$, is nonzero if $i$ is adjacent to $j$, and zero otherwise; however, $\xi(G)$ restricts to those matrices $A$ in $\mathcal{S}(G)$ with the Strong Arnold Property, which means $X=O$ is the only symmetric matrix that satisfies $A\circ X=O$, $I\circ X=O$, and $AX=O$. This paper introduces zero forcing parameters $Z_{\mathrm{SAP}}(G)$ and $Z_{\mathrm{vc}}(G)$, and proves that $Z_{\mathrm{SAP}}(G)=0$ implies every matrix $A\in \mathcal{S}(G)$ has the Strong Arnold Property and that the inequality $M(G)-Z_{\mathrm{vc}}(G)\leq \xi(G)$ holds for every graph $G$. Finally, the values of $\xi(G)$ are computed for all graphs up to $7$ vertices, establishing $\xi(G)=\lfloor Z\rfloor(G)$ for these graphs.