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29 March 2024
 
  » arxiv » 1601.1341

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Using a new zero forcing process to guarantee the Strong Arnold Property
Jephian C.-H. Lin ;
Date 6 Jan 2016
AbstractThe 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, $i eq j$, 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 $Acirc X=O$, $Icirc X=O$, 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 $Ain mathcal{S}(G)$ 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 $xi(G)=lfloor Z floor(G)$ for these graphs.
Source arXiv, 1601.1341
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