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Article overview
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Using a new zero forcing process to guarantee the Strong Arnold Property | Jephian C.-H. Lin
; | Date: |
6 Jan 2016 | 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, $i
eq j$, 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 | Services: | Forum | Review | PDF | Favorites |
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