用一种新的迫零过程保证强Arnold性质
Using a new zero forcing process to guarantee the Strong Arnold Property
摘要：
最大零度$M(G)$和Colin de Verdière型参数$ξ(G)$都考虑了$\mathcal{S}(G)$中矩阵的最大可能零度，$\mathcal{S}(G)$是实对称矩阵族，其$i,j$ -项$i\neq j$如果$i$与$j$相邻则为非零，否则为零;然而，$ξ(G)$仅限于$\mathcal{S}(G)$中具有强Arnold性质的那些矩阵$A$，这意味着$X=O$是满足$A\circ X=O$、$I\circ X=O$和$AX=O$的唯一对称矩阵。本文引入迫零参数$Z_{\mathrm{SAP}}(G)$和$Z_{\mathrm{vc}}(G)$，证明了$Z_{\mathrm{SAP}}(G)=0$蕴涵每个矩阵$A\in \mathcal{S}(G)$具有强Arnold性质，并证明了不等式$M(G)-Z_{\mathrm{vc}}(G)\leq ξ(G)$对每个图$G$成立.最后，计算所有图的$ξ(G)$的值，直到$7$顶点，为这些图建立$ξ(G)=\lfloor Z\rfloor(G)$。
英文摘要：
The maximum nullity $M(G)$ and the Colin de Verdière type parameter $ξ(G)$ 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, $ξ(G)$ 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 $M(G)-Z_{\mathrm{vc}}(G)\leq ξ(G)$ holds for every graph $G$. Finally, the values of $ξ(G)$ are computed for all graphs up to $7$ vertices, establishing $ξ(G)=\lfloor Z\rfloor(G)$ for these graphs.
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