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arXiv 0909.5266math.CO
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匹配多项式非零根的广义$D$ -图

Generalized $D$-graphs for Nonzero Roots of the Matching Polynomial

Cheng Yeaw Ku, Kok Bin Wong

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22 pages
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摘要：

最近，Bauer等人（J Graph Theory 55（4）（2007），343- 358）介绍了一种图算子$D(G)$，称为$G$的$D$ -图，其在研究具有完美匹配的$G$中的最大Tutte集的结构方面是有用的。在其他结果中，他们证明了最大Tutte集在图$D(G)$中的最大独立集和$G$中的最大极值集方面的特征。这后来被Busch等人扩展到没有完美匹配的图（Discrete Appl.Math.155（2007），2487- 2495）。设$θ$为实数，$μ(G,x)$为图$G$的匹配多项式.令$\textnormal{mult} (θ, G)$是作为$μ(G,x)$的根的$θ$的重数。我们观察到$D$ -图的概念与$θ=0$是隐式相关的.本文对任意实数$θ$给出了$G$的$D$ -图的一个自然推广，并用$D_θ(G)$表示这个新算子，使得当$θ=0$时，$D_θ(G)$与$D(G)$重合.证明了极大$θ$ -Tutte集是$G$中极大Tutte集的$θ$ -模拟集的一个特征.特别地，我们证明了对于任何$X \subseteq V(G)$，$|X|>1$和任何实数$θ$，$\m(θ, G \setminus X)=\m(θ, G)+|X|$当且仅当$\m(θ, G \setminus uv)=\m(θ, G)+2$对于任何$u, v \in X$，$u \not = v$，从而扩展了Bauer等人和Busch等人先前的工作，建立了$θ=0$情况下的结果。

英文摘要：

Recently, Bauer et al. (J Graph Theory 55(4) (2007), 343--358) introduced a graph operator $D(G)$, called the $D$-graph of $G$, which has been useful in investigating the structural aspects of maximal Tutte sets in $G$ with a perfect matching. Among other results, they proved a characterization of maximal Tutte sets in terms of maximal independent sets in the graph $D(G)$ and maximal extreme sets in $G$. This was later extended to graphs without perfect matchings by Busch et al. (Discrete Appl. Math. 155 (2007), 2487--2495). Let $θ$ be a real number and $μ(G,x)$ be the matching polynomial of a graph $G$. Let $\textnormal{mult} (θ, G)$ be the multiplicity of $θ$ as a root of $μ(G,x)$. We observe that the notion of $D$-graph is implicitly related to $θ=0$. In this paper, we give a natural generalization of the $D$-graph of $G$ for any real number $θ$, and denote this new operator by $D_θ(G)$, so that $D_θ(G)$ coincides with $D(G)$ when $θ=0$. We prove a characterization of maximal $θ$-Tutte sets which are $θ$-analogue of maximal Tutte sets in $G$. In particular, we show that for any $X \subseteq V(G)$, $|X|>1$, and any real number $θ$, $\m(θ, G \setminus X)=\m(θ, G)+|X|$ if and only if $\m(θ, G \setminus uv)=\m(θ, G)+2$ for any $u, v \in X$, $u \not = v$, thus extending the preceding work of Bauer et al. and Busch et al. which established the result for the case $θ=0$.

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