标题： 图的独立数为三的独立吸引子的连通性 显示英文标题

标题： Connectedness of independence attractors of graphs with independence number three

摘要： 一个简单图$G$中的独立集是$G$中两两不相邻的顶点集合。 独立多项式$G$，记为$I_G$，定义为$1 + a_1 z + a_2 z^2+\cdots+a_d z^{d}$，其中$a_i$表示大小为$i$的独立集的数量，$d$是$G$中最大独立集的大小。 这个$d$被称为$G$的独立数。 令$G^m$表示$G$与自身的$m$次字典积。 独立吸引子$G$，记为$\mathcal{A}(G)$，定义为$\mathcal{A}(G) = \lim\limits_{m\rightarrow \infty} \{z: I_{G^m}(z)=0\}$，其中极限是相对于平面所有紧子集空间上定义的Hausdorff度量取到的。 本文研究了所有独立数为三的图的独立吸引子的连通性。 设图$G$的独立多项式为$1+a_1 z +a_2 z^2 +a_3 z^3$。 对于$a_1 =3$，$\mathcal{A}(G)$最终成为$ \{-1\} \cup \{z: |z+1|=1\} $。 对于$a_1 >3$，我们证明以下内容。 如果$a_2 ^2 \leq 3 a_1 a_3$，或者$3 a_1 a_3 < a_2 ^2 < 4a_3 (a_1 -1)$，那么$\mathcal{A}(G)$是完全不连通的。 对于$a_2 ^2 =4a_3 (a_1 -1) $，当$a_1 =5$时$\mathcal{A}(G)$是连通的，而在其他所有$a_1$的值下是不连通的但不是完全不连通的。 如果$a_2 ^2 > 4a_3 (a_1 -1)$则$\mathcal{A}(G)$可以是连通的、完全不连通的或不连通但非完全不连通的，这取决于涉及$a_1, a_2$和$a_3$的进一步条件。提供了展示所有可能性的图示例。 显示英文摘要

摘要： An independent set in a simple graph $G$ is a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$, denoted by $I_G$ is defined as $1 + a_1 z + a_2 z^2+\cdots+a_d z^{d}$, where $a_i$ denotes the number of independent sets with cardinality $i$ and $d$ is the cardinality of a largest independent set in $G$. This $d$ is known as the independence number of $G$. Let $G^m$ denote the $m$-times lexicographic product of $G$ with itself. The independence attractor of $G$, denoted by $\mathcal{A}(G)$ is defined as $\mathcal{A}(G) = \lim\limits_{m\rightarrow \infty} \{z: I_{G^m}(z)=0\}$, where the limit is taken with respect to the Hausdorff metric defined on the space of all compact subsets of the plane. This paper investigates the connectedness of the independence attractors of all graphs with independence number three. Let the independence polynomial of $G$ be $1+a_1 z +a_2 z^2 +a_3 z^3$. For $a_1 =3$, $\mathcal{A}(G)$ turns out to be $ \{-1\} \cup \{z: |z+1|=1\} $. For $a_1 >3$, we prove the following. If $a_2 ^2 \leq 3 a_1 a_3$, or $3 a_1 a_3 < a_2 ^2 < 4a_3 (a_1 -1)$ then $\mathcal{A}(G)$ is totally disconnected. For $a_2 ^2 =4a_3 (a_1 -1) $, $\mathcal{A}(G)$ is connected when $a_1 =5$ and is disconnected but not totally disconnected for all other values of $a_1$. If $a_2 ^2 > 4a_3 (a_1 -1)$ then $\mathcal{A}(G)$ can be connected, totally disconnected or disconnected but not totally disconnected depending on further conditions involving $a_1, a_2$ and $a_3$. Examples of graphs exhibiting all the possibilities are provided.