AI 中文总结
本文针对独立数为5的图，利用独立多项式在图字典积下的封闭性及不连通图的独立多项式分解性质，对其独立多项式进行分类并刻画不连通构型。
AI 中文摘要
简单图$G$的独立多项式为$I_G(z) = i_0 + i_1 z + i_2 z^2 + \cdots + i_\alpha z^\alpha$，其中$i_\alpha$表示最大独立集的大小，也称为图的独立数。该独立多项式具有在图复合（字典积）下本质上封闭的显著特性。本文确定了独立数为5的图的独立多项式。对于不连通图$G$，利用$I_G(z)$可分解为$G$的各连通分量的独立多项式乘积的性质，进一步对这类不连通图$G$可能出现的所有独立多项式进行分类，并通过考察其分量结构，刻画了可能出现的不连通构型。
英文摘要
The independence polynomial of a simple graph $G$ is given by \( I_G(z) = i_0 + i_1 z + i_2 z^2 + \cdots + i_αz^α\), where \( i_α\) denotes the size of a maximum independent set, also called the independence number of the graph. The independence polynomial has the notable feature of being essentially closed under graph composition (lexicographic product). In this paper, we determine the independence polynomials of size five. For a disconnected graph $G$, we exploit the fact that $I_G(z)$ factors as the product of the independence polynomials of the connected components of $G$. Furthermore, we classify all independence polynomials that can occur for such a disconnected graph $G$ and, by examining their component structures, we characterize the disconnected configurations that may arise.
Comments18 pages. Comments are welcome
