论文标题
关于集团多项式的矩阵矩阵和衍生物
On Clique Incidence Matrices and Derivatives of Clique Polynomials
论文作者
论文摘要
由$ c（g，x）$表示的完整子图（cliques）数量的普通生成函数称为图$ g $的集合多项式。在本文中，我们首先引入了与简单图$ g $关联的发病率矩阵作为$ g $ $ g $的经典顶点 - 边缘矩阵的概括。然后，使用这些集团的入射矩阵，我们获得了两个集团计数身份，可用于为第一个和第二个多项式的第一个和第二个衍生物得出两个组合公式。最后，我们以几个开放性问题和猜想的结论，涉及我们主要结果的可能扩展，以提高集团多项式的较高衍生物。
The ordinary generating function of the number of complete subgraphs (cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of the graph $G$. In this paper, we first introduce some \emph{clique} incidence matrices associated by a simple graph $G$ as a generalization of the classical vertex-edge incidence matrix of $G$. Then, using these clique incidence matrices, we obtain two clique-counting identities that can be used for deriving two combinatorial formulas for the first and the second derivatives of clique polynomials. Finally, we conclude the paper with several open questions and conjectures about possible extensions of our main results for higher derivatives of clique polynomials.
