给定点的局部平均控制集的研究

The Local Average Order of Dominating Sets Containing a Fixed Vertex

摘要: 图$G$ 的平均控制集$avd\left(G\right)$ 是其所有控制集的顶点数的平均值。类似地，局部平均控制集$av{d}_v\left(G\right)$ 是所有含给定点$v$ 的控制集的顶点数的平均值。$G$ 的控制多项式$D\left(G,x\right)$ 是其控制集的生成函数，由于平均控制集是控制多项式的对数导数在1处的取值，在本文中，我们利用平均控制集和控制多项式之间的关系给出了$n\left(n\ge 3\right)$ 阶连通图中度为$n-2$ 的给定点的局部平均控制集的公式。

Abstract: The global average order of dominating sets of a graph $avd\left(G\right)$ is the average number of vertices of its dominating sets. Analogously, the local average order of dominating sets $av{d}_v\left(G\right)$ is the average number of vertices of its dominating sets containing a fixed vertex $v$ . The domination polynomial $D\left(G,x\right)$ of a graph $G$ is the generating function of its dominating sets. The global average order of dominating sets is the value of the logarithmic derivative of the domination polynomial at $x=1$ . In this paper, we derive a formula for the local average order of dominating sets in the case where the degree of the fixed vertex is $n-2$ .