发表机构
Guangdong University of Technology; Guangdong Polytechnic Normal University; West Virginia University（广东工业大学; 广东技术师范大学; 西弗吉尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图的控制集局部平均阶，证明其下界为$\frac{n+1}{2}$、无孤立顶点图的上界为$\frac{5n-1}{6}$，还给出度数为$n-2$时的精确公式及l-茎顶点的上界。
AI 中文摘要
图的（全局）控制集平均阶是其所有控制集的顶点数的平均值。类似地，控制集的局部平均阶是包含一个固定顶点的控制集的顶点数的平均值。本文证明，n个顶点的图的控制集局部平均阶至少为$\frac{n+1}{2}$，当且仅当固定顶点的度数为$n-1$时取等号。此外，对于n个顶点且无孤立顶点的图，证明控制集局部平均阶的上界为$\frac{5n-1}{6}$。另外，给出固定顶点度数为$n-2$时控制集局部平均阶精确公式的证明，并确定固定顶点为l-茎（$l \geq 2$）时控制集局部平均阶的上界。
英文摘要
The (global) average order of dominating sets of a graph is the average number of vertices of its dominating sets. Analogously, the local average order of dominating sets is the average number of vertices of its dominating sets containing a fixed vertex. In this paper, we show that the local average order of dominating sets of a graph with $n$ vertices is at least $\frac{n+1}{2}$, with equality if and only if the degree of the fixed vertex is $n-1$. Furthermore, for a graph on $n$ vertices without isolated vertices, we show that $\frac{5n-1}{6}$ is an upper bound for the local average order of dominating sets. Additionally, we give a proof of an exact formula for the local average order of dominating sets when the degree of the fixed vertex is $n-2$, and determine an upper bound for the local average order of dominating sets when the fixed vertex is an $l$-stem ($l \geq 2$).
