Probabilistic zero forcing is a coloring game played on a graph where the goal is to color every vertex blue starting with an initial blue vertex set. As long as the graph is connected, if at least one vertex is blue then eventually all of the vertices will be colored blue. The most studied parameter in probabilistic zero forcing is the expected propagation time starting from a given vertex of $G.$ In this paper we improve on upper bounds for the expected propagation time by Geneson and Hogben and Chan et al. in terms of a graph's order and radius. In particular, for a connected graph $G$ of order $n$ and radius $r,$ we prove the bound $\text{ept}(G) = O(r\log(n/r)).$ We also show using Doob's Optional Stopping Theorem and a combinatorial object known as a cornerstone that $\text{ept}(G) \le n/2 + O(\log n).$ Finally, we derive an explicit lower bound $\text{ept}(G)\ge \log_2 \log_2 n.$
翻译：概率零强迫是一个在图形上播放的彩色游戏, 目标是从最初的蓝色顶点设置开始, 给每个顶点上色, 从最初的蓝色顶点设置开始 。 只要图形连接, 如果至少一个顶点是蓝色的, 那么最终所有顶点都将是蓝色的。 概率零强迫中研究最多的参数是从给定的顶点G美元开始的预期传播时间。 本文中, 我们用图表的顺序和半径来改进Geneson和Hogben及Chan等人的预期传播时间的上界。 特别是, 对于一个连接的图表, $G$G$( 美元) 和半径为$r, 我们用 $\ text{ ept} (G) = O(r\log (n/r) 。 我们还用 Doob 的选项停止 Theorem 和一个被称作“ $text{ept} (G)\ n/2 + O(\log n) $, 我们用一个清晰的底框$\ text_\\\\\\\\ g} g) 显示一个基点的基点。 最后, 我们以明确的 $\\\\\\\\\\\\ g\\\\\\ g\ g\ g\ g\\\\\ g\\\\\\ g\ g\ g\ g\ g\\\\\\\\\ g\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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