Zero forcing is a graph coloring process that was defined as a tool for bounding the minimum rank and maximum nullity of a graph. It has also been used for studying control of quantum systems and monitoring electrical power networks. One of the problems from the 2017 AIM workshop "Zero forcing and its applications" was to explore edge-weighted probabilistic zero forcing, where edges have weights that determine the probability of a successful force if forcing is possible under the standard zero forcing coloring rule.
In this paper, we investigate the expected time to complete the weighted zero forcing coloring process, known as the expected propagation time, as well as the time for the process to be completed with probability at least $α$, known as the $α$-confidence propagation time. We demonstrate how to find the expected and confidence propagation times of any edge-weighted graph using Markov matrices. We also determine the expected and confidence propagation times for various families of edge-weighted graphs including complete graphs, stars, paths, and cycles.
迫零是一种在图上玩的着色游戏，十多年前在几个不同的应用程序中引入。我们的目标是通过重复使用（确定性的）颜色更改规则，将所有顶点着色为蓝色。概率迫零是由Kang和Yi在[Probably zero forcing in graphs，Bull. Inst. Combin. Appl.67（2013），9- 16]，并且产生离散动力系统，其对于某些应用是更好的模型。由于在一个连通图中，任何一个顶点最终都可以使用概率迫零将整个图染成蓝色，因此完成此操作的预期时间是一个自然的参数。我们确定的路径和周期的预期传播时间，建立恒星的渐近值，并提出渐近的上限和下界的任何图形的半径和顺序。我们应用这些结果来获得$\ell$轮概率迫零，概率迫零的节流数，传播时间的置信水平的值和界限。
Zero forcing propagation time intervals and graphs with fixed propagation time
Texas State University（德克萨斯州立大学）；American Institute of Mathematics（美国数学研究所）；Iowa State University（爱荷华州立大学）；Purdue University（普渡大学）；Winston-Salem State University（温斯顿-塞勒姆州立大学）
The zero forcing numbers and propagation times of gear graphs and helm graphs
University of North Texas（北德克萨斯大学）；Rice University（莱斯大学）；Trinity College（三一学院）；University of Denver（丹佛大学）；Southeast Missouri State University（密苏里东南州立大学）
Power propagation time and lower bounds for power domination number
我们提出了一个反例的下界的权力控制数在廖，权力支配与有限的时间限制，J。31（2016）725-742。我们还使用Liao中的功率支配传播思想和Hogben等人的Propagation time for zero forcing on a graph，Discrete Appl.Math.160（2012）1994-2005中的（迫零）传播时间来定义功率传播时间。
