Bounds on expected propagation time of probabilistic zero forcing

TL;DR

This paper establishes new upper and lower bounds on the expected propagation time in probabilistic zero forcing on graphs, improving understanding of how graph properties influence the process's speed.

Contribution

It introduces improved bounds on expected propagation time based on graph order and radius, using probabilistic and combinatorial techniques.

Findings

Expected propagation time is O(r log(n/r)) for graphs of order n and radius r.

Expected propagation time is at most n/2 + O(log n).

Lower bound on expected propagation time is log_2 log_2 n.

Abstract

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 $\text{ept}(G)\ge \log_2 \log_2 n.$