Some Bounds on the Zero Forcing Number of a Graph

A set $Z$ of vertices of a graph $G$ is a zero forcing set of $G$ if initially labeling all vertices in $Z$ with $1$ and all remaining vertices of $G$ with $0$, and then, iteratively and as long as possible, changing the label of some vertex $u$ from $0$ to $1$ if $u$ is the only neighbor with label $0$ of some vertex with label $1$, results in the entire vertex set of $G$. The zero forcing number $Z(G)$, defined as the minimum order of a zero forcing set of $G$, was proposed as an upper bound of the corank of matrices associated with $G$, and was also considered in connection with quantum physics and logic circuits. In view of the computational hardness of the zero forcing number, upper and lower bounds are of interest. Refining results of Amos, Caro, Davila, and Pepper, we show that $Z(G)\leq \frac{\Delta-2}{\Delta-1}n$ for a connected graph $G$ of order $n$ and maximum degree $\Delta$ at least $3$ if and only if $G$ does not belong to $\{ K_{\Delta+1},K_{\Delta,\Delta},K_{\Delta-1,\Delta},G_1,G_2\}$, where $G_1$ and $G_2$ are two specific graphs of orders $5$ and $7$, respectively. For a connected graph $G$ of order $n$, maximum degree $3$, and girth at least $5$, we show $Z(G)\leq \frac{n}{2}-\Omega\left(\frac{n}{\log n}\right)$. Using a probabilistic argument, we show $Z(G)\leq \left(1-\frac{H_r}{r}+o\left(\frac{H_r}{r}\right)\right)n$ for an $r$-regular graph $G$ of order $n$ and girth at least $5$, where $H_r$ is the $r$-th harmonic number. Finally, we show $Z(G)\geq (g-2)(\delta-2)+2$ for a graph $G$ of girth $g\in \{ 5,6\}$ and minimum degree $\delta$, which partially confirms a conjecture of Davila and Kenter.