The zero forcing polynomial of a graph

TL;DR

This paper introduces the zero forcing polynomial to count zero forcing sets in graphs, characterizes its extremal coefficients, and explores its structural properties across different graph families.

Contribution

It defines the zero forcing polynomial, derives formulas for specific graph families, and analyzes its structural properties such as multiplicativity and unimodality.

Findings

Closed-form expressions for zero forcing polynomials of certain graphs

Characterization of extremal coefficients of the polynomial

Insights into structural properties like unimodality and multiplicativity

Abstract

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph $G$ of order $n$ as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is the number of zero forcing sets of $G$ of size $i$. We characterize the extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of $\mathcal{Z}(G;x)$, including multiplicativity, unimodality, and uniqueness.