# Maximum Nullity and Forcing Number on Graphs with Maximum Degree at most Three

## TL;DR

This paper characterizes all graphs with maximum degree at most three that have a forcing number of three, and classifies these graphs according to their maximum nullity, linking dynamic coloring and algebraic properties.

## Contribution

It provides a complete characterization of graphs with maximum degree three and forcing number three, and classifies these graphs based on their maximum nullity, a novel connection between graph dynamics and linear algebra.

## Key findings

- Identifies all graphs with maximum degree ≤ 3 and forcing number 3.
- Classifies these graphs according to their maximum nullity.
- Establishes a relationship between forcing number and maximum nullity for these graphs.

## Abstract

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $F$ of colored vertices, with all remaining vertices being non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $F$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. The forcing number of a graph $G$, denoted by $F(G)$, is the cardinality of a minimum forcing set of $G$. The maximum nullity of $G$, denoted by $M(G)$, is defined to be the largest possible nullity over all real symmetric matrices $A$ whose $a_{ij} \neq 0$ for $i \neq j$, whenever two vertices $u_{i}$ and $u_{j}$ of $G$ are adjacent. In this paper, we characterize all graphs $G$ of order $n$, maximum degree at most three, and $F(G)=3$. Also we classify these graphs with their maximum nullity.