[Paper Review] Maximum nullity and zero forcing number on cubic graphs

Saieed Akbari, Ebrahim Vatandoost+1 more|arXiv (Cornell University)|May 27, 2017

Graph theory and applications4 references3 citations

TL;DR

This paper characterizes cubic graphs with zero forcing number 3 and proves that for this family, the maximum nullity equals 3. It introduces a family of cubic graphs where maximum nullity and zero forcing number are both 4, and presents an algorithm linking maximum nullity to the number of leaves in a spanning tree. The work contributes to the open problem of identifying graphs where M(G) = Z(G).

ABSTRACT

Let $G$ be a graph. The maximum nullity of $G$, denoted by $M(G)$, is defined to be the largest possible nullity over all real symmetric matrices $A$ whose $a_{ij} eq 0$ for $i eq j$, whenever two vertices $u_i$ and $u_j$ of $G$ are adjacent. In this paper, we characterize all cubic graphs with zero forcing number $3$. As a corollary, it is shown that if the zero forcing number is $3$, then $M(G)=3$. In addition, we introduce a family of cubic graphs containing graphs $G$ with $M(G)=Z(G)=4$. Also, we provide an algorithm which make a relation between maximum nullity of $G$ and the number of leaves in a spanning tree of $G$.

Motivation & Objective

• To characterize all cubic graphs with zero forcing number 3.
• To establish a connection between maximum nullity and the number of leaves in a spanning tree of a cubic graph.
• To provide a family of cubic graphs where M(G) = Z(G) = 4.
• To contribute to the open problem of identifying graphs for which M(G) = Z(G).
• To challenge and provide a counterexample to Conjecture 1.2 on zero forcing number bounds in cubic graphs.

Proposed method

• Uses the zero forcing process: a black vertex with exactly one white neighbor forces that neighbor black, repeated until no more changes occur.
• Identifies zero forcing sets of size 3 in cubic graphs and characterizes graphs admitting such sets.
• Applies the color-change rule to derive the zero forcing number Z(G) and relates it to structural properties of cubic graphs.
• Constructs a spanning tree from the graph structure and uses the number of leaves to bound the maximum nullity.
• Employs spectral graph theory to derive a lower bound for maximum nullity using eigenvalues.
• Analyzes the Heawood graph as a case study to verify M(G) = Z(G) in a known cubic graph.

Experimental results

Research questions

• RQ1 Which cubic graphs have zero forcing number 3?
• RQ2 For cubic graphs with Z(G) = 3, is M(G) = 3?
• RQ3 Can a family of cubic graphs be constructed such that M(G) = Z(G) = 4?
• RQ4 Does the number of leaves in a spanning tree of a cubic graph provide a bound on M(G)?
• RQ5 Is Conjecture 1.2 (Z(G) ≤ n/3 + 2 for connected cubic graphs) valid, or can a counterexample be found?

Key findings

• All cubic graphs with zero forcing number 3 are completely characterized in the paper.
• For cubic graphs with Z(G) = 3, it is proven that M(G) = 3, confirming M(G) = Z(G) in this family.
• A family of cubic graphs is constructed where M(G) = Z(G) = 4.
• A counterexample to Conjecture 1.2 is presented: a cubic graph of order 16 with Z(G) = 8, exceeding n/3 + 2 = 16/3 + 2 ≈ 7.33.
• An algorithm is developed that relates the maximum nullity M(G) to the number of leaves in a spanning tree of G.
• The Heawood graph is shown to satisfy M(G) = Z(G), supporting the broader conjecture that M(G) = Z(G) holds for certain cubic graphs.