Bounds for the Zero Forcing Number of Graphs with Large Girth

• Main Authors: Randy Davila, Franklin Kenter
• Format: Article
• Language: English
• Published: Georgia Southern University 2015-01-01
• Series: Theory and Applications of Graphs
• Subjects: zero forcing set; zero forcing number; k-forcing set; k-forcing number; girth; triangle-free graphs
• Online Access: https://digitalcommons.georgiasouthern.edu/tag/vol2/iss2/1

The zero-forcing number, Z(G) is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the graph. A simple lower bound is δ ≤ Z(G) where δ is the minimum degree. An improvement of this bound is provided in the case that G has girth of at least 5. In particular, it is shown that 2δ − 2 ≤ Z(G) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Lastly, a conjecture is made regarding a lower bound for Z(G) as a function of the girth, g, and δ; this conjecture is proved in a few cases and numerical evidence is provided.