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.
Davila, Randy and Kenter, Franklin
"Bounds for the Zero Forcing Number of Graphs with Large Girth,"
Theory and Applications of Graphs:
2, Article 1.
Available at: http://digitalcommons.georgiasouthern.edu/tag/vol2/iss2/1