Bounds for the Zero Forcing Number of Graphs with Large Girth

Authors

Randy Davila, Rice UniversityFollow
Franklin Kenter, Rice UniversityFollow

Publication Date

2015

Abstract

The zero-forcing number, Ζ(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 ≤ Ζ(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 ≤ Ζ(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 Ζ(G) as a function of the girth, and δ; this conjecture is proved in a few cases and numerical evidence is provided.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Davila, Randy and Kenter, Franklin (2015) "Bounds for the Zero Forcing Number of Graphs with Large Girth," Theory and Applications of Graphs: Vol. 2: Iss. 2, Article 1.
DOI: 10.20429/tag.2015.020201
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol2/iss2/1