Hicks, Illya V2016-01-072016-01-072015-052015-04-16May 2015Davila, Randy R. "Bounding the Forcing Number of a Graph." (2015) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/87761">https://hdl.handle.net/1911/87761</a>.https://hdl.handle.net/1911/87761The forcing number, denoted F(G), is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the simple graph G. Simple lower and upper bounds are δ ≤ F(G) where δ is the minimum degree and F (G) ≤ n − 1 where n is the order of the graph. This thesis provides improvements on the minimum degree lower bound in the case that G has girth of at least 5. In particular, it is shown that 2δ − 2 ≤ F (G) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Further, this thesis also conjectures a lower bound on F(G) as a function of the girth, g, and δ.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Zero Forcing Numberk-Forcing NumberThesis2016-01-07