Abstract
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.
Highlights
IntroductionThe problem of determining the minimum rank of a graph seeks the minimum rank over all symmetric matrices whose sparsity pattern is determined by the graph
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
The problem of determining the minimum rank of a graph seeks the minimum rank over all symmetric matrices whose sparsity pattern is determined by the graph
Summary
The problem of determining the minimum rank of a graph seeks the minimum rank over all symmetric matrices whose sparsity pattern is determined by the graph. The zero forcing process, and the associated zero forcing number, were introduced by [1] and [6] in order to bound the minimum rank of a graph (and the maximum nullity). Given the minimum and maximum degree of a graph, δ and ∆ respectively, the zero forcing number on a graph with n vertices can be as low as δ. The only result concerning minimum rank or zero forcing and triangle-free graphs is given by Deaett [10] where it is shown that the semidefinite minimum rank of a triangle-free graph is bounded below by half the number of vertices.
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