Skew propagation time

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2015-01-01
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Kingsley, Nicole
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Leslie Hogben
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Mathematics
Abstract

The zero forcing number has long been used as a tool for determining the maximum nullity of a graph, and has since been extended to skew zero forcing number, which is the zero forcing number used when the matrices corresponding to the graphs in question are required to have zeros as their diagonal entries. Skew zero forcing is based on the following specific color change rule: for a graph $G$ (that does not contain any loops), where some vertices are colored blue and the rest of the vertices are colored white, if a vertex has only one white neighbor, that vertex forces its white neighbor to be blue. The minimum number of blue vertices that it takes to force the graph using this color change rule is called the skew zero forcing number. A set of blue vertices of order equal to the skew zero forcing number of the graph, such that when the skew zero forcing process is carried out to completion the entire graph is colored blue, is called a minimum skew zero forcing set. More recently, the concept of propagation time of a graph was introduced. Propagation time of a graph is how fast it is possible to force the entire graph blue over all possible minimum zero forcing sets. In this thesis, the concept of propagation time is extended to skew propagation time. We discuss the tools used to study extreme skew propagation time, and examine the skew propagation time of several common families of graphs. Finally, we include a brief discussion on loop graph propagation time, where the graphs in question are allowed to contain loops (in other words, these graphs are loop graphs, not simple graphs, as in the skew zero forcing case).

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Thu Jan 01 00:00:00 UTC 2015