#### Publication Date

5-2006

#### Advisor(s) - Committee Chair

Claus Ernst

#### Degree Program

Department of Mathematics

#### Degree Type

Master of Science

#### Abstract

Applications of graph theory abound in the real world, and often in studying graphs that represent real-world situations, practical solutions to existing problems are found. Consider a graph in which the vertices represent computer stations and the edges represent connections between stations. At one extreme, removing a single station could cause disruption of communication among all other stations. At the other, removal of a single station could leave the rest of the network unaffected. The first network is perceived as very susceptible to disruption, while the second is not very susceptible. There are several different measures of this susceptibility, or vulnerability, of graphs.

One such measure is *integrity*. The integrity of a graph involves two variables. The first is the number of vertices that are removed from the graph, and the second is the size of the largest component that remains. A finer measure of vulnerability that is more closely examined in this thesis is the mean integrity, which looks at the weighted average of all remaining components, not just the size of the largest component. Both integrity and mean integrity are very difficult to computer. Thus, there are oly a few graph families for which the exact value of the integrity or mean integrity is known. For example, for complete graphs of *n *vertices, both the integrity and mean integrity are known to be *n*. Since removing no vertices leaves the original graph, one connected component of size *n, *the largest possible value for each of the integrities is *n*, which is achieved by the complete graphs. The path *P _{n}* ( a path of length

*n*) represents the other end of the spectrum, that is, a path is a connected graph of

*n*vertices in which both the mean integrity and the integrity are very small. The integrity of

*P*is known to be [2√

_{n}*n*+1] – 2. Surprisingly, there is no exact formula for the mean integrity in the literature. The construction and proof of such a formula is the main topic of this thesis.

#### Disciplines

Mathematics | Physical Sciences and Mathematics

#### Recommended Citation

Rountree, Beth, "The Mean Integrity of a Path" (2006). *Masters Theses & Specialist Projects.* Paper 3432.

https://digitalcommons.wku.edu/theses/3432

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