#### Date of Award

2023

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Puck Rombach

#### Abstract

Centrality measures assign numbers or rankings to network nodes that reflect their importance. There are many types of centrality measures, each suitable for different types of networks and applications. In Chapter 2, we consider a model of astronaut health during a space mission. Katz centrality is commonly used to measure the influence of nodes in social and biological networks. We motivate its use in this application to estimate the expected quality time lost due to the progression of medical conditions. In Chapter 3, we find dominating sets in satellite networks. To do this, we use the Shapley value, a centrality measure that originates in game theory and is computed based only on local network information. We demonstrate that the Shapley value is an effective and time-efficient tool for finding small dominating sets in various random graph families and a set of real-world networks. In Chapter 4, we introduce a novel algorithm for categorizing which nodes are nearest the boundary, called boundary nodes, in a network that uses Chvátal’s definition of a line in a graph. We test this algorithm on random geometric graphs and compare its effectiveness to other known methods for boundary node detection.

In Chapter 5, for certain sets S and equations eq, we look for the smallest number of colors rb(S, eq) such that for every surjective rb(S, eq)-coloring of S, there exists a solution to eq where every element of the solution set is assigned a distinct color. We show that rb([n], x_1 + x_2 = x_3) = ⌊log_2(m) + 2⌋ and rb([m] × [n], x_1 + x_2 = x_3) = m + n + 1 for m, n > 1. In Chapter 6, a graph G is H-semi-saturated if adding an edge e to G that is not currently in G causes H to appear as a subgraph in G that contains e. We say that G is H-saturated if G does not contain H as a subgraph before adding e. The smallest number of edges in an H-semi-saturated (H-saturated) graph is called the semi-saturation number of H (saturation number of H). We show that the saturation and semi-saturation numbers differ by at least 1 for a disjoint union of paths called a linear forest. Additionally, we find graph families for which the saturation number is monotonic with respect to the subgraph relation.

#### Language

en

#### Number of Pages

113 p.

#### Recommended Citation

Rehm, Hunter Dane, "Applications of Centrality Measures and Extremal Combinatorics" (2023). *Graduate College Dissertations and Theses*. 1702.

https://scholarworks.uvm.edu/graddis/1702