Spectral graph theory is a branch of graph theory which finds relationships between structural properties of graphs and eigenvalues of matrices corresponding to graphs. In this thesis, I obtain sufficient eigenvalue conditions for the existence of edge-disjoint spanning trees in regular graphs, and I show this is best possible. The vertex toughness of a graph is defined as the minimum value of [special characters omitted], where S runs through all subsets of vertices that disconnect the graph, and c(G\S ) denotes the number of components after deleting S. I obtain sufficient eigenvalue conditions for a regular graph to have toughness at least 1, and I show this is best possible. Furthermore, I determine the toughness value for many families of graphs, and I classify the subsets S of each family for when this value is obtained.
|Advisor:||Cioaba, Sebastian M.|
|Commitee:||Cioaba, Sebastian, Coulter, Robert, Lazebnik, Felix, Shader, Bryan|
|School:||University of Delaware|
|Department:||Department of Mathematical Sciences|
|School Location:||United States -- Delaware|
|Source:||DAI-B 75/01(E), Dissertation Abstracts International|
|Keywords:||Eigenvalues, Graph theory, Graph toughness, Regular graphs, Spanning trees, Spectral graph theory|
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