Spanning trees : eigenvalues, special numbers, and the tree-cover ratios, asymptotes and areas of graphs.
This dissertation deals with spanning trees associated with graphs. The number of spanning trees of a graph can be found by considering the eigenvalues of the Laplacian matrix associated with that graph. Special classes of graphs are also considered, such as fan and wheel graphs, where their spanning tree numbers are connected to special numbers, like the Lucas and Fibonacci numbers. We use the eigenvalues of the complete graph and its associated circulant matrix to create a unit-trigonometric equation which generates a sequence and diagram similar to that of the famous Farey sequence. A new ratio is introduced: the tree-cover ratio involving spanning trees and vertex coverings and is motivated by the fact that such a ratio, associated with complete graphs, has the asymptotic convergence identical to that of the secretary problem. We use this ratio to introduce the idea of tree-cover asymptotes and areas and determine such values for known classes of graphs. This ratio, in communication networks, allows for the investigation of the outward social connectivity from a vertex covering to the rest of the network when a large number of vertices are involved.