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Full-Text Articles in Physical Sciences and Mathematics
Contractible Theta Complexes Of Graphs, Chelsea Marian Mcamis
Contractible Theta Complexes Of Graphs, Chelsea Marian Mcamis
Masters Theses
We examine properties of graphs that result in the graph having a contractible theta complex. We classify such properties for tree graphs and graphs with one loop and we introduce examples of graphs with such properties for tree graphs and graphs with one or two loops. For more general graphs, we show that having a contractible theta complex is not an elusive property, and that any skeleton of a graph with at least three loops can be made to have a contractible theta complex by strategically adding vertices to its skeleton.
Diagonal Entry Restrictions In Minimum Rank Matrices, And The Inverse Inertia And Eigenvalue Problems For Graphs, Curtis G. Nelson
Diagonal Entry Restrictions In Minimum Rank Matrices, And The Inverse Inertia And Eigenvalue Problems For Graphs, Curtis G. Nelson
Theses and Dissertations
Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in …