Ultraconnected and Critical Graphs

Jason Grout
We investigate the ultraconnectivity condition on graphs, and provide further connections between critical and ultraconnected graphs in the positive definite partial matrix completion problem. We completely characterize when the join of graphs is ultraconnected, and prove that ultraconnectivity is preserved by Cartesian products. We completely characterize when adding a vertex to an ultraconnected graph preserves ultraconnectivity. We also derive bounds on the number of vertices which guarantee ultraconnectivity of certain classes of regular graphs. We...
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