[PAST EVENT] Honors Thesis Defense - Jacob Zimmerman
Title: "The Minimum Number of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph is a Tree"
Let T be a tree, let S(T) denote the set of real symmetric matrices whose graph is T, and let U(T) be the minimum number of eigenvalues with multiplicity 1 among matrices in S(T). We show that if T ? is a tree constructed by adding a pendent vertex to a diameter 5 tree T such that d(T) = d(T) ? , then U(T ? ) ? U(T). We then take this result as motivation to show results that are consistent with and suggest that if T ? is a tree constructed by adding a pendent vertex to a tree T such that d(T) = d(T) ? , then U(T ? ) ? U(T). In particular, we show that the result is true for the case where T and T ? are both linear trees, the case where T is a linear tree and T ? is a nonlinear tree, and the case where T and T ? are both minimally nonlinear trees such that U(T) = 2. We then construct a partial proof for the same and similar results where T and T ? are general trees.