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The Minimum Number of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph is a Tree
Zimmerman, Jacob
Zimmerman, Jacob
Abstract
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.
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2022-05-01
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Mathematics
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