# Graduate Seminar Series

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Tue., Nov. 7, 2023 11:30 a.m.
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Location: ED 438 and Live Stream
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**Speaker**: Johnna Parenteau

**Title:** A Smörgåsbord of Cycles, Matchings, and Multiplicities (72 kB)

**Abstract:**

One of the most important historical contributions to the inverse eigenvalue problem associated with trees is the celebrated Parter-Wiener Theorem. This theorem states for any tree, *T*, and any associated real symmetric matrix, *A*, with a corresponding eigenvalue λ of multiplicity at least two, there exists some vertex *v* in *T* that, upon deletion from *T*, results in a strict increase in the multiplicity of the eigenvalue λ; however, there is no analogous result that can be extended beyond trees using the characteristic polynomial. In this presentation, we study a different polynomial, called the weighted matching polynomial, and its related graphical properties such as matchings and vertex-deleted subgraphs in order to derive properties on the roots and their multiplicities. For graphs containing a Hamilton path, we are able to establish a special case of the extended Parter-Wiener Theorem for hollow matrices.

**Live Stream:**