Ordered multiplicity inverse eigenvalue problem for graphs on six vertices

John Ahn; Christine Alar; Beth Bjorkman; Steve Butler; Joshua Carlson; Audrey Goodnight; Haley Knox; Casandra Monroe; Michael C. Wigal

arXiv · 1708.02438

Ordered multiplicity inverse eigenvalue problem for graphs on six vertices

John Ahn·Christine Alar·Beth Bjorkman·Steve Butler·Joshua Carlson·Audrey Goodnight·Haley Knox·Casandra Monroe·Michael C. Wigal

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Abstract

For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\min(m,n)\ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.

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John Ahn, Christine Alar, Beth Bjorkman, Steve Butler, Joshua Carlson, Audrey Goodnight, Haley Knox, Casandra Monroe, Michael C. Wigal. 2017-08-08. Ordered multiplicity inverse eigenvalue problem for graphs on six vertices. https://arxiv.org/abs/1708.02438

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