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Subdivision and Graph Eigenvalues. (arXiv:2303.10488v1 [math.CO])

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This paper investigates the asymptotic nature of graph spectra when some
edges of a graph are subdivided sufficiently many times. In the special case
where all edges of a graph are subdivided, we find the exact limits of the
$k$-th largest and $k$-th smallest eigenvalues for any fixed $k$. It is
expected that after subdivision, most eigenvalues of the new graph will lie in
the interval $[-2,2]$. We examine the eigenvalues of the new graph outside this
interval, and we prove several results that might be of independent interest.

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