# Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues (with E. Woei), *Linear Algebra and its Applications*, vol.481, pp.263-279, 2015.

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Abstract

We developed a procedure of reducing the number of vertices and edges of a given
tree, which we call the "tree simplification procedure," without changing
its topological information.
Our motivation for developing this procedure was to reduce computational
costs of graph Laplacian eigenvalues of such trees.
When we applied this procedure to a set of trees representing dendritic
structures of retinal ganglion cells of a mouse and computed their graph
Laplacian eigenvalues, we observed two "plateaux" (i.e., two sets of multiple
eigenvalues) in the eigenvalue distribution of each such simplified tree.
In this article, after describing our tree simplification procedure,
we analyze why such eigenvalue plateaux occur in a simplified tree, and
explain such plateaux can occur in a more general graph if it satisfies a
certain condition, identify these two eigenvalues specifically as well as
the lower bound to their multiplicity.
**Keywords:**
vertex reduction; graph Laplacian eigenvalues;
eigenvalue multiplicity; monic polynomials with integer coefficients

Get the full paper: PDF file (revised on 04/28/15).
Get the official version via doi:10.1016/j.laa.2015.05.004.

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