On Laplacian spectrum of dendrite trees

TL;DR

This paper investigates the spectral properties of dendrite trees, revealing a relationship between eigenvalue spikes and T-junctions, and validates predictions with biological data.

Contribution

It provides a theoretical analysis linking eigenvalue spikes to T-junctions in uniform trees, explaining previously mysterious spectral phenomena in dendrite graphs.

Findings

Exact formulas relating spikes to T-junctions

Predictions match observed spike counts in real dendrite graphs

Theoretical insights explain spectral phase transitions

Abstract

For dendrite graphs from biological experiments on mouse's retinal ganglion cells, a paper by Nakatsukasa, Saito and Woei reveals a mysterious phase transition phenomenon in the spectra of the corresponding graph Laplacian matrices. While the bulk of the spectrum can be well understood by structures resembling starlike trees, mysteries about the spikes, that is, isolated eigenvalues outside the bulk spectrum, remain unexplained. In this paper, we bring new insights on these mysteries by considering a class of uniform trees. Exact relationships between the number of such spikes and the number of T-junctions are analyzed in function of the number of vertices separating the T-junctions. Using these theoretic results, predictions are proposed for the number of spikes observed in real-life dendrite graphs. Interestingly enough, these predictions match well the observed numbers of spikes, thus confirm the practical meaningness of our theoretical results.