  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.
