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For dendrite graphs from biological experiments on mouses 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.
In this paper, we address the question of comparison between populations of trees. We study an statistical test based on the distance between empirical mean trees, as an analog of the two sample z statistic for comparing two means. Despite its simpli
Many machine learning tools for regression are based on recursive partitioning of the covariate space into smaller regions, where the regression function can be estimated locally. Among these, regression trees and their ensembles have demonstrated im
Since their inception in the 1980s, regression trees have been one of the more widely used non-parametric prediction methods. Tree-structured methods yield a histogram reconstruction of the regression surface, where the bins correspond to terminal no
In this paper, we use a new and correct method to determine the $n$-vertex $k$-trees with the first three largest signless Laplacian indices.
Brouwers Conjecture states that, for any graph $G$, the sum of the $k$ largest (combinatorial) Laplacian eigenvalues of $G$ is at most $|E(G)| + binom{k+1}{2}$, $1 leq k leq n$. We present several interrelated results establishing Brouwers conjecture