Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues

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

Mysteries around the graph Laplacian eigenvalue 4

We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue distribution for such a tree is a smooth bell-shaped curve starting from the eigenvalue 0 up to 4. Then, at the eigenvalue 4, there is a sudden jump. Interestingly, the eigenvectors corresponding to the eigenvalues below 4 are semi-global oscillations (like Fourier modes) over the entire tree or one of the branches; on the other hand, those corresponding to the eigenvalues above 4 are much more localized and concentrated (like wavelets) around junctions/branching vertices. For a special class of trees called starlike trees, we obtain a complete understanding of such phase transition phenomenon. For a general graph, we prove the number of the eigenvalues larger than 4 is bounded from above by the number of vertices whose degrees is strictly higher than 2. Moreover, we also prove that if a graph contains a branching path, then the magnitudes of the components of any eigenvector corresponding to the eigenvalue greater than 4 decay exponentially from the branching vertex toward the leaf of that branch.Comment: 22 page

Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues

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

Genome-Wide Structural Variation Detection by Genome Mapping on Nanochannel Arrays.

Comprehensive whole-genome structural variation detection is challenging with current approaches. With diploid cells as DNA source and the presence of numerous repetitive elements, short-read DNA sequencing cannot be used to detect structural variation efficiently. In this report, we show that genome mapping with long, fluorescently labeled DNA molecules imaged on nanochannel arrays can be used for whole-genome structural variation detection without sequencing. While whole-genome haplotyping is not achieved, local phasing (across >150-kb regions) is routine, as molecules from the parental chromosomes are examined separately. In one experiment, we generated genome maps from a trio from the 1000 Genomes Project, compared the maps against that derived from the reference human genome, and identified structural variations that are >5 kb in size. We find that these individuals have many more structural variants than those published, including some with the potential of disrupting gene function or regulation

Genome-Wide Structural Variation Detection by Genome Mapping on Nanochannel Arrays

Comprehensive whole-genome structural variation detection is challenging with current approaches. With diploid cells as DNA source and the presence of numerous repetitive elements, short-read DNA sequencing cannot be used to detect structural variation efficiently. In this report, we show that genome mapping with long, fluorescently labeled DNA molecules imaged on nanochannel arrays can be used for whole-genome structural variation detection without sequencing. While whole-genome haplotyping is not achieved, local phasing (across >150-kb regions) is routine, as molecules from the parental chromosomes are examined separately. In one experiment, we generated genome maps from a trio from the 1000 Genomes Project, compared the maps against that derived from the reference human genome, and identified structural variations that are >5 kb in size. We find that these individuals have many more structural variants than those published, including some with the potential of disrupting gene function or regulation