Eigenvalues of Graph Laplacians Via Rank-One Perturbations

Eigenvalues of Transmission Graph Laplacians

The standard notion of the Laplacian of a graph is generalized to the setting of a graph with the extra structure of a “transmission” system. A transmission system is a mathematical representation of a means of transmitting (multi-parameter) data along directed edges from vertex to vertex. The associated transmission graph Laplacian is shown to have many of the former properties of the classica...

Analysis of Neuronal Dendrite Patterns Using Eigenvalues of Graph Laplacians

We report our current effort on extracting morphological features from neuronal dendrite patterns using the eigenvalues of their graph Laplacians and clustering neurons using those features into different functional cell types. Our preliminary results indicate the potential usefulness of such eigenvalue-based features, which we hope to replace the morphological features extracted by methods tha...

A Majorization Bound for the Eigenvalues of Some Graph Laplacians

Grone and Merris [5] conjectured that the Laplacian spectrum of a graph is majorized by its conjugate vertex degree sequence. In this paper, we prove that this conjecture holds for a class of graphs including trees. We also show that this conjecture and its generalization to graphs with Dirichlet boundary conditions are equivalent.

Rank one perturbations of self-adjoint operators

Optimizing Extremal Eigenvalues of Weighted Graph Laplacians and Associated Graph Realizations

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