Tree simplification and the ‘plateaux’ phenomenon of graph Laplacian eigenvalues

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 ...

Graph Embeddings and Laplacian Eigenvalues

Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n×n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix Γ; the best possible bound based on this embedding is n/λmax(Γ Γ). Howeve...

Laplacian eigenvalues and optimality: II. The Laplacian of a graph

Bounds for Laplacian Graph Eigenvalues

Let G be a connected simple graph whose Laplacian eigenvalues are 0 = μn (G) μn−1 (G) · · · μ1 (G) . In this paper, we establish some upper and lower bounds for the algebraic connectivity and the largest Laplacian eigenvalue of G . Mathematics subject classification (2010): 05C50, 15A18.

Convex Optimization of Graph Laplacian Eigenvalues

We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function)...