Graph toughness from Laplacian eigenvalues

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...

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.

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

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)...

On Laplacian Eigenvalues of a Graph

Let G be a connected graph with n vertices and m edges. The Laplacian eigenvalues are denoted by μ1(G) ≥ μ2(G) ≥ ·· · ≥ μn−1(G) > μn(G) = 0. The Laplacian eigenvalues have important applications in theoretical chemistry. We present upper bounds for μ1(G)+ · · ·+μk(G) and lower bounds for μn−1(G)+ · · ·+μn−k(G) in terms of n and m, where 1 ≤ k ≤ n−2, and characterize the extremal cases. We also ...