Weighted graph Laplacians and isoperimetric inequalities

Weighted Graph Laplacians and Isoperimetric Inequalities

We consider a weighted Cheeger’s constant for a graph and we examine the gap between the first two eigenvalues of Laplacian. We establish several isoperimetric inequalities concerning the unweighted Cheeger’s constant, weighted Cheeger’s constants and eigenvalues for Neumann and Dirichlet conditions .

On weighted isoperimetric and Poincaré-type inequalities

Weighted isoperimetric and Poincaré-type inequalities are studied for κ-concave probability measures (in the hierarchy of convex measures).

Weighted Laplacians and the Sigma Function of a Graph

We consider a general notion of the Laplacian of a graph. The weight of an edge reflects both the width and the length of an edge. Further, we allow the edge weights to vary in order to minimize the maximum eigenvalue, and using this minimum we construct the so-called σ−function of a graph. We consider a geometric interpretation of the σ−function, in particular as it applies to the detection of...

Isoperimetric Inequalities and Eigenvalues

Isoperimetric Inequalities and Eigenvalues

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...