The solution of the eigenvalue problem of the Laplacian on a general homogeneous space G/H is given. Here G is a compact, semi-simple Lie group, H is a closed subgroup of G, and the rank of H is equal to the rank of G. It is shown that the multiplicity of the lowest eigenvalue of the Laplacian on G/H is just the degeneracy of the lowest Landau level for a particle moving on G/H in the presence ...

Detecting protein complexes is an important way to discover the relationship between network topological structure and its functional features in protein-protein interaction (PPI) network. The spectral clustering method is a popular approach. However, how to select its optimal Laplacian matrix is still an open problem. Here, we analyzed the performances of three graph Laplacian matrices (unnorm...

A graph can be associated with a matrix in several ways. For instance, by associating the vertices of the graph to the rows/columns and then using 1 to indicate an edge and 0 otherwise we get the adjacency matrix A. The combinatorial Laplacian matrix is defined by L = D − A where D is a diagonal matrix with diagonal entries the degrees and A is again the adjacency matrix. Both of these matrices...

In this class, we introduced the random walk on graphs. The last lecture shows Perron-Frobenius theory to the analysis of primary eigenvectors which is the stationary distribution. In this lecture we will study the second eigenvector. To analyze the properties of the graph, we construct two matrices: one is (unnormalized) graph Laplacian and the other is normalized graph Laplacian. In the first...

Let Ω ⊂ R2 be an arbitrary simply connected domain in the plane. We define RΩ = supz∈Ω dΩ(z) (the inradius of the domain) where dΩ(z) is the distance from z to the boundary of Ω. Let σΩ(z) be the density of the hyperbolic metric in Ω and let σΩ = infz∈Ω σΩ(z). Finally, denote by λ1 the lowest eigenvalue for the Dirichlet Laplacian in Ω and denote by τΩ the first exit time of Brownian motion fro...

We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any ε, δ ∈ (0, 1), given cost and demand graphs with edge weights C,D : ( V 2 ) → R+ respectively, we can find a set T ⊆ V with C(T,V \T ) D(T,V \T ) at most 1+ε δ times the optimal non-uniform sparsest cut value, in time 2 poly(n) provided λr ≥ Φ∗/(1 − δ). Here λr is the r’th smallest generalized ...

The analogue of Pólya’s conjecture is shown to fail for the fractional Laplacian (−∆) on an interval in 1-dimension, whenever 0 < α < 2. The failure is total: every eigenvalue lies below the corresponding term of the Weyl asymptotic. In 2-dimensions, the fractional Pólya conjecture fails already for the first eigenvalue, when 0 < α < 0.984. Introduction. The Weyl asymptotic for the n-th eigenva...

Abstract We study the asymptotics of Dirichlet eigenvalues and eigenfunctions fractional Laplacian $$(-\Delta )^s$$ (-Δ)s in bounded open Lipschitz sets small order limit $$s \rightarrow 0^+$$ xmlns:mml...

We construct two infinite families of trees that are pairwise cospectral with respect to the normalized Laplacian. We also use the normalized Laplacian applied to weighed graphs to give new constructions of cospectral pairs of bipartite unweighted graphs.

2 Eigenvalues of graphs 5 2.1 Matrices associated with graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The largest eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3...