One of the fundamental problems is to study the eigenvalue problem for the differential operator in geometric analysis. In this article, we introduce the recent developments of the eigenvalue problem for the Finsler Laplacian. M.S.C. 2010: 53C60; 35P30; 35J60.

In this paper, we investigate connected nonregular graphs with four distinct Laplacian eigenvalues. We characterize all such graphs which are bipartite or have exactly one multiple Laplacian eigenvalue. Other examples of interest are also presented.

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 ...

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...

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...

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...

The family G of connected graphs with second largest Laplacian eigenvalue at most θ, where θ = 3.2470 is the largest root of the equation μ−5μ+6μ−1 = 0, is characterized by Wu, Yu and Shu [Y.R. Wu, G.L. Yu and J.L. Shu, Graphs with small second largest Laplacian eigenvalue, European J. Combin. 36 (2014) 190–197]. Let G(a, b, c, d) be a graph with order n = 2a + b + 2c + 3d + 1 that consists of ...

Let G be a connected simple graph. The relationship between the third smallest eigenvalue of the Laplacian matrix and the graph structure is explored. For a tree the complete description of the eigenvector corresponding to this eigenvalue is given and some results about the multiplicity of this eigenvalue are given.

Spectral techniques are often used to partition the set of vertices a graph, or form clusters. They based on Laplacian matrix. These allow easily integrate weights edges. In this work, we introduce p-Laplacian, generalized matrix with potential, which also allows us take into account vertices. vertex independent edge weights. way, can cluster importance vertices, assigning more weight some than...