Consider the discrete Laplacian operator A acting on l2(Z). It is well known from classical literature that essential spectrum of a compact interval. In this article, we give an elementary proof for result, using finite-dimensional truncations An A. We do not rely symbol analysis or any infinite-dimensional arguments. Instead, consider eigenvalue-sequences and make use filtration techniques due...

Brouwer's Conjecture states that, for any graph G , the sum of k largest (combinatorial) Laplacian eigenvalues is at most | E ( ) + 1 2 ≤ n . We present several interrelated results establishing conjecture a wide range graphs and parameters In particular, we show that (1) true low-arboricity graphs, in particular planar when ≥ 11 ; (2) whenever variance degree sequence not very high, generalizi...

The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. S...

A graph is said to be Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. Using combinatorial and matrix-theoretic techniques, we identify, up to isomorphism, the 21 connected Laplacian integral graphs of maximum degree 3 on at least 6 vertices.

This paper focuses on optimal estimators of the magnitude spectrum for speech enhancement. We present an analytical solution for estimating in the MMSE sense the magnitude spectrum when the clean speech DFT coefficients are modeled by a Laplacian distribution and the noise DFT coefficients are modeled by a Gaussian distribution. Furthermore, we derive the MMSE estimator under speech presence un...

Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible neg...

Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible neg...

This paper presents an alternative means of deriving and discretizing spectral distances and kernels on a 3D shape by filtering its Laplacian spectrum. Through the selection of a filter map, we design new spectral kernels and distances, whose smoothness and encoding of both local and global properties depend on the convergence of the filtered Laplacian eigenvalues to zero. Approximating the dis...