In this article, we study a class of matrix functions the combinatorial Laplacian that preserve its structure, i.e., define matrices, which are positive semidefinite and have zero row-sum nonpositive off-diagonal entries. This formulation has merit presenting different incarnations appeared in recent literature unified framework. For first time, apply family to consensus theory, show they leave...

A new distance measure between probability density functions (pdfs) is introduced, which we refer to as the Laplacian pdf distance. The Laplacian pdf distance exhibits a remarkable connection to Mercer kernel based learning theory via the Parzen window technique for density estimation. In a kernel feature space defined by the eigenspectrum of the Laplacian data matrix, this pdf distance is show...

The problem of fitting a union of subspaces to a collection of data points drawn from multiple subspaces is considered in this paper. In the traditional low rank representation model, the dictionary used to represent the data points is chosen as the data points themselves and thus the dictionary is corrupted with noise. This problem is solved in the low rank subspace clustering model which deco...

Let G be a weighted graph with Laplacian matrix L and signless Laplacian matrix Q. In this note, block representations for the group inverse of L and Q are given. The resistance distance in a graph can be obtained from the block representation of the group inverse of L.

Recall that, given a graph G, the matrix Q = D + A is called the signless Laplacian, where A is the adjacency matrix and D is the diagonal matrix of vertex degrees. The matrix L = D − A is known as the Laplacian of G. Graphs with the same spectrum of an associated matrix M are called cospectral graphs with respect to M , or M–cospectral graphs. A graph H cospectral with a graph G, but not isomo...

Let G be a simple graph with order n and size m. The quantity $$M_1(G)=\sum _{i=1}^{n}d^2_{v_i}$$ is called the first Zagreb index of G, where $$d_{v_i}$$ degree vertex $$v_i$$ , for all $$i=1,2,\dots ,n$$ . signless Laplacian matrix $$Q(G)=D(G)+A(G)$$ A(G) D(G) denote, respectively, adjacency diagonal degrees G. $$q_1\ge q_2\ge \dots \ge q_n\ge 0$$ eigenvalues largest eigenvalue $$q_1$$ spectr...

This paper presents a new voice activity detection (VAD) method using the Laplacian distribution and a uniformly most powerful (UMP) test. The UMP test is employed to derive the new decision rule based on likelihood ratio test (LRT). The proposed method provide the decision rule by comparing the sum of magnitude of real and imaginary parts of the noisy spectral component to the adaptive thresho...

We introduce the tensor numerical method for solution of d-dimensional optimal control problems (d=2,3) with spectral fractional Laplacian type operators in constraints discretized on large n?d tensor-product Cartesian grids. The approach is based rank-structured approximation matrix valued functions corresponding finite difference Laplacian. solve equation function, where system includes sum a...

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. After deriving and recalling relevant properties of its first eigenfunction for p > 1, we investigate the limit problem as p → 1.

Given a C-function f on a compact riemannian manifold (X, g) we give a set of frequencies L = Lf (ε) depending on a small parameter ε > 0 such that the relative L-error ‖f−f ‖ ‖f‖ is bounded above by ε, where f L denotes the L-partial sum of the Fourier series f with respect to an orthonormal basis of L(X) constituted by eigenfunctions of the Laplacian operator ∆ associated to the metric g.