A Laplacian matrix, L = (lij) ∈ R , has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with − 1 n ≤ lij ≤ 0 at j 6= i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrice...

This paper presents a survey as well as a systematic empirical comparison of seven graph kernels and two related similarity matrices (simply referred to as graph kernels), namely the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regularized Laplacian kernel, the commute-time kernel, the random-walk-with-restart similarity matrix,...

in this paper main aim is to focus on to remove impulse noise from corrupted image. Here present a method for removing noise from digital images corrupted with additive, multiplicative, and mixed noise. Here used hybrid graph Laplacian regularized regression to perform progressive image recovery using unified framework. by using laplacian pyramid here build multi-scale representation of input i...

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with ...

We present an iterative algorithm for solving a class of nonlinear Laplacian system of equations in Õ(km log(kn/ǫ)) iterations, where k is a measure of nonlinearity, n is the number of variables, m is the number of nonzero entries in the graph Laplacian L, ǫ is the solution accuracy and Õ() neglects (non-leading) logarithmic terms. This algorithm is a natural nonlinear extension of the one by o...

Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define corona graphs. Given a small simple connected graph which we call basic graph, corona graphs are defined by taking corona product of the basic ...

The theory of dynamical networks is concerned with systems of dynamical units coupled according to an underlying graph structure. It therefore investigates the interplay between dynamics and structure, between the temporal processes going on at the individual units and the static spatial structure linking them. In order to analyse that spatial structure, formalized as a graph, we discuss an ess...

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Two graphs of the same order are said to be equienergetic if their energies are equal. We point out the following two open problems for equienergetic graphs. (1) Although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...

This paper investigates the effect of Laplacian normalization in graph-based semi-supervised learning. To this end, we consider multi-class transductive learning on graphs with Laplacian regularization. Generalization bounds are derived using geometric properties of the graph. Specifically, by introducing a definition of graph cut from learning theory, we obtain generalization bounds that depen...

The eigenspectrum of a graph Laplacian encodes smoothness information over the graph. A natural approach to learning involves transforming the spectrum of a graph Laplacian to obtain a kernel. While manual exploration of the spectrum is conceivable, non-parametric learning methods that adjust the Laplacian’s spectrum promise better performance. For instance, adjusting the graph Laplacian using ...