This paper studies a generalization of the standard continuous-time consensus protocol, obtained by replacing the Laplacian matrix of the communication graph with the so-called deformed Laplacian. The deformed Laplacian is a second-degree matrix polynomial in the real variable s which reduces to the standard Laplacian for s equal to unity. The stability properties of the ensuing deformed consen...

Let G be a graph and L(G) be the Laplacian matrix of G. In this article, we first point out that the sequence of the moduli of Laplacian coefficients of G is log-concave and hence unimodal. Using this fact, we provide an upper bound for the partial sums of the Laplacian eigenvalues of G, based on coefficients of its Laplacian characteristic polynomial. We then obtain some lower bounds on the al...

This paper studies a generalization of the standard continuous-time consensus protocol, obtained by replacing the Laplacian matrix of the communication graph with the so-called deformed Laplacian. The deformed Laplacian is a second-degree matrix polynomial in the real variable s which reduces to the standard Laplacian for s equal to unity. The stability properties of the ensuing deformed consen...

The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...

The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...

In this paper we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and we call them graph Laplacian quantum states. Then we obtain structural properties of these graphs such that the corresponding graph Laplacian states have zero quantum discord by investigating structural properties o...

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. At the end we will also discuss the nonlinear Laplacian on a graph, a nonlinear generalization of the graph Laplacian as its name implies. These generalized Laplacians will be constructed out of coboun...

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...

Parallel to the signless Laplacian spectral theory, we introduce and develop the nonlinear spectral theory of signless 1-Laplacian on graphs. Again, the first eigenvalue μ1 of the signless 1-Laplacian precisely characterizes the bipartiteness of a graph and naturally connects to the maxcut problem. However, the dual Cheeger constant h+, which has only some upper and lower bounds in the Laplacia...

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...