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...

Abstract. A signed graph Γ = (G, σ) consists of an unsigned graph G = (V, E) and a mapping σ : E → {+,−}. Let Γ be a connected signed graph and L(Γ),L(Γ) be its Laplacian matrix and normalized Laplacian matrix, respectively. Suppose μ1 ≥ · · · ≥ μn−1 ≥ μn ≥ 0 and λ1 ≥ · · · ≥ λn−1 ≥ λn ≥ 0 are the Laplacian eigenvalues and the normalized Laplacian eigenvalues of Γ, respectively. In this paper, ...

We define the product distance matrix of a tree and obtain formulas for its determinant and inverse. The results generalize known formulas for the exponential distance matrix. When we restrict the number of variables to two, we are naturally led to define a bivariate analogue of the laplacian matrix of an arbitrary graph. We also define a bivariate analogue of the Ihara-Selberg zeta function an...

In this paper we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are una...

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...

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...

This paper describes a successful application of Matting Laplacian Matrix to the problem of generating high-resolution range images. The Matting Laplacian Matrix in this paper exploits the fact that discontinuities in range and coloring tend to co-align, which enables us to generate high-resolution range image by integrating regular camera image into the range data. Using one registered and pot...

In this paper, we give some properties of the left and right finite Caputo derivatives. Such derivatives lead to finite Riesz type fractional derivative, which could be considered as the fractional power of the Laplacian operator modelling the dynamics of many anomalous phenomena in super-diffusive processes. Finally, the exact solutions of certain fractional diffusion partial differential equa...

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

connections between integral kernels of positivity preserving semigroups and suitable Lp contractivity properties are established. Then these questions are studied for the semigroups generated by -A + V and H,, the Dirichlet Laplacian for an open, connected region Q. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to 352, of the form /n(x)1 ,< coo(x) lJHk,nl...