A concept related to the spectrum of a graph is that of energy. The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of G . The Laplacian energy of a graph G is equal to the sum of distances of the Laplacian eigenvalues of G and the average degree d(G) of G. In this paper we introduce the concept of Laplacian energy of fuzzy graphs. ...

This paper, which is the follow-up to part I, concerns the equation (−Δ)sv + G′(v) = 0 in Rn, with s ∈ (0, 1), where (−Δ)s stands for the fractional Laplacian—the infinitesimal generator of a Lévy process. When n = 1, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits ±1 at ±∞) if and only if the potential G has only two absolute minima in [−1,...

The absorption of acoustic wave propagation in a broad variety of lossy media is characterized by an empirical power law function of frequency, y ω α0 . It has long been noted that exponent y ranges from 0 to 2 for diverse media. Recently, the present author developed a fractional Laplacian wave equation to accurately model the power law dissipation, which can be further reduced to the fraction...

This paper presents a novel approach to quantifying the information flow on a graph. The proposed approach is based on the solution of a wave equation, which is defined using the edge-based Laplacian of the graph. The initial condition of the wave equation is a Gaussian wave packet on a single edge of the graph. To measure the information flow on the graph, we use the average return time of the...

This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of enhanced diffusing movements of random turbulent particle resulting from nonlinear inertial interactions. A combined effect of the inertial interactions and the molecule Brownian diffusivities is found to be the bi-fractal mechanism behind multifractal scaling in the inertial range of scales of moderate Rey...

Abstract This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue value problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the bound...