Given an undirected graph, the resistance distance between two nodes is the resistance one would measure between these two nodes in an electrical network if edges were resistors. Summing these distances over all pairs of nodes yields the so-called Kirchhoff index of the graph, which measures its overall connectivity. In this work, we consider Erdős-Rényi random graphs. Since the graphs are rand...

1 Laplacian Methods: An Overview 2 1.1 De…nition: The Laplacian operator of a Graph . . . . . . . . . . 2 1.2 Properties of the Laplacian and its Spectrum . . . . . . . . . . . 4 1.2.1 Spectrum of L and e L: Graph eigenvalues and eigenvectors: 4 1.2.2 Other interesting / useful properties of the normalized Laplacian (Chung): . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Laplacians of Weight...

This paper concerns itself with Besov space solutions of the 2-D quasi-geostrophic (QG) equation with dissipation induced by a fractional Laplacian (−1)α . The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov space Br 2,q [J. Wu, Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. Math. Anal. 36 (2004–2005) 1014...

1. Background. Intuitive arguments drawn from quantum mechanics and optics suggest that there should be some relation between the closed geodesies (periodic "particles") on a compact Riemannian manifold X and the eigenvalues (periodic "waves") of the Laplace-Beltrami operator A^. Indeed, in 1959, Huber [8] proved, for X a surface of constant negative curvature, that the set of lengths of closed...

Since recent development and dissemination of ICTs activate information exchange on social networks, the dynamics for describing propagation of activities on the networks has became an interesting research object. This paper proposes an oscillation model describing the propagation of activities on social and information networks. In order to analyze such dynamics, we generally need to model asy...

In this paper we provide‎ ‎existence results for positive solution to‎ ‎Dirichlet p(t)-Laplacian boundary value problems‎. ‎The sublinear and‎ ‎superlinear cases are considerd‎.

According to recent advances in digital image processing techniques, interest in high-quality images has been increased. This paper presents a resolution enhancement (RE) algorithm based on the pyramid structure, in which Laplacian histogrammatching is utilized for high-frequency image prediction. The conventional RE algorithms yield blurring near-edge boundaries, degrading image details. In or...

We consider the oscillatory property of the following p(t)-Laplacian equations −(|u′|p(t)−2u′)′ = 1/tθ(t)g(t,u), t > 0. Since there is no Picone-type identity for p(t)Laplacian equations, it is an unsolved problem that whether the Sturmian comparison theorems for p(x)-Laplacian equations are valid or not. We obtain sufficient conditions of the oscillatory of solutions for p(t)-Laplacian equations.

The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the n-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the n-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian pe...

Abstract. Inspired by the graph Laplacian and the point integral method, we introduce a novel weighted graph Laplacian method to compute a smooth interpolation function on a point cloud in high dimensional space. The numerical results in semi-supervised learning and image inpainting show that the weighted graph Laplacian is a reliable and efficient interpolation method. In addition, it is easy ...