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 ...

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...

Let G be a graph of order n such that ∑n i=0(−1)iaiλn−i and ∑n i=0(−1)ibiλn−i are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that ai ≥ bi for i = 0,1, . . . , n. As a consequence, we prove that for any α, 0 < α ≤ 1, if q1, . . . , qn and μ1, . . . ,μn are the signless Laplacian and the Laplacian eigenvalues of G, respectively,...

Laplacian eigenmap algorithm is a typical nonlinear model for dimensionality reduction in classical machine learning. We propose an efficient quantum Laplacian eigenmap algorithm to exponentially speed up the original counterparts. In our work, we demonstrate that the Hermitian chain product proposed in quantum linear discriminant analysis (arXiv:1510.00113,2015) can be applied to implement qua...

In this paper, we study the multiplicity of positive solutions for the Laplacian systems with sign-changing weight functions. Using the decomposition of the Nehari manifold, we prove that an elliptic system has at least two positive solutions.

Let G = (V,E) be an undirected graph, LG ∈ R ×V be the associated Laplacian matrix, and b ∈ R V be a vector. Solving the Laplacian system LGx = b has numerous applications in theoretical computer science, machine learning, and network analysis. Recently, the notion of the Laplacian operator LF : R V → 2 V for a submodular transformation F : 2 → RE+ was introduced, which can handle undirected gr...

The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and the structure of ‘local’ subgraphs of the network. We call a subgraph local when it is induced by the set of nodes obtained from a breath-first search (BFS)...

In directed network, algebraic connectivity is defined as the second smallest eigenvalue of graph Laplacian, and it captures the most conservative estimate of convergence rate and synchronicity of networked systems. In this paper, distributed estimation of algebraic connectivity of directed and connected graphs is studied using a decentralized power iteration scheme. Specifically, the proposed ...

Real-world data are seldom unstructured, yet traditional Matrix Factorization (MF) models, as one of the most powerful collaborative filtering approaches, generally rely on this assumption to recover the low-rank structures for recommendation. However, few of them are able to explicitly consider structured constraint with the underlying low-rank assumption to model complex user interests. To so...

We present a computer-assisted approach to coarse graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a (fixed) network structure. The existence of a spectral gap for the coupling network graph Laplacian suggests that the graph dynamics may quickly become low dimensional. Our first choice of coarse variables consists of the components of the oscillator sta...