Let L(G) be the Laplacian matrix of G. In this paper, we characterize all of the connected graphs with second largest Laplacian eigenvalue no more than l; where l . = 3.2470 is the largest root of the equation μ3 − 5μ2 + 6μ − 1 = 0. Moreover, this result is used to characterize all connected graphs with second largest Laplacian eigenvalue no more than three. © 2013 Elsevier Ltd. All rights rese...

We show the existence of simply-connected unbounded planar domains for which the second nodal line of the Dirichlet Laplacian does not touch the boundary.

We study the optimal sets Ω∗ ⊂ R for spectral functionals F ( λ1(Ω), . . . , λp(Ω) ) , which are bi-Lipschitz with respect to each of the eigenvalues λ1(Ω), . . . , λp(Ω) of the Dirichlet Laplacian on Ω, a prototype being the problem min { λ1(Ω) + · · ·+ λp(Ω) : Ω ⊂ R, |Ω| = 1 } . We prove the Lipschitz regularity of the eigenfunctions u1, . . . , up of the Dirichlet Laplacian on the optimal se...

We consider scattering by an obstacle in Rd, d ≥ 3 odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius ρ does, then the obstacle is a ball of radius ρ. We give related results for obstacles which are disjoint unions of several balls of the same radius.

We provide an analysis of the expected meeting time of two independent random walks on a regular graph. For 1-D circle and 2-D torus graphs, we show that the expected meeting time can be expressed as the sum of the inverse of non-zero eigenvalues of a suitably defined Laplacian matrix. We also conjecture based on empirical evidence that this result holds more generally for simple random walks o...

Let G be a graph with n vertices. We denote the largest signless Laplacian eigenvalue of G by q1(G) and Laplacian eigenvalues of G by μ1(G) > · · · > μn−1(G) > μn(G) = 0. It is a conjecture on Laplacian spread of graphs that μ1(G)−μn−1(G) 6 n − 1 or equivalently μ1(G) + μ1(G) 6 2n − 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, μ1(G)μ1(G) 6 n(n − ...

We consider the critical dissipative SQG equation in bounded domains, with the square root of the Dirichlet Laplacian dissipation. We prove global a priori interior C and Lipschitz bounds for large data.

In this paper, we develop a regularization framework for image deblurring based on a new definition of the normalized graph Laplacian. We apply a fast scaling algorithm to the kernel similarity matrix to derive the symmetric, doubly stochastic filtering matrix from which the normalized Laplacian matrix is built. We use this new definition of the Laplacian to construct a cost function consisting...

This is the third part of our work with a common title. The first [11] and the second part [12] will be also referred in the sequel as Part I and Part II, respectively. This third part was not planned at the beginning, but a lot of recently published papers on the signless Laplacian eigenvalues of graphs and some observations of ours justify its preparation. By a spectral graph theory we unders...