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...

We briefly review known results about the signed edge domination number of graphs. In the case of bipartite graphs, the signed edge domination number can be viewed in terms of its bi-adjacency matrix. This motivates the introduction of the signed domination number of a (0, 1)-matrix. We investigate the signed domination number for various classes of (0, 1)-matrices, in particular for regular an...

We consider a one-phase free boundary problem involving a fractional Laplacian (−∆), 0 < α < 1, and we prove that “flat free boundaries” are C . We thus extend the known result for the case α = 1/2.

Rjk = R̄jk + ∇̄mD jk − ∇̄jD mk −Dn jkD nm −Dn mkD nj . (To see why this is true, just compute in local normal coordinates for ḡ!) From now on all components are tensorial (as opposed to local coordinate components.) The terms involving second covariant derivatives of the metric are: 1 2 g(−∇̄m∇̄lgjk +∇̄m∇̄kgjl +∇̄m∇̄jglk +∇̄j∇̄lgmk−∇̄j∇̄kgml−∇̄j∇̄mglk). The first term is a kind of Laplacian. The third and six...

Let D be a convex planar domain, symmetric about both the xand y-axes, which is strictly contained in (−a, a) × (−b, b) = Γ. It is proved that, unless D is a certain kind of rectangle, the difference (gap) between the first two eigenvalues of the Dirichlet Laplacian in D is strictly larger than the gap for Γ. We show how to give explicit lower bounds for the difference of the gaps.

We show that the zeta function of a regular graph admits a representation as a quotient of a determinant over a L 2-determinant of the combinatorial Laplacian.

We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The ...