The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order.

The method of iterated conformal maps for the study of diffusion limited aggregates (DLA) is generalized to the study of Laplacian growth patterns and related processes. We emphasize the fundamental difference between these processes: DLA is grown serially with constant size particles, while Laplacian patterns are grown by advancing each boundary point in parallel, proportional to the gradient ...

Laplacian Abelian Projection is discussed. This term refers to the use of a new (“Laplacian”) gauge fixing prescription for implementing the Abelian Projection of QCD. The gauge condition is based on the lowest-lying eigenvector of the covariant Laplacian operator in the adjoint representation. This Laplacian gauge fixing procedure is free of the ambiguities which plague lattice simulations whi...

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.

This paper is devoted to the study of the eigenpairs of the Dirichlet Laplacian on a family of triangles where two vertices are fixed and the altitude associated with the third vertex goes to zero. We investigate the dependence of the eigenvalues on this altitude. For the first eigenvalues and eigenfunctions, we obtain an asymptotic expansion at any order at the scale cube root of this altitude...

A Hermite type formula is introduced and used to study the zeta function over the real and complex n-projective space. This approach allows to compute the residua at the poles and the value at the origin as well as the value of the derivative at the origin, that gives the regularized determinant of the associated Laplacian operator.

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators’ domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of “almost convex” domains, which apply to do...