Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are respon...

In this paper, we compute universal estimates of eigenvalues a coupled system elliptic differential equations in divergence form on bounded domain Euclidean space. As an application, show interesting case rigidity inequalities the Laplacian, more precisely, consider countable family domains Gaussian shrinking soliton that makes behavior known Laplacian invariant by first-order perturbation Lapl...

We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical Z, triangular and honeycomb lattice at the critical temperature. Fisher [Fis66] introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understand...

We introduce an (equi-)affine invariant geometric structure by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to evaluate a new form of geodesic distances and to construct an invariant Laplacian from which local and global diffusion geometry is constructed. Applications of the proposed fram...

The action of the isometry algebra Uh(sl(2)) on the h–deformed Lobachevsky plane is found. The invariant distance and the invariant 2–point functions are shown to agree precisely with the classical ones. The propagator of the Laplacian is calculated explicitely. It is invariant only after adding a “non–classical” sector to the Hilbert space. PACS classification 11.10.Kk, 03.65.Fd John.Madore@th...

The Laplacian and normalized Laplacian energy of G are given by expressions EL(G) = ∑n i=1 |μi − d|, EL(G) = ∑n i=1 |λi − 1|, respectively, where μi and λi are the eigenvalues of Laplacian matrix L and normalized Laplacian matrix L of G. An interesting problem in spectral graph theory is to find graphs {L,L}−noncospectral with the same E{L,L}(G). In this paper, we present graphs of order n, whi...

In the first part of this paper, we study the heat equation and the heat kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group invariant setting. In particular, this Laplacian gives rise to a Feller-Markov semigroup on a fundamental alcove of the affine Weyl group. The second part of the paper is devoted to the Segal-Bargmann transform in our context. A Hilbert space of ...

1. Calculus on spheres 2. Spherical Laplacian from Euclidean 3. Eigenvectors for the spherical Laplacian 4. Invariant integrals on spheres 5. L spectral decompositions on spheres 6. Sup-norms of spherical harmonics on Sn−1 7. Pointwise convergence of Fourier-Laplace series 8. Irreducibility of representation spaces for O(n) 9. Hecke’s identity • Appendix: Bernstein’s proof of Weierstraß approxi...

We study the Laplacian-∞ path as an extreme case of the Laplacian-α random walk. Although, in the finite α case, there is reason to believe that the process converges to SLEκ, with κ = 6/(2α+ 1), we show that this is not the case when α = ∞. In fact, the scaling limit depends heavily on the lattice structure, and is not conformal (or even rotational) invariant.