Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs wh...

Abstract We derive an analytical expression for the mean load at each node of arbitrary undirected graph non-uniform multicommodity flow problem under weighted random routing. show node, net its demand and normalized by (weighted) degree, is a constant equal to trace product two matrices: Laplacian matrix generalized inverse Laplacian. For case uniform demand, this reduces sum inverses non-zero...

In this paper, we consider the Lévy Laplacian acting on multiple Wiener integrals by the Lévy process, and give a necessary and sufficient condition for eigenfunctions of the Lévy Laplacian. Moreover we give a decomposition by eigenspaces consisting of multiple Wiener integrals by the Lévy process in terms of the Lévy Laplacian.

Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group Φ(G), obtained from the Smith normal form of M , and whose order is the number of spanning trees of G. We provide some general results on the relationship between the eigenvalues of M and the structure of Φ(G), and address the question of how often the group Φ(G) is cyclic.

The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.

We study the Robin Laplacian in a domain with two corners of the same opening, and we calculate the asymptotics of the two lowest eigenvalues as the distance between the corners increases to infinity.

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...