Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with ...

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial o...

Recently, Sorkine et al. proposed a least squares based representation of meshes, which is suitable for compression and modeling. In this paper we look at this representation from the viewpoint of Tikhonov regularization. We show that this viewpoint yields a smoothing algorithm, which can be seen as an approximation of the shape using weighted geometry aware bases, where the weighting factor is...

Abstract. Subordination of a killed Brownian motion in a bounded domain D ⊂ R via an α/2-stable subordinator gives a process Zt whose infinitesimal generator is −(− |D), the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Zt in a Lipschitz domain D by comparing the process with the rotationally invariant α-stable process killed upon exi...

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite p and investigate the limit problem as p → ∞.

Necessary conditions for an undirected graph G to contain a graph H as induced subgraph involving the smallest ordinary or the largest normalized Laplacian eigenvalue of G are presented.

Eigenfunction methods for mapping high dimensional data sets into lower dimensional spaces are useful in a broad range of applications. In particular, a recent paper by Jones et al [2] shows that eigenfunctions of the Laplacian operator give a good local coordinate system under very general conditions. Here I outline the proof of the theorem and explore its use in cryo-electron microscopy.

In this article, we are interested in the exponentially small eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian ∆ f,h, in the general framework of p-forms, on a connected compact Riemannian manifold without boundary. Our purpose is to notice that the knowledge of (the asymptotic formulas for) the smallest non zero eigenvalues of the self adjoint realization of ∆ ...

It is known that there an alternative characterization of characteristic vertices for trees with positive weights on their edges via Perron values and branches. Moreover, the algebraic connectivity a tree edge can be expressed in terms value. In this article, we consider matrix edges. More precisely, are interested following classes weights: definite weights, lower (or upper) triangular diagona...

In this paper, all connected graphs with the fourth largest Laplacian eigenvalue less than two are determined, which are used to characterize all connected graphs with exactly three Laplacian eigenvalues no less than two. Moreover, we determine bipartite graphs such that the adjacency matrices of their line graphs have exactly three nonnegative eigenvalues. © 2003 Elsevier Ltd. All rights reser...