is the diagonal matrix of vertex degrees of G and A(G) is the adjacency matrix ofG. The eigenvalues of L(G) are called the Laplacian eigenvalues and denoted by λ1 ≥ λ2 ≥ · · · ≥ λn = 0. It is well known that λ1 ≤ n. We denote the number of spanning trees (also known as complexity) of G by κ(G). The following formula in terms of the Laplacian eigenvalues of G is well known (see, for example, [2]...

In this paper, we address the problem of surface tracking in multiple camera environments and over time sequences. In order to fully track a surface undergoing significant deformations, we cast the problem as a mesh evolution over time. Such an evolution is driven by 3D displacement fields estimated between meshes recovered independently at different time frames. Geometric and photometric infor...

Let Ω be a C∞-smooth bounded domain of R, n ≥ 1, and let the matrix a ∈ C(Ω;R 2 ) be symmetric and uniformly elliptic. We consider the L(Ω)-realization A of the operator −div(a∇·) with Dirichlet boundary conditions. We perturb A by some real valued potential V ∈ C∞ 0 (Ω) and note AV = A + V . We compute the asymptotic expansion of tr ( eV − e ) as t ↓ 0 for any matrix a whose coefficients are h...

We give several bounds on the second smallest eigenvalue of the weighted Laplacian matrix of a finite graph and on the second largest eigenvalue of its weighted adjacency matrix. We establish relations between the given Cheeger-type bounds here and the known bounds in the literature. We show that one of our bounds is the best Cheeger-type bound available.

For almost all graphs the answer to the question in the title is still unknown. Here we survey the cases for which the answer is known. Not only the adjacency matrix, but also other types of matrices, such as the Laplacian matrix, are considered. © 2003 Elsevier Inc. All rights reserved.

This paper presents a new method for estimating the eigenvalues of the Laplacian matrix associated with the graph describing the network topology of a multi-agent system. Given an approximate value of the average of the initial condition of the network state and some intermediate values of the network state when performing a Laplacian-based average consensus, the estimation of the Laplacian eig...

In spectral graph theory, the Grone-Merris Conjecture asserts that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of this graph. We give a complete proof for this conjecture. The Laplacian of a simple graph G with n vertices is a positive semi-definite n×n matrix L(G) that mimics the geometric Laplacian of a Riemannian manifold; see §1 for d...

Abstract. We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes ∆, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Lapl...

We study random graphs with possibly different edge probabilities in the challenging sparse regime of bounded expected degrees. Unlike in the dense case, neither the graph adjacency matrix nor its Laplacian concentrate around their expectations due to the highly irregular distribution of node degrees. It has been empirically observed that simply adding a constant of order 1/n to each entry of t...

Let L be the Laplacian matrix of a tree. We present a graph-theoretic interpretation of the cofactors of order 2 of L. From this, we deduce a description for the inverse of the rooted Laplacian, reflecting the geometry of a branch. Defining the thickness of a branch as the Perron root of this matrix, we present a minimaxcharacterization of the characteristic center of the tree based on the thic...