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 Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl, and dk−j+1 be the degree of vertices in the level j. In this paper we study the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl) for all positive integer l, and give some results about the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl). By using thes...

In Lecture 10, we introduced a fundamental object of spectral graph theory: the graph Laplacian, and established some of its basic properties. We then focused on the task of estimating the value of eigenvalues of Laplacians. In particular, we proved the Courant-Fisher theorem that is instrumental in obtaining upper-bounding estimates on eigenvalues. Today, we continue by showing a technique – s...

In directed network, algebraic connectivity is defined as the second smallest eigenvalue of graph Laplacian, and it captures the most conservative estimate of convergence rate and synchronicity of networked systems. In this paper, distributed estimation of algebraic connectivity of directed and connected graphs is studied using a decentralized power iteration scheme. Specifically, the proposed ...

The purpose of this paper is to develop a “calculus” on graphs that allows graph theory to have new connections to analysis. For example, our framework gives rise to many new partial differential equations on graphs, most notably a new (Laplacian based) wave equation (see [FTa, FTc]); this wave equation gives rise to a partial improvement on the Chung-Faber-Manteuffel diameter/eigenvalue bound ...

Abstract. For any bounded regular domain Ω of a real analytic Riemannian manifold M , we denote by λk(Ω) the k-th eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λk and as a functional upon the set of domains of fixed volume in M . We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient condit...

1 he leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {(x, y)l 1x1 " / yl < 11 and for 4 + lxlD I yj4 all of which are non-Weyl because of infinite phase space volumes are computed. Along the way, a general inequality on quantum partition functions coriputed in a kind of Born-Oppenheimer approximation is proved.

Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n×n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix Γ; the best possible bound based on this embedding is n/λmax(Γ Γ). Howeve...

One of the more challenging problems in spectral theory and mathematical physics today is to understand the statistical distribution of eigenvalues of the Laplacian on a compact manifold. There are now several challenging conjectures about these, originating in the physics literature. In this survey, a version of a talk delivered as the Colloquio De Giorgi at the Scuola Normale Superiore of Pis...