Showing papers on "Spectral graph theory published in 1999"
TL;DR: In this article, rank one perturbations of symmetric matrices which change only one eigenvalue are described. And the result is applied to study how the Laplacian spectrum of a graph changes when adding an edge.
Abstract: Characterization of rank one perturbations of symmetric matrices which change only one eigenvalue are given. Then the result is applied to study how the Laplacian spectrum of a graph changes when adding an edge.
64 citations
TL;DR: In this paper, the relationship between the spectrum of the Schrodinger operator on a covering graph and that on a finite graph was investigated, and the analyticity of the bottom of the spectrum with respect to magnetic flow was shown.
38 citations
TL;DR: This book defines the Laplacian of a graph, a matrix closely related to the adjacency matrix, in analogy with the continuous case and studies the eigenvalues of this Laplace, which are related to many other more "discrete" invariants.
Abstract: Specifying a graph is equivalent to specifying its adjacency relation, which may be encoded in the form of a matrix. This suggests that study of the adjacency matrix from a linear-algebraic point of view might yield valuable information about graphs. In particular, any invariant associated to the matrix is also an invariant associated to the graph, and might have combinatorial meaning. Spectral graph theory is the study of the relationship between a graph and the eigenvalues of matrices (such as the adjacency matrix) naturally associated to that graph. This book looks at the subject from a geometric point of view, exploiting an analogy between a graph and a Riemannian manifold: Chung defines the Laplacian of a graph, a matrix closely related to the adjacency matrix, in analogy with the continuous case and studies the eigenvalues of this Laplacian.There are several reasons that these eigenvalues may be of interest. On the purely mathematical level, the eigenvalues have the advantage of being an extremely natural invariant which behaves nicely under operations such as Cartesian product and disjoint union. From a combinatorial point of view, the eigenvalues of a graph are related to many other more "discrete" invariants. From a geometric point of view, there are many respects in which the eigenvalues of a graph behave like the spectrum of a compact Riemannian manfiold. For the computationally-minded, the eigenvalues of a graph are easy to compute, and their relationship to other invariants can often yields good approximations to less tractible computations.
19 citations
10 Jan 1999
18 citations
TL;DR: While no method has a definite advantage when bounding above the diameter, the use of the Laplacian matrix seems better when dealing with the mean distance of a graph.
17 citations
TL;DR: In this paper, a graph theoretic analogue of Cheng's eigenvalue comparison theorems for the Laplacian of complete Riemannian manifolds is given.
Abstract: We give a graph theoretic analogue of Cheng's eigenvalue comparison theorems for the Laplacian of complete Riemannian manifolds. As its applications, we determine the infimum of the (essential) spectrum of the discrete Laplacian for infinite graphs.
17 citations
5 citations
TL;DR: In this paper, the spectrum of the Neumann Laplacian for a graph with boundary is studied and the optimal upper and lower bounds of the first eigenvalue of the NEUMB problem of the combinatorial LaplACian for graphs with boundary are given.
Abstract: In this paper, the spectrum of the Neumann Laplacian for a graph with boundary is studied. Two comparison theorems of the Neumann Laplacian for a graph are shown. Namely, the optimal upper and lower bounds of the first eigenvalue of the Neumann boundary problem of the combinatorial Laplacian for a graph with boundary are given.
3 citations