Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Review of Spectral Graph Theory: by Fan R. K. Chung

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.

The tau constant and the discrete Laplacian matrix of a metrized graph

Abstract: We express the tau constant of a metrized graph in terms of the discrete Laplacian matrix and its pseudo-inverse.

Similarity-aware spectral sparsification by edge filtering

Abstract: In recent years, spectral graph sparsification techniques that can compute ultra-sparse graph proxies have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications.

Curvature and higher order Buser inequalities for the graph connection Laplacian

Abstract: We study the eigenvalues of the connection Laplacian on a graph with an orthogonal group or unitary group signature. We establish higher order Buser type inequalities, i.e., we provide upper bounds for eigenvalues in terms of Cheeger constants in the case of nonnegative Ricci curvature. In this process, we discuss the concepts of Cheeger type constants and a discrete Ricci curvature for connection Laplacians and study their properties systematically. The Cheeger constants are defined as mixtures of the expansion rate of the underlying graph and the frustration index of the signature. The discrete curvature, which can be computed efficiently via solving semidefinite programming problems, has a characterization by the heat semigroup for functions combined with a heat semigroup for vector fields on the graph.