Integral graph

Abstract: Spectral graph theory is the study of the eigenvalues and eigenvectors of matrices associated with graphs. In this tutorial, we will try to provide some intuition as to why these eigenvectors and eigenvalues have combinatorial significance, and will sitn'ey some of their applications.

Abstract: This paper is mainly concerned with the realizability of a set of n integers as the degrees of vertices of an n-vertex linear graph. Other related problems, such as when a set of integers is realizable as a connected graph, connected graph without “parallel” elements, separable graph, and nonseparable graph, are considered. The relationship between this problem and the problem of isomers in the organic chemistry is described. A similar problem in weighted graphs is also studied.

Abstract: We study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs. A frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity. Smoothly varying signals defined on the nodes are of particular interest in various applications, and tend to be approximately bandlimited in the frequency basis. Sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples. Most approaches to this problem require a computation of the frequency basis (i.e., the eigenvectors of the variation operator), followed by a search procedure using the basis elements. This can be impractical, in terms of storage and time complexity, for real datasets involving very large graphs. We circumvent this issue in our formulation by introducing quantities called graph spectral proxies, defined using the powers of the variation operator, in order to approximate the spectral content of graph signals. This allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements. We show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited. Furthermore, the proposed approach is valid for any choice of the variation operator, thereby covering a wide range of graphs and applications. We demonstrate its effectiveness through various numerical experiments.

Abstract: In the study of the spectra of power law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law while the other predicts that the eigenvalues follow a power law distributions. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will show that (under certain conditions) the eigenvalues of the (normalized) Laplacian of a random power law graph follow the semicircle law while the spectrum of the adjacency matrix of a power law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent β where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as randomized algorithms that involve rapidly mixing Ma...