Spectral graph theory - Wikipedia
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.
The aim of this book is to understand the spectral grpah theory. We combine all classic sources, e.g. Fan Chung’s book, Dan Spielman and Luca Trevisan’s graduate courses.
We begin with basic but necessary de nitions in graph theory that are important to both describe and prove results in spectral graph theory. Then, we move to topics in linear algebra that are required to understand spectral graph theory, such as the Rayleigh quotient and spectral theorem.
Spectral graph theory is the study of the eigenvalues and eigen-vectors of matrices associated with graphs. This paper is a review of Cvetkovic's GRAPHS AND THEIR SPECTRA [1], and builds up to a proof of Kirchho 's Matrix Tree Theorem, which provides a formula for computing the number of spanning trees in a graph.
2016年9月27日 · A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY BOGDAN NICA – INTRODUCTION – Spectral graph theory starts by associating matrices to graphs, notably, the adja-cency matrix and the laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenval-
Eigenvalues and the Laplacian of a graph. 1.1. Introduction. 1.2. The Laplacian and eigenvalues. 1.3. Basic facts about the spectrum of a graph. 1.4. Eigenvalues of weighted graphs. 1.5. Eigenvalues and random walks. Chapter 2. Isoperimetric problems. 2.1. History. 2.2. The Cheeger constant of a graph. 2.3. The edge expansion of a graph. 2.4.
Spectral graph theory is a vast and expanding area of combinatorics. We start these notes by introducing and motivating classical matrices associated with a graph, and then show how to derive combinatorial properties of a graph from the eigenvalues of these matrices.
Lectures #11: Spectral Graph Theory, I Tim Roughgarden & Gregory Valiant∗ May 7, 2024 Spectral graph theory is the powerful and beautiful theory that arises from the following question: What properties of a graph are exposed/revealed if we 1) represent the graph as a matrix, and 2) study the eigenvectors/eigenvalues of that matrix.
In the eld of spectral graph theory we relate combinatorial properties of graphs to their algebraic properties.
D. J. Kelleher Spectral graph theory. Spectral Theorem Spectral Theorem If Ais a real symmetric n n-matrix, then each eigenvalue is real, and there is an orthonormal basis of Rn of eigenfunctions (eigenvectors) of A. fe jgn j=1 is orthonormal if e j e k = jk = (0 if j6= k 1 if j= k: