Lecture 7: More on Graph Eigenvalues, and the Power Method
نویسنده
چکیده
We will discuss a few basic facts about the distribution of eigenvalues of the adjacency matrix, and some applications. Then we discuss the question of computing the eigenvalues of a symmetric matrix. 1 Eigenvalue distribution Let us consider a d-regular graph G on n vertices. Its adjacency matrix AG is an n× n symmetric matrix, with all of its eigenvalues lying in [−d, d]. How are the eigenvalues distributed in the interval [−d, d]? Are there always many negative eigenvalues? What is the typical magnitude of the eigenvalues? The key to answering these questions is the simple fact that the trace of a matrix is the sum of its eigenvalues. Since all the diagonal entries of AG are 0 The trace, denoted Tr(·), is defined to be the sum of the diagonal entries of a matrix. (the graph has no self loops), we have that
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