Cs-621 Theory Gems
نویسندگان
چکیده
In Lecture 10, we introduced a fundamental object of spectral graph theory: the graph Laplacian, and established some of its basic properties. We then focused on the task of estimating the value of eigenvalues of Laplacians. In particular, we proved the Courant-Fisher theorem that is instrumental in obtaining upper-bounding estimates on eigenvalues. Today, we continue by showing a technique – so-called graph inequalities – that enables us to establish eigenvalue lower-bounding estimates. Then, we use these upperand lower-bounding tools to obtain estimates on second-smallest eigenvalues of some important graphs. Later, we discuss how the second-smallest eigenvalue of a (normalized) Laplacian of a graph relates to the mixing time of a random walk in that graph, as well as, its connectivity structure.
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