6.1 About these notes These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. The notes written after class way what I wish I said. I think that these notes are mostly correct. 6.2 Overview As the title suggests, in this lecture I will introduce conductance, a measure of the quality of a cut, and the normalized Laplacian matrix of a graph. I will then prove Cheeger’s inequality, which relates the second-smallest eigenvalue of the normalized Laplacian to the conductance of a graph. Cheeger [Che70] first proved his famous inequality for manifolds. Many discrete versions of Cheeger’s inequality were proved in the late 80’s [SJ89, LS88, AM85, Alo86, Dod84, Var85]. Some of these consider the walk matrix (which we will see in a week or two) instead of the normalized Laplacian, and some consider the isoperimetic ratio instead of conductance. The proof that I present today follows an approach developed by Luca Trevisan [Tre11]. 6.3 Conductance Back in Lecture 2, we related to isoperimetric ratio of a subset of the vertices to the second eigenvalue of the Laplacian. We proved that for every S ⊂ V where s = |S | / |V | and θ(S) ≥ λ2(1 − s), θ(S) de
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