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Unformatted text preview: April 21, 2010 Wensong(Tony) Xu Cheeger inequality 1 Introduction In this lecture, we will introduce the Cheeger ratio and the Cheeger inequality. 2 Cheeger ratio The isoperimetric problem is to find a plane figure which maximizes its area, given a fixed length of boundary. Now we consider a discrete version of isoperimetric problem. Suppose G is a undi- rected graph. For all S V ( G ) , fix Vol S , we want to find a S which minimizes the vertex bound- ary S , or the edge boundary S . Usually these two minimizations gives similar results, but in some cases they are quite different. For example, for graph Q n , to minimize the vertex boundary, we shall look at it as a hamming ball and cut it at some level. On the other hand, to minimize the edge boundary, we shall regard it as two Q n- 1 s and cut between them. For a subset S V ( G ) , define h ( S ) = | S | Vol S . The Cheeger constant h G of a graph G is defined to be h G = min S V, Vol S 1 2 Vol G h ( S ) In some applications, it would also be useful to define...
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This note was uploaded on 09/30/2011 for the course MATH 262 taught by Professor Aterras during the Fall '08 term at UCSD.
- Fall '08