Diameter , Probability , and Concentration of Measure
نویسنده
چکیده
The diameter of a graph is the maximum over vertices u and v of the distance between u and v. It is easy to show that graphs with high conductance must have low diameter. If the set of edges leaving a set must be large relative to the size of the set, then after taking neighborhoods a few times, one must encounter most of the edges. We now make this intuition precise. Recall from Lecture 7 the definion of the sparsity of a set S ⊆ V : sp(S) def = |∂(S)| min d(S), d(¯ S) , where we recall that d(S) is the sum of the degrees of vertices in S. We now bound the diameter of G in terms of sp G def = min S sp(S).
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