By Emmanuel Kowalski

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**Extra resources for Expander graphs**

**Example text**

2. Random walks The definition of expansion constant (and consequently of expander graphs) does not provide an easy or direct way of computing h(Γ). In terms of the number of vertices, which is a natural parameter coding the size of a graph of bounded valency, the exact determination of h(Γ) requires looking at all subsets containing at most half of the vertices, a number of sets which is exponentially large in terms of |Γ|. In this section, we will describe another invariant that can be estimated, in practice, much more easily, and which controls to some extent the expansion constant.

11. (1) As usual, we will often drop the subscript Γ when the context is clear. (2) Note that if Γ is k-regular for some k 1, the measure µΓ becomes the uniform counting measure defined by |W | µΓ (W ) = |V | 43 for all W ⊂ V . This will be the case for most applications so the reader may read this section first with this case in mind. 11) v+ |V | v− |V | for all W ⊂ V , where v− = min val(x), v+ = max val(x). x∈V x∈V (3) If Γ is an infinite graph, we can not define a probability measure in the same way (since N typically diverges), but we can still speak of the measure with weight val(x), and of the associated L2 -space.

5) h(Γ) = min(h0 , h1 ), hi = min |W | W ⊂Vi 1 |W | |Vi |/2 (note that all the vertices in the boundary of a subset of Vi are in the other part). 16. For Γ = (V, E) a finite bipartite graph with a decomposition V = V0 ∪ V1 and with maximal valency v 1. Assume that |V0 | = |V1 |. We have ˇ h(Γ) −1 ˇ h(Γ) v h(Γ). 2 Proof. The upper-bound is easy and left as an exercise. For the lower-bound, we ˇ ˇ can assume that h(Γ) 1, say h(Γ) = 1 + δ with δ 0; we must check that h(Γ) δ/2. Let then W ⊂ V be any subset with 1 |W | |V |/2.

### Expander graphs by Emmanuel Kowalski

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