By Emmanuel Kowalski

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Let Γ be a finite non-empty connected graph. 2) diam(Γ) log |Γ| 2 2 log 1 + h(Γ) v +3 where v = max val(x). x∈V The intuitive idea is the following: to “join” x to y with a short path, we look at how many elements there are at increasing distance from x and y; the definition of the expansion ratio gives a geometric lower-bound on the number of new elements when we increase the distance by one, and at some point the sets which can be reached in n steps from both sides are so big that they have to intersect, giving a distance at most 2n by the triangle inequality.

N log( −1 Γ ) Hence, for n larger than some (possibly large, depending on how close Γ is to 1) multiple of log |V |, a random walk (Xn ) becomes significantly well-distributed. This type of considerations turns out to play an important role in understanding the arguments of the next chapter. We see from this corollary that Γ controls the rate of convergence of a random walk on Γ to the normalized graph measure µΓ . The intuition that the speed of convergence should be greater when the expansion constant is also large leads to the suspicion that bounding Γ away from 1 should be related to bounding h(Γ) away from 0.

If γ : P2 −→ Γ is any path of length 2 with γ(0) = x, γ(2) = y, it follows that ϕ(x) = −ϕ(γ(1)) = ϕ(y). Similarly, y = γ(2k) for a path of even length 2k. Now we fix some x0 ∈ V , and let W be the set of vertices in Γ which are the other extremity of a path γ : P2k −→ Γ of even length with γ(0) = x0 (in particular, x0 ∈ W using a path of length 0). We see that ϕ is constant, equal to ϕ(x0 ), on all of W . If W = V , it follows that ϕ is constant, hence M ϕ = ϕ = −ϕ, so ϕ = 0. On the other hand, if W = V , we claim that V0 = W , V1 = V − W is a bipartite partition of V .