# Probability that a random graph is an expander

I have a random graph $G = (V, E)$ and each edge is in the graph with probability $p$.

I need to show that the probability that $G$ is $\delta$-edge-expander* when $\delta= \frac{np}{4}$ goes to $1$ as $n=|V|\to\infty$.

*A graph is $\delta$-edge-expander if for every set of nodes $S$ of size at most $\frac{n}{2}$ it holds that the number of edges from $S$ to $S'$ (where $S'$ is all the nodes that don't appear in $S$) is at least $\delta$. You can see the definition in this wiki page.

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please elaborate on what is a δ−edge−expander – K.Ghosh Dec 27 '12 at 8:16

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It's unclear what you mean by the event that $G$ is "$\delta--edge--expander$. And you should mention that $n$ is the number of vertices of the graph $G$, if that's what $n$ is here. – coffeemath Dec 27 '12 at 8:18

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The definition on the wiki page requires the edge boundary to be at least $\mathrm{const}\,|S|$, and you wrote "at least $\delta$", without $|S|$. – user53153 Dec 27 '12 at 17:29
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