# Isoperimetric constant on random graph

I have the following problem.

Show that there is a constant $$c=c(p) > 0$$ such that almost all graphs in $$\mathcal{G}_{n,p}$$ verify the following property : for each subset $$X \in V(G)$$ with cardinality $$|X|\leq n/2$$, $$e(X, V\backslash X) \geq c |X|$$

Where $$e(X, V\backslash X)$$ denotes the number of edges connecting $$X$$ and $$V\backslash X$$.

I found a "standard" solution, using combinatorial arguments (the constant $$c(p)=p$$ satisfies the condition), but I have the feeling that a "more elegant" proof should exist.

My intuition is as follow : the problem is equivalent to proving, with $$i(G)$$ the isoperimetric constant : $$\forall p, \ \exists c(p)>0, \ s.t. \ \mathbb{P}[ i(G) > c] \rightarrow 1 \text{ when }n\rightarrow \infty$$

And for any graph $$G$$, we know a lower bound for $$i(G)$$: $$\mu_2 /2 \leq i(G)$$ With $$\mu_2$$ being the second smallest eigenvalue of the Laplacian matrix of $$G$$. Now if I can find some bound for $$\mu_2$$ I should be able to do something.

I know that for fixed $$p$$ the probability that $$G$$ is connected tends to 1, hence $$\mu_2 > 0$$. But can we find similar argument for $$\mu_2 > c$$ for some constant $$c$$ ? in term of connectivity surely? Thanks for the help!

From this paper we can estimate the normalized Laplacian spectrum of $$\mathcal G_{n,p}$$ from the spectrum of an "average-case Laplacian" $$\bar{L} = I - \bar{D}^{-1/2} \bar{A}\bar{D}^{-1/2}$$ where $$\bar{D} = (n-1)pI$$ is the expected value of $$D$$, and $$\bar{A} = p(J-I)$$ is the expected value of $$A$$. (As usual, $$J$$ is the all-ones matrix.) Since $$\bar{D}$$ commutes with everything, we can rewrite this as $$I - \bar{D}^{-1}\bar{A}$$ which simplifies to$$\frac{n}{n-1}I - \frac{1}{n-1}J.$$ The eigenvalues of $$J$$ are $$n,0,0,\dots,0,0$$ and so the eigenvalues of $$\bar{L}$$ are $$0, \frac{n}{n-1}, \frac{n}{n-1}, \dots, \frac{n}{n-1}$$.
This looks like a rather stupid estimate that doesn't depend on $$p$$, but the dependence on $$p$$ is hidden in the error bound. For all constant $$\epsilon>0$$, we have $$|\lambda_i(L) - \lambda_i(\bar{L})| \le \mathcal O\left(\sqrt{\frac{\log(n/\epsilon)}{(n-1)p}}\right)$$ provided $$(n-1)p > k(\epsilon)\log n$$. In other words, as long as $$p$$ is not a too-small function of $$n$$, we have $$\lambda_2(L) \ge 1 - o(1)$$ with high probability.