Spectral gap of $G(n,\alpha n,{\alpha }^{2}n)$ graphs and the giant component theorem

The spectrum of a graph $G$ is the set of the eigenvalues of its adjacency matrix. It turns out that one can say a lot about a graph with the only knowledge being the spectrum of this graph. In this paper we obtain new results about the spectrum of $G(n,\alpha n,{\alpha }^{2}n)$ graphs. We then apply these results to get a giant component theorem for them.