Spectral gap of G(n,αn,α2n) 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 , α n , α 2 n ) graphs. We then apply these results to get a giant component theorem for them.