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The preferential attachment graph is a random graph formed by adding a new vertex at each time-step, with a single edge which points to a vertex selected at random with probability proportional to its degree. Every m steps the most recently added m vertices are contracted into a single vertex, so at time t there are roughly t/m vertices and exactly t edges. This process yields a graph which has been proposed as a simple model of the World Wide Web [Barabási and Albert 99]. For any constant k, let Δ1 ≥ Δ2 ≥ … ≥ Δk be the degrees of the k highest degree vertices. We show that at time t, for any function ƒ with ƒ(t)→ ∞ as , and for , with high probability (whp). We use this to show that at time t the largest k eigenvalues of the adjacency matrix of this graph have λ k = (1 ± o(1))Δ k 1/2 whp

High Degree Vertices and Eigenvalues in the Preferential Attachment Graph