The top eigenvalue of uniformly random trees

Let 𝐓_n be a uniformly random tree with vertex set [n]={1,…,n}, let Δ_𝐓_n be the largest vertex degree in 𝐓_n, and let λ_1(𝐓_n) be the largest eigenvalue of its adjacency matrix. We prove that |λ_1(𝐓_n)-√(Δ_𝐓_n)| → 0 in expectation as n →∞, and additionally prove probability tail bounds for |λ_1(𝐓_n)-√(Δ_𝐓_n)|. The proof is based on the trace method and thus on counting closed walks in a random tree. To this end, we develop novel combinatorial tools for encoding walks in trees that we expect will find other applications. In order to apply these tools, we show that uniformly random trees – after appropriate "surgery" – satisfy, with high probability, the properties required for the combinatorial bounds to be effective.