Derive it, check it,
write it into your paper.

2  Preliminaries

Definition 1.An Erdös-RényiErdős–Rényi random graph G(n, p) is a graph on n vertices where each posible edgepossible edge {u, v} is included with probability pindependent, independently of all other edges.

Let A be the adjacency matrix of a graph G be a graph on n vertices. Its adjacency matrix A = A(G) is an n × n matrix where A_uv = 1 if {u, v} is an edge in G. The eigenvalues of, and A_uv = 0 otherwise. Since Aare denotedis symmetric, its eigenvalues are real and are denoted by λ₁ ≥ λ₂ ≥ ⋯ ≥ λₙ.

3  Spectral Gap in Random Regular Graphs

For a d-regular graph G (where every vertex has degree d), it is well-known that the largest eigenvalue of its adjacency matrix is λ₁ = d. The spectral gap, defined as d − λ₂, plays a crucial role in the expansion properties of the graph.

Theorem 2 (Alon–Boppana).For a d-regular graph on n vertices, λ₂ is largeλ₂ ≥ 2√(d−1) − o(1) as n → ∞. A d-regular graph meeting this bound with equality is called a good expanderRamanujan graph.

Proof. It follows from counting walks.Count closed walks of length 2k rooted at a fixed vertex. The number of such walks in the d-regular tree is the Catalan-weighted moment C_k (d−1)^k, and comparing with tr(A^2k) = Σ_i λ_i^2kas n → ∞ forces λ₂ ≥ 2√(d−1) − o(1).∎