Proposition 4.6. Assuming that:

Then the following are equivalent:

(i) $G$ is a $(n, d, λ)$ -graph
(ii) $λ_{k} (A_{G}) ∋ [- λ, λ]$ , for $1 \leq k \leq n - 1$
(iii) $λ_{k} (L_{G}) \in [d - λ, d + λ]$ for $2 \leq k \leq n$
(iv) $λ_{k} ({\tilde{L}}_{G}) \in [1 - \frac{λ}{d}, 1 + \frac{λ}{d}] = [1 - 𝜀, 1 + 𝜀]$ for $1 \leq k \leq n$
(v) $(1 - 𝜀) \frac{d}{n} L_{K_{n}} ≼ L_{G} ≼ (1 + 𝜀) \frac{d}{n} L_{K_{n}}$
(vi) $G$ is $(d, 𝜀)$ -expander (also $(d, \frac{λ}{d})$ -expander)
(vii) $∥ L_{G} - \frac{d}{n} L_{K_{n}} ∥ \leq 𝜀 d = λ$
(viii) $∥ A_{G} - \frac{d}{n} J ∥ \leq 𝜀 d = λ$