Diameters and eigenvalues

We derive a new upper bound for the diameter of a k k -regular graph G G as a function of the eigenvalues of the adjacency matrix. Namely, suppose the adjacency matrix of G G has eigenvalues λ 1, λ 2, …, λ n ₁, ₂, , ₙ with | λ 1 | ≥ | λ 2 | ≥ ⋯ ≥ | λ n | | { ₁} | | { ₂} | | { ₙ} | where λ 1 = k ₁ = k, λ = | λ 2 | = | { ₂} |. Then the diameter