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 {λ _1},{λ _2}, … ,{λ _n} with | λ 1 | ≥ | λ 2 | ≥ ⋯ ≥ | λ n | | {{λ _1}} | ≥ | {{λ _2}} | ≥ ⋯ ≥ | {{λ _n}} | where λ 1 = k {λ _1} = k , λ = | λ 2 | λ = | {{λ _2}} | . Then the diameter