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Constructing New Realisable Lists from Old in the NIEP

Given a list of complex numbers σ:=(λ_1,λ_2,...,λ_m), we say that σ is realisable if σ is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (or NIEP) is the problem of categorising all realisable lists. Given a realisable list (ρ,λ_2,λ_3,...,λ_m), where ρ is the Perron eigenvalue and λ_2 is real, we find families of lists (μ_1,μ_2,...,μ_n), for which (μ_1,μ_2,...,μ_n,λ_3,λ_4,...,λ_m) is realisable. In addition, given a realisable list (ρ,α+iβ,α-iβ,λ_4,λ_5,...,λ_m), where ρ is the Perron eigenvalue and α and β are real, we find families of lists (μ_1,μ_2,μ_3,μ_4), for which (μ_1,μ_2,μ_3,μ_4,λ_4,λ_5,...,λ_m) is realisable.