Existence of a not necessarily symmetric matrix with given distinct eigenvalues and graph

Abstract For given distinct numbers λ 1 ± μ 1 i , λ 2 ± μ 2 i , … , λ k ± μ k i ∈ C ∖ R and γ 1 , γ 2 , … , γ l ∈ R , and a given graph G with a matching of size at least k , we will show that there is a real matrix whose eigenvalues are the given numbers and its graph is G . In particular, this implies that any real matrix with distinct eigenvalues is similar to a real, irreducible, tridiagonal matrix.