Guo perturbation for symmetric nonnegative circulant matrices

Guo[W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261–270] sets the question: if the list Λ={λ1,λ2,…,λn} is symmetrically realizable (that is, Λ is the spectrum of a symmetric nonnegative matrix), and t>0, whether or not the list Λt={λ1+t,λ2±t,λ3,…,λn} is also symmetrically realizable. In this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant. We also give a necessary and sufficient condition for Λ to be the spectrum of a nonnegative left circulant matrix.