  The nonnegative inverse eigenvalue problem (NIEP) is to characterize the
spectra of entrywise nonnegative matrices. A finite multiset of complex numbers
is called realizable if it is the spectrum of an entrywise nonnegative matrix.
Monov conjectured that the k\textsuperscript{th}-moments of the list of
critical points of a realizable list are nonnegative. Johnson further
conjectured that the list of critical points must be realizable. In this work,
Johnson's conjecture, and consequently Monov's conjecture, is established for a
variety of important cases including Ciarlet spectra, Sule\u{\i}manova spectra,
spectra realizable via companion matrices, and spectra realizable via
similarity by a complex Hadamard matrix. Additionally we prove a result on
differentiators and trace vectors, and use it to provide an alternate proof of
a result due to Malamud and a generalization of a result due to Kushel and
Tyaglov on circulant matrices. Implications for further research are discussed.
