  An $n$-list $\lambda:=\left(r; \lambda_2, \ldots, \lambda_n\right)$ of
complex numbers with $r>0,$ is said to be realizable if $\lambda$ is the
spectrum of $n\times n$ nonnegative matrix $A$ and in this case $A$ is said to
be a nonnegative realization of $\lambda$. If, in addition, each row and column
sum of $A$ equals $r$, then $\lambda$ is said to be doubly stochastically
realizable and in such case $A$ is said to be a doubly stochastic realization
for $\lambda$. In 1997, Guo proved that if $\left(\lambda_2,\ldots,
\lambda_n\right)$ is any list of complex numbers which is closed under complex
conjugation then there exists a least real number $\lambda_0$ with
$\max\limits_{2\leq j\leq n}|\lambda_j|\leq\lambda_0\leq 2n\max\limits_{2\leq
j\leq n}|\lambda_j|$ such that the list of complex numbers $\{
\lambda_1,\lambda_2,...,\lambda_n\}$ is realizable if and only if
$\lambda_1\geq \lambda_0$. In 2020, Julio and Soto showed that the upper bound
may be reduced to $(n-1)\max\limits_{2\leq j\leq n}|\lambda_j|$ in the case
when at least one of the $\lambda_i$ is real.
  In this paper, we first describe an algorithm for passing from a nonnegative
realization to a doubly stochastic realization. As applications, we give a new
sufficient condition for a stochastic matrix $A$ to be cospectral to a doubly
stochastic matrix $B$ and in this case $B$ is shown to be the unique closest
doubly stochastic matrix to $A$ with respect to the Frobenius norm. Then, our
next results slightly improve the upper bound for the nonnegative realization
presented by Julio and Soto and in the case when none of the $\lambda_i$ is
real, we also give an improvement of Guo's bound. Then, for doubly stochastic
realizations, we obtain an upper bound that improves Guo's bound as well.
Finally, for certain particular cases, we give a further improvement of our
last bound for doubly stochastic realization.
