  The study on parity signed graphs was initiated by Acharya and Kureethara
very recently and then followed by Zaslavsky etc.. Let $(G,\sigma)$ be a signed
graph on $n$ vertices. If $(G,\sigma)$ is switch-equivalent to $(G,+)$ at a set
of $\lfloor \frac{n}{2} \rfloor$ many vertices, then we call $(G,\sigma)$ a
parity signed graph and $\sigma$ a parity-signature. $\Sigma^{-}(G)$ is defined
as the set of the number of negative edges of $(G,\sigma)$ over all possible
parity-signatures $\sigma$. The $rna$ number $\sigma^-(G)$ of $G$ is given by
$\sigma^-(G)=\min \Sigma^{-}(G)$. In other words, $\sigma^-(G)$ is the smallest
cut size that has nearly equal sides.
  In this paper, all graphs considered are finite, simple and connected. We
apply switch method to the characterization of parity signed graphs and the
study on the $rna$ number. We prove that: for any graph $G$,
$\Sigma^{-}(G)=\left\{\sigma^{-}(G)\right\}$ if and only if $G$ is $K_{1, n-1}
$ with $n$ even or $K_{n}$. This confirms a conjecture proposed in [M. Acharya
and J.V. Kureethara. Parity labeling in signed graphs. J. Prime Res. Math., to
appear. arXiv:2012.07737]. Moreover, we prove a nontrivial upper bound for the
$rna$ number: for any graph $G$ on $m$ edges and $n$ ($n\geq 4$) vertices,
$\sigma^{-}(G)\leq \lfloor \frac{m}{2}+\frac{n}{4} \rfloor$. We show that
$K_n$, $K_n-e$ and $K_n-\triangle$ are the only three graphs reaching this
bound. This is the first upper bound for the $rna$ number so far. Finally, we
prove that: for any graph $G$, $\sigma^-(G)+\sigma^-(\overline{G})\leq
\sigma^-(G\cup \overline{G})$, where $\overline{G}$ is the complement of $G$.
This solves a problem proposed in [M. Acharya, J.V. Kureethara and T.
Zaslavsky. Characterizations of some parity signed graphs. 2020,
arXiv:2006.03584v3].
