A study on certain bounds of the rna number and some characterizations of the parity signed graphs

For a given graph $G$, let $f:V(G)\to \{1,2,\ldots,n\}$ be a bijective mapping. For a given edge $uv \in E(G)$, $\sigma(uv)=+$, if $f(u)$ and $f(v)$ have the same parity and $\sigma(uv)=-$, if $f(u)$ and $f(v)$ have opposite parity. The resultant signed graph is called a parity signed graph. Let us denote a parity signed graph $S=(G,\sigma)$ by $G_\sigma$. Let $E^-(G_\sigma)$ be a set of negative edges in a parity signed graph and let $Si(G)$ be the set of all parity signatures for the underlying graph $G$. We define the \emph{rna} number of $G$ as $\sigma^-(G)=\min\{p(E^-(G_\sigma)):\sigma \in Si(G)\}$. In this paper, we prove a non-trivial upper bound in the case of trees: $\sigma^-(T)\leq \left \lceil \frac{n}{2}\right \rceil$, where $T$ is a tree of order $n+1$. We have found families of trees whose \emph{rna} numbers are bounded above by $\left \lceil \frac{\Delta}{2}\right \rceil$ and also we have shown that for any $i\leq \left \lceil \frac{n}{2}\right \rceil$, there exists a tree $T$ (of order $n+1$) with $\sigma^-(T)=i$. This paper gives the characterizations of graphs with \emph{rna} number 1 in terms of its spanning trees and characterizations of graphs with \emph{rna} number 2.