On the rna number of powers of cycles

Abstract

A signed graph $(G,\sigma)$ on $n$ vertices is called a \textit{parity signed graph} if there is a bijective mapping $f \colon V(G) \rightarrow \{1,\ldots,n\}$ such that $f(u)$ and $f(v)$ have same parity if $\sigma(uv)=1$, and opposite parities if $\sigma(uv)=-1$ for each edge $uv$ in $G$. The \emph{rna} number $\sigma^{-}(G)$ of $G$ is the least number of negative edges among all possible parity signed graphs over $G$. In other words, $\sigma^{-}(G)$ is the smallest size of an edge-cut of $G$ such that the sizes of two sides differ at most one. Let $C_n^{d}$ be the $d\text{th}$ power of a cycle of order $n$. Recently, Acharya, Kureethara and Zaslavsky proved that the \emph{rna} number of a cycle $C_n$ on $n$ vertices is $2$. In this paper, we show for $2 \leq d < \lfloor \frac{n}{2} \rfloor$ that $2d \leq \sigma^{-}(C_n^{d}) \leq d(d+1)$. Moreover, we prove that the graphs $C_n^{2}$ and $C_n^{3}$ achieve the upper bound of $d(d+1)$.