on the rna number of generalized petersen graphs

A signed graph $(g,sigma)$ is called a parity signed graph if there exists a bijective mapping $f colon v(g) rightarrow {1,ldots,|v(g)|}$ such that for each edge $uv$ in $g$, $f(u)$ and $f(v)$ have same parity if $sigma(uv)=+1$, and opposite parity if $sigma(uv)=-1$. the emph{rna} number $sigma^{-}(g)$ of $g$ is the least number of negative edges among all possible parity signed graphs over $g$. equivalently, $sigma^{-}(g)$ is the least size of an edge-cut of $g$ that has nearly equal sides.in this paper, we show that for the generalized petersen graph $p_{n,k}$, $sigma^{-}(p_{n,k})$ lies between $3$ and $n$. moreover, we determine the exact value of $sigma^{-}(p_{n,k})$ for $kin {1,2}$. the emph{rna} numbers of some famous generalized petersen graphs, namely, petersen graph, d& urer graph, m& obius-kantor graph, dodecahedron, desargues graph and nauru graph are also computed. recently, acharya, kureethara and zaslavsky characterized the structure of those graphs whose emph{rna} number is $1$. we use this characterization to show that the smallest order of a $(4n+1)$-regular graph having emph{rna} number $1$ is $8n+6$. we also prove the smallest order of $(4n-1)$-regular graphs having emph{rna} number $1$ is bounded above by $12n-2$. in particular, we show that the smallest order of a cubic graph having emph{rna} number $1$ is 10.