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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 $d(d+1)$., Comment: 10 pages

For $m \geq 3$ and $n \geq 1$, the $m$-cycle book graph $B(m,n)$ consists of $n$ copies of the cycle $C_m$ with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed $B(m,n)$ is $n+1$, and (b) the chromatic number of a signed $B(m,n)$ is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs., Comment: Substantial text overlap with arXiv:1812.08382

A signed graph $\Sigma=(G,\sigma)$ is said to be parity signed if there exists a bijection $f : V(G) \rightarrow \{1,2,...,|V(G)|\}$ such that $\sigma(uv)=+$ if and only if $f(u)$ and $f(v)$ are of same parity, where $uv$ is an edge of $G$. The rna number of a graph $G$, denoted $\sigma^{-}(G)$, is the minimum number of negative edges among all possible parity signed graphs over $G$. The rna number is also equal to the minimum cut size that has nearly equal sides. In this paper, for generalized Petersen graph $P(n,k)$, we prove that $3 \leq \sigma^{-}(P(n,k)) \leq n$ and these bounds are sharp. The exact value of $\sigma^{-}(P(n,k))$ is determined for $k=1,2$. Some famous generalized Petersen graphs namely, Petersen graph $P(5,2)$, Durer graph $P(6,2)$, Mobius-Kantor graph $P(8,3)$, Dodecahedron $P(10,2)$, Desargues graph $P(10,3)$ and Nauru graph $P(12,5)$ are also treated. We show that the minimum order of a $(4n-1)$-regular graph having rna number one is bounded above by $12n-2$. The sharpness of this upper bound is also shown for $n=1$. We also show that the minimum order of a $(4n+1)$-regular graph having rna number one is $8n+6$. Finally, for any simple connected graph of order $n$, we propose an $O(2^n + n^{\lfloor \frac{n}{2} \rfloor})$ time algorithm for computing its rna number.

Two signed graphs are called switching isomorphic to each other if one is isomorphic to a switching of the other. The wheel $W_n$ is the join of the cycle $C_n$ and a vertex. For $0 \leq p \leq n$, $\psi_{p}(n)$ is defined to be the number of switching non-isomorphic signed $W_n$ with exactly $p$ negative edges on $C_n$. The number of switching non-isomorphic signed $W_n$ is denoted by $\psi(n)$. In this paper, we compute the values of $\psi_{p}(n)$ for $p=0,1,2,3,4,n-4,n-3,n-2,n-1,n$ and of $\psi(n)$ for $n=4,5,...,10$. Our method of obtaining $\psi_{p}(n)$ not only count the switching non-isomorphic signed wheels but also generates them.

In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $\Sigma = (G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset $D \subseteq V(\Sigma)$ is a double dominating set of $\Sigma$ if it satisfies the following conditions: (i) $D$ is a double dominating set of $G$, and (ii) $\Sigma[D:V \setminus D]$ is balanced, where $\Sigma[D:V \setminus D]$ is the subgraph of $\Sigma$ induced by the edges of $\Sigma$ with one end point in $D$ and the other end point in $V \setminus D$. The cardinality of a minimum double dominating set of $\Sigma$ is the double domination number $\gamma_{\times 2}(\Sigma)$. In this paper, we give bounds for the double domination number of signed cubic graphs. We also obtain some bounds on the double domination number of signed generalized Petersen graphs and signed I-graphs., Comment: 13 pages

A \textit{signed graph} is a simple graph whose edges are labelled with positive or negative signs. A cycle is \textit{positive} if the product of its edge signs is positive. A signed graph is \textit{balanced} if every cycle in the graph is positive. The \textit{frustration index} of a signed graph is the minimum number of edges whose deletion makes the graph balanced. The \textit{maximum frustration} of a graph is the maximum frustration index over all sign labellings. In this paper, first, we prove that the maximum frustration of generalized Petersen graphs $P_{n,k}$ is bounded above by $\left\lfloor \frac{n}{2} \right\rfloor + 1$ for $\gcd(n,k)=1$, and this bound is achieved for $k=1,2,3$. Second, we prove that the maximum frustration of $P_{n,k}$ is bounded above by $d\left\lfloor \frac{n}{2d} \right\rfloor + d + 1$, where $\gcd(n,k)=d\geq2$.

A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs. It is also known that, up to switching isomorphism, there are two signed $K_3$'s, three signed $K_4$'s, and seven signed $K_5$'s. In this paper, we prove that there are sixteen signed $K_6$'s upto switching ismomorphism., Comment: 8 Pages