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Let $S=(G, \sigma)$ be a signed graph where $G=(V, E)$ is a graph called the underlying graph of $S$ and $\sigma:E(G) \rightarrow \{+,~-\}$. Let $f:V(G) \rightarrow \{1,2,\dots,|V(G)|\}$ such that $\sigma(uv)=+$ if and only if $f(u)$ and $f(v)$ are of same parity and $\sigma(uv)=-$ if and only if $f(u)$ and $f(v)$ are of opposite parity. Under $f$ we get a signed graph $G_f$ denoted as $S$, which is a parity signed graph. In this paper, we initiate the study of parity labeling in signed graphs and we define and find `rna' number denoted as $\sigma^-(S)$ for some classes of signed graphs. We also characterize some signed graphs which are parity signed graphs. Some directions for further research are also suggested., Comment: 10 pages, 1 figure

Recently, Bennett et al. introduced the vertex-induced weighted Tur\'an problem. In this paper, we consider their open Tur\'an problem under sum-edge-weight function and characterize the extremal structure of $K_{l}$-free graphs. Based on these results, we propose a generalized version of the Erd\H{o}s-Stone theorem for weighted graphs under two types of vertex-induced weight functions.