Turán problem for $C_{2k+1}^{-}$-free signed graph

Abstract: In this paper, we study the Tur\'{a}n problem for $C_{2k+1}^{-}$. Suppose that $\dot{G}$ is an unbalanced signed graph of order $n$ with $e(\dot{G})$ edges. Let $\lambda_{1} (\dot{G})$ be the largest eigenvalue of $\dot{G}$, and $C_{2k+1}^{-}$ be the set of the negative cycle with length $2k+1$($3 \le k \le \frac{n}{15}$). We prove that if $\dot{G}$ is a $C_{2k+1}^{-}$-free unbalanced signed graph, then $e(\dot{G}) \le e(C_{3}^{-} \cdot K_{n-2})$ and $\lambda_{1}(\dot{G}) \le \lambda_{1}(C_{3}^{-} \cdot K_{n-2})$, with equality holding if and only if $\dot{G}$ is switching equivalent to $C_{3}^{-} \cdot K_{n-2}$.