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

Abstract

In this paper, we study the Turán 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 $C_3^{-}\cdot K_{n-2}$.