In this paper, we study the Tur\'{ a}n problem of signed graphs version. Suppose that $\dot{ G}$ is a connected unbalanced signed graph of order $n$ with $e(\dot{ G})$ edges and $e^-(\dot{ G})$ negative edges, and let $\rho(\dot{ G})$ be the spectral radius of $\dot{ G}.$ The signed graph $\dot{ G}^{ s,t}$ ($s+t=n-2$) is obtained from an all-positive clique $(K_{ n-2},+)$ with $V(K_{ n-2})=\{ u_1,\dots,u_s,v_1,\dots,v_t\}$ ($s,t\ge 1$) and two isolated vertices $u$ and $v$ by adding negative edge $uv$ and positive edges $uu_1,\dots,uu_s,vv_1,\dots,vv_t.$ Firstly, we prove that if $\dot{ G}$ is $C_3^-$-free, then $e(\dot{ G})\le \frac{ n(n-1)}{ 2}-(n-2),$ with equality holding if and only if $\dot{ G}\sim \dot{ G}^{ s,t}.$ Moreover, $e^-(\dot{ G}^{ s,t})\le \lfloor\frac{ n-2}{ 2}\rfloor\lceil\frac{ n-2}{ 2}\rceil+n-2,$ with equality holding if and only if $\dot{ G}^{ s,t}= \dot{ G}_U^{ \lfloor\frac{ n-2}{ 2}\rfloor,\lceil\frac{ n-2}{ 2}\rceil},$ where $\dot{ G}_U^{ \lfloor\frac{ n-2}{ 2}\rfloor,\lceil\frac{ n-2}{ 2}\rceil}$ is obtained from $\dot{ G}^{ \lfloor\frac{ n-2}{ 2}\rfloor,\lceil\frac{ n-2}{ 2}\rceil}$ by switching at vertex set $U=\{ v,u_1,\dots,u_{ \lfloor\frac{ n-2}{ 2}\rfloor}\}.$ Secondly, we prove that if $\dot{ G}$ is $C_3^-$-free, then $\rho(\dot{ G})\le \frac{ 1}{ 2}( \sqrt{ n^2-8}+n-4),$ with equality holding if and only if $\dot{ G}\sim \dot{ G}^{ 1,n-3}.$
Categories: math.CO