Extremed signed graphs for triangle

Abstract

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 $\dot{G}\sim \dot{G}^{1,n-3}.$