Extremed signed graphs for triangle

Abstract

In this paper, we study the Turán 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 $u{u}_1,\dots ,u{u}_s,v{v}_1,\dots ,v{v}_t.$ 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 $\dot{G}\sim {\dot{G}}^{1,n-3}.$

19 pages, 1 figures