论文标题
三角形的极端签名图
Extremed signed graphs for triangle
论文作者
论文摘要
在本文中，我们研究了签名图版本的Turán问题。假设$ \ dot {g} $是$ e（\ dot {g}）$ edges和$ e^ - （\ dot {g}）$负面的连接$ n $连接的签名图$ n $，然后让$ρ（\ dot {g}）$是$ \ dot cagrable of $ \ dot ragraph（\ dot {g}）$ $ \ dot {g}^{s，t} $（$ s+t = n-2 $）是从$ v（k_ {n-2}）= \ {k_ {n-2}）= \ {u_1，\ dots，v_1，v_1，v_1，v_t $ s，v_ two p（k_ {n-2}）的全阳性clique $（k_ {n-2}，+）$中获得的$（$ s+t = n-2 $）通过添加负边缘$ uv $和正边缘$ uu_1，\ dots，uu_s，uu_s，vv_1，\ dots，vv_t。$首先，我们证明，如果$ \ dot {g} $是$ C_3^ - $ E（$ c_3^ - free），那么dots $ u $和$ v $和$ v $。 \ frac {n（n-1）} {2} - （n-2），$均等时，仅当$ \ dot {g} \ sim \ dot {g} {g}^{s，t}。 \ lfloor \ frac {n-2} {2} \ rfloor \ lceil \ lceil \ frac {n-2} {2} {2} \ rceil+n-2，$ equality holdity in equality holding if，horl仅当$ \ dot {g}^g}^{s，t} = \ dot {g} _u^{\ lfloor \ frac {n-2} {2} {2} \ rfloor，\ lceil \ frac {n-2} {2} {2} \ rceil}，$ $ \ dot {g} _u^{\ lfloor \ frac {n-2} {2} {2} \ rfloor，\ lceil \ frac {n-2} {2} {2} \ rceil} $是从中获得的$ \ dot {g}^{\ lfloor \ frac {n-2} {2} {2} \ rfloor，\ lceil \ frac {n-2} {2} {2} \ rceil} $通过在verte set上切换$ u = \ { \ frac {1} {2}（\ sqrt {n^2-8}+n-4），$具有等值时，只有$ \ dot {g} {g} \ sim \ sim \ dot {g}^{1，n-3}。
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 $ρ(\dot{G})$ 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 $ρ(\dot{G})\le \frac{1}{2}( \sqrt{ n^2-8}+n-4),$ with equality holding if and only if $\dot{G}\sim \dot{G}^{1,n-3}.$
