Turán problem for ${\mathcal{K}}_4^{-}$-free signed graphs

Abstract

Suppose that $\dot{G}$ is an unbalanced signed graph of order $n$ with $e(\dot{G})$ edges. Let $\rho (\dot{G})$ be the spectral radius of $\dot{G}$, and ${\mathcal{K}}_4^{-}$ be the set of the unbalanced $K_4$. In this paper, we prove that if $\dot{G}$ is a ${\mathcal{K}}_4^{-}$-free unbalanced signed graph of order $n$, then $e(\dot{G})⩽\frac{n(n-1)}{2}-(n-3)$ and $\rho (\dot{G})⩽n-2$. Moreover, the extremal graphs are completely characterized.