Spectral Turán problem for ${\mathcal{K}}_{3,3}^{-}$-free signed graphs

Abstract

The classical spectral Turán problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n$. Zhai and Wang [Linear Algebra Appl, 437 (2012) 1641-1647] determined the maximum spectral radius of $C_4$-free graphs of given order. Additionally, Nikiforov obtained spectral strengthenings of the Kővari-Sós-Turán theorem [Linear Algebra Appl, 432 (2010) 1405-1411] when the forbidden graphs are complete bipartite. The spectral Turán problem concerning forbidden complete bipartite graphs in signed graphs has also attracted considerable attention. Let ${\mathcal{K}}_{s,t}^{-}$ be the set of all unbalanced signed graphs with underlying graphs $K_{s,t}$. Since the cases where $s=1$ or $t=1$ do not conform to the definition of ${\mathcal{K}}_{s,t}^{-}$, it follows that $s,t\ge 2$. Wang and Lin [Discrete Appl. Math, 372 (2025) 164-172] have solved the case of $s=t=2$ since ${\mathcal{K}}_{2,2}^{-}$ is ${\mathcal{C}}_4^{-}$ in this situation. This paper gives an answer for $s=t=3$ and completely characterizes the corresponding extremal signed graphs.