Spectral Tur\'{a}n problem for $\mathcal{{K}}_{3,3}^{-}$-free signed graphs

Abstract

The classical spectral Tur\'{a}n 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 K\H{o}vari-S\'{o}s-Tur\'{a}n theorem [Linear Algebra Appl, 432 (2010) 1405-1411] when the forbidden graphs are complete bipartite. The spectral Tur\'{a}n 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 $s,t\geq 2$. 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.