The index of $t{\mathcal{C}}_3^{-}$-free signed graphs

The classical spectral Turán problem is to determine the maximum spectral radius of an $F$-free graph of order $n$. This paper extends this framework to signed graphs. Let ${\mathcal{C}}_r^{-}$ be the set of all unbalanced signed graphs with underlying graphs $C_r$. Wang, Hou and Li [Linear Algebra Appl, 681 (2024) 47-65] previously determined the spectral Turán number of ${\mathcal{C}}_3^{-}$. In the present work, we characterize the extremal graphs that achieve the maximum index among all unbalanced signed graphs of order $n$ that are $t\mathcal{C}{3}^{-}$-free for $t\ge 2$. Furthermore, for $t\ge 3$, we identify the graphs with the second maximum index among all $t\mathcal{C}{3}^{-}$-free unbalanced signed graphs of fixed order $n$.