Maxima of the index: forbidden unbalanced cycles

This paper aims to address the problem: what is the maximum index among all $\mathcal{C}^-_r$-free unbalanced signed graphs, where $\mathcal{C}^-_r$ is the set of unbalanced cycle of length $r$. Let $\Gamma_1 = C_3^- \bullet K_{n-2}$ be a signed graph obtained by identifying a vertex of $K_{n-2}$ with a vertex of $C_3^-$ whose two incident edges in $C_3^-$ are all positive, where $C_3^-$ is an unbalanced triangle with one negative edge. It is shown that if $\Gamma$ is an unbalanced signed graph of order $n$, $r$ is an integer in $\{4, \cdots, \lfloor \frac{n}{3}\rfloor + 1 \}$, and $$\lambda_{1}(\Gamma) \geq \lambda_{1}(\Gamma_1),$$ then $\Gamma$ contains an unbalanced cycle of length $r$, unless $\Gamma \sim \Gamma_1$. \indent It is shown that the result are significant in spectral extremal graph problems. Because they can be regarded as a extension of the spectral Tur{\'a}n problem for cycles [Linear Algebra Appl. 428 (2008) 1492--1498] in the context of signed graphs. Furthermore, our result partly resolved a recent open problem raised by Lin and Wang [arXiv preprint arXiv:2309.04101 (2023)].