The largest eigenvalue of ${\mathcal{C}}_4^{-}$-free signed graphs

Let ${\mathcal{C}}_k^{-}$ be the set of all negative $C_k$. For odd cycle, Wang, Hou and Li [29] gave a spectral condition for the existence of negative $C_3$ in unbalanced signed graphs. For even cycle, we determine the maximum index among all ${\mathcal{C}}_4^{-}$-free unbalanced signed graphs and completely characterize the extremal signed graph in this paper. This could be regarded as a signed graph version of the results by Nikiforov [23] and Zhai and Wang [37].