Maximum index of unbalanced signed graphs without a positive odd cycle

Let ${\mathcal{C}}_r^{+}$ be the set of all balanced signed graphs with underlying graphs $C_r$. A signed graph is called ${\mathcal{C}}_r^{+}$-free if it contains no subgraph isomorphic to any member of ${\mathcal{C}}_r^{+}$. In this paper, we characterize the extremal graphs that achieve the maximum index among all ${\mathcal{C}}_{2k+1}^{+}$-free unbalanced signed graphs of order $n$ when $k\ge 2$ and $n\ge 187k$.