Maximum index of unbalanced signed graphs without a positive odd cycle

Source abstract

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 $n\ge 187k$.