Connected signed graphs with given inertia indices and given girth

Suppose that $\mathrm{\Gamma }=(G,\sigma )$ is a connected signed graph with at least one cycle. The number of positive, negative and zero eigenvalues of the adjacency matrix of $\mathrm{\Gamma }$ are called positive inertia index, negative inertia index and nullity of $\mathrm{\Gamma }$, which are denoted by $i_{+}(\mathrm{\Gamma })$, $i_{-}(\mathrm{\Gamma })$ and $\eta (\mathrm{\Gamma })$, respectively. Denoted by $g$ the girth, which is the length of the shortest cycle of $\mathrm{\Gamma }$. We study relationships between the girth and the negative inertia index of $\mathrm{\Gamma }$ in this article. We prove $i_{-}(\mathrm{\Gamma })\ge \lceil \frac{g}{2}\rceil -1$ and extremal signed graphs corresponding to the lower bound are characterized. Furthermore, the signed graph $\mathrm{\Gamma }$ with $i_{-}(\mathrm{\Gamma })=\lceil \frac{g}{2}\rceil$ for $g\ge 4$ are given. As a by-product, the connected signed graphs with given positive inertia index, nullity and given girth are also determined, respectively.