Some classes of connected signed graphs with girth $g$ and negative inertia index $\lceil \frac{g}{2}\rceil +1$

Let $\mathrm{\Gamma }$ be a signed graph. The number of negative eigenvalues of the adjacency matrix of $\mathrm{\Gamma }$ is called the negative inertia index of $\mathrm{\Gamma }$, which is denoted by $i_{-}(\mathrm{\Gamma })$. The length of the shortest cycle contained in $\mathrm{\Gamma }$ is called the girth of $\mathrm{\Gamma }$, and it is denoted by $g$. In this paper, we give some classes of connected signed graphs $\mathrm{\Gamma }$ which satisfy the condition $i_{-}(\mathrm{\Gamma })=\lceil \frac{g}{2}\rceil +1$.