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

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