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