Characterizing the positive inertia index of connected signed graphs in terms of girth

Let $G^\sigma=(G,\sigma)$ be a connected signed graph and $A(G^\sigma)$ be its adjacency matrix. The positive inertia index of $G^\sigma$, denoted by $p^{+}(G^\sigma)$, is defined as the number of positive eigenvalues of $A(G^\sigma)$. Assume that $G^\sigma$ contains at least one cycle, and let $g_{r}$ be its girth. In this paper, we prove $p^{+}(G^\sigma) \geq \lceil \frac {g_{r}}{2} \rceil-1$ for a signed graph $G^\sigma$. The extremal signed graphs corresponding to $p^{+}(G^\sigma) = \lceil \frac {g_{r}}{2} \rceil-1$ and $p^{+}(G^\sigma) =\lceil \frac {g_{r}}{2} \rceil$ are characterized, respectively. The results presented in this article extend the recent work on ordinary graphs by Duan and Yang (Linear Algebra Appl., 2024) to the context of signed graphs.