Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices

Let $\mathrm{\Gamma }=(G,\sigma )$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $\sigma :E(G)\to \{+,-\}$. Denoted by $i_{+}(\mathrm{\Gamma })$, $\theta (\mathrm{\Gamma })$ and $p(\mathrm{\Gamma })$ the positive inertia index, the cyclomatic number and the number of pendant vertices of $\mathrm{\Gamma }$, respectively. In this article, we prove that $i_{+}(\mathrm{\Gamma })$, $\theta (\mathrm{\Gamma })$ and $p(\mathrm{\Gamma })$ are related by the inequality $i_{+}(\mathrm{\Gamma })\ge \frac{n-p(\mathrm{\Gamma })}{2}-\theta (\mathrm{\Gamma })$. Furthermore, we completely characterize the signed graph $\mathrm{\Gamma }$ for which $i_{+}(\mathrm{\Gamma })=\frac{n-p(\mathrm{\Gamma })}{2}-\theta (\mathrm{\Gamma })$. As a by-product, the inequalities $i_{-}(\mathrm{\Gamma })\ge \frac{n-p(\mathrm{\Gamma })}{2}-\theta (\mathrm{\Gamma })$ and $\eta (\mathrm{\Gamma })\le p(\mathrm{\Gamma })+2\theta (\mathrm{\Gamma })$ are also obtained, respectively.