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

Abstract

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