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

TL;DR

This paper establishes inequalities relating inertia indices, cyclomatic number, and pendant vertices in signed graphs, and characterizes graphs where equality holds, advancing understanding of spectral properties in graph theory.

Contribution

It introduces a new inequality linking positive inertia index, cyclomatic number, and pendant vertices, and characterizes the graphs achieving equality.

Findings

Proves that the positive inertia index satisfies a specific lower bound.

Characterizes all signed graphs where the bound is tight.

Derives additional inequalities involving negative inertia index and inertia-related parameters.

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.