# The relation between the independence number and rank of a signed graph

## TL;DR

This paper explores the relationship between the independence number, rank of the adjacency matrix, and cyclomatic number in signed graphs, establishing bounds and characterizing extremal cases.

## Contribution

It introduces new bounds linking independence number, rank, and cyclomatic number in signed graphs and investigates conditions for extremal cases.

## Key findings

- Established bounds: 2n - 2c(G) ≤ r(G, σ) + 2α(G) ≤ 2n
- Characterized signed graphs reaching the lower bound
- Enhanced understanding of structural properties of signed graphs

## Abstract

A signed graph $(G, \sigma)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $(G, \sigma)$. Let $c(G)$, $\alpha(G)$ and $r(G, \sigma)$ be the cyclomatic number, the independence number and the rank of the adjacency matrix of $(G, \sigma)$, respectively. In this paper, we study the relation among the independence number, the rank and the cyclomatic number of a signed graph $(G, \sigma)$ with order $n$, and prove that $2n-2c(G) \leq r(G, \sigma)+2\alpha(G) \leq 2n$. Furthermore, the signed graphs that reaching the lower bound are investigated.