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

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.