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 $2n-2c(G)\le r(G,\sigma )+2\alpha (G)\le 2n$. Furthermore, the signed graphs that reaching the lower bound are investigated.