The rank of a complex unit gain graph in terms of the rank of its underlying graph

Let $$\Phi =(G, \varphi )$$ Φ = ( G , φ ) be a complex unit gain graph (or $$\mathbb {T}$$ T -gain graph) and $$A(\Phi )$$ A ( Φ ) be its adjacency matrix, where G is called the underlying graph of $$\Phi $$ Φ . The rank of $$\Phi $$ Φ , denoted by $$r(\Phi )$$ r ( Φ ) , is the rank of $$A(\Phi )$$ A ( Φ ) . Denote by $$\theta (G)=|E(G)|-|V(G)|+\omega (G)$$ θ ( G ) = | E ( G ) | - | V ( G ) | + ω ( G ) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and $$\omega (G)$$ ω ( G ) are the number of edges, the number of vertices and the number of connected components of G, respectively. In this paper, we investigate bounds for $$r(\Phi )$$ r ( Φ ) in terms of r(G), that is, $$r(G)-2\theta (G)\le r(\Phi )\le r(G)+2\theta (G)$$ r ( G ) - 2 θ ( G ) ≤ r ( Φ ) ≤ r ( G ) + 2 θ ( G ) , where r(G) is the rank of G. As an application, we also prove that $$1-\theta (G)\le \frac{r(\Phi )}{r(G)}\le 1+\theta (G)$$ 1 - θ ( G ) ≤ r ( Φ ) r ( G ) ≤ 1 + θ ( G ) . All corresponding extremal graphs are characterized.