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

Let $Phi=(G, varphi)$ be a complex unit gain graph (or $mathbb{T}$-gain graph) and $A(Phi)$ be its adjacency matrix, where $G$ is called the underlying graph of $Phi$. The rank of $Phi$, denoted by $r(Phi)$, is the rank of $A(Phi)$. Denote by $theta(G)=|E(G)|-|V(G)|+omega(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $omega(G)$ 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 $r(Phi)$ in terms of $r(G)$, that is, $r(G)-2theta(G)leq r(Phi)leq r(G)+2theta(G)$, where $r(G)$ is the rank of $G$. As an application, we also prove that $1-theta(G)leqfrac{r(Phi)}{r(G)}leq1+theta(G)$. All corresponding extremal graphs are characterized.