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

Let Φ = (G, φ) be a complex unit gain graph (or T-gain graph) and A(Φ) be its adjacency matrix, where G is called the underlying graph of Φ. The rank of Φ , denoted by r(Φ) , is the rank of A(Φ). Denote by θ(G) = | E(G) | - | V(G) | + ω(G) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and 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(Φ) in terms of r(G), that is, 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-θ(G)≤r(Φ)r(G)≤1+θ(G). All corresponding extremal graphs are characterized.