Bounds for the rank of a complex unit gain graph in terms of the independence number

A complex unit gain graph (or -gain graph) is a triple (or for short) consisting of a simple graph G with , as the underlying graph of , the set of unit complex numbers and a gain function with the property that . The adjacency matrix of is , where if is adjacent to and otherwise. The rank of , denoted by , is the rank of . Let and be the independence number and the cyclomatic number of G, respectively. In this paper, we prove that . And the properties of the complex unit gain graphs that attain the lower bound are characterized. Furthermore, the lower and upper bounds on , and are identified. These results generalize the corresponding known results about undirected graphs, mixed graphs, oriented graphs and signed graphs.