Bounds for the rank of a complex unit gain graph in terms of its maximum degree

Let Φ=(G,φ) be a T-gain graph with maximum degree Δ. Zhou et al. (2018) [35] gave a lower bound of the rank of simple graphs in items of maximum degree. Motivated by above result, in this paper, we extend this result to T-gain graphs. We obtain that r(Φ)≥nΔ for a T-gain graph. All the corresponding extremal graphs are characterized. Furthermore, we also obtain some other results about the rank of T-gain graphs.