The gap between the rank of a complex unit gain graph and its underlying graph

Let Phi = (G, G , phi) be a complex unit gain graph (or T-gain graph) and A (Phi) be its adjacency matrix, where G is the underlying graph of Phi . The rank r (Phi) of Phi is the rank of A (Phi). Lu et al. (2019) proved that r (G) - 2c(G) c (G) <= r (Phi) <= r (G) + 2c(G), c (G), where c (G) = | E (G) | - | V (G) | + omega (G) is the dimension of cycle space of G , omega (G) is the number of connected components of G . In this paper, we prove that no T-gain graphs Phi with the rank r (Phi) = r (G) + 2c(G) c (G) - 1. We also prove that no T-gain graphs Phi with the rank r (Phi) = r (G) - 2c(G) c (G) + 1, when T-gain cycles (if any) in Phi are not of Type E . For a given c (G), we obtain that there are infinitely many connected T-gain graphs with rank r (Phi) = r (G) + 2c(G) c (G) - s , where s is an element of [ 0 , 4c(G)], c (G) ] , s not equal 1 and 4c(G) c (G) - 1. These results can be applied to signed graphs and mixed graphs. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.