Complex unit gain graphs with rank 2n(G)−2α(G)−2c(G)+1

Let r ( G φ ) be the rank of a complex unit gain graph (or T -gain graph) G φ , and let n ( G ) , α ( G ) , and c ( G ) denote the order, independence number, and cyclomatic number of G , respectively. In 2025, Wu et al. proved that no T -gain graph G φ satisfies r ( G φ ) = 2 n ( G ) − 2 α ( G ) − 2 c ( G ) + 1 if every cycle in G φ is of neither Type C nor Type D . Motivated by this result, in this paper, we completely characterize all T -gain graphs G φ satisfying r ( G φ ) = 2 n ( G ) − 2 α ( G ) − 2 c ( G ) + 1 .