Bounds for the rank of a complex unit gain graph in terms of the independence number. Issue 7 (3rd May 2022) - He, Shengjie
Abstract : A complex unit gain graph (or T -gain graph) is a triple Φ = ( G, T, ϕ ) (or ( G, ϕ ) for short) consisting of a simple graph G with | V ( G ) | = n, as the underlying graph of ( G, ϕ ), the set of unit complex numbers T = { z ∈ C : | z | = 1 } and a gain function ϕ : E → → T with the property that ϕ ( e i j ) = ϕ ( e j i ) − 1 = ϕ ( e j i ) ¯ . The adjacency matrix of ( G, ϕ ) is A ( G, ϕ ) = ( a i j ) n × n, where a i j = ϕ ( e i j ) if v i is adjacent to v j and a i j = 0 otherwise. The rank of ( G, ϕ ), denoted by r ( G, ϕ ), is the rank of A ( G, ϕ ) . Let α ( G ) and c ( G ) be the independence number and the cyclomatic number of G, respectively. In this paper, we prove that 2 n − 2 c ( G ) ≤ r ( G, ϕ ) + 2 α ( G ) ≤ 2 n . And the properties of the complex unit gain graphs that attain the lower bound are characterized. Furthermore, the lower and upper bounds on r ( G, ϕ ) + α ( G ), r ( G, ϕ ) − α ( G ) and r ( G, ϕ ) α ( G ) are identified. These results generalize the corresponding known results about undirected graphs, mixed graphs, oriented graphs and signed graphs.