The rank of a complex unit gain graph in terms of the matching number

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, 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(ei,j)=φ(ej,i)−1. In this paper, we prove that 2m(G)−2c(G)≤r(G,φ)≤2m(G)+c(G), where r(G,φ), m(G) and c(G) are the rank of the Hermitian adjacency matrix H(G,φ), the matching number and the cyclomatic number of G, respectively. Furthermore, the complex unit gain graphs (G,T,φ) with r(G,φ)=2m(G)−2c(G) and r(G,φ)=2m(G)+c(G) are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that 2m(G−V0)≤r(G,φ)≤2m(G)+S holds for any S⊂V(G) such that G−S is bipartite and any subset V0 of V(G) such that G−V0 is acyclic.