Relation between the H-rank of a mixed graph and the rank of its underlying graph

Given a simple graph G = (V-G, E-G) with vertex set V-G and edge set E-G , the mixed graph (C) over tilde is obtained from G by orienting some of its edges. Let H(G)over tilde>) denote the Hermitian adjacency matrix of (G)over tilde> and A(G) be the adjacency matrix of G. The H-rank (resp. rank) of (G ) over tilde (resp. G), written as rk(G) (resp. r(G)), is the rank of H(G) (resp. A(G)). Denote by d(G) the dimension of cycle space of G, that is d(G) = vertical bar E-G vertical bar - vertical bar V-G vertical bar + omega(G), where omega(G) denotes the number of connected components of G. In this paper, we concentrate on the relation between the H-rank of (G) over tilde and the rank of G. We first show that -2d(G) <= rk((G) over tilde) - r(G) <= 2d(G) for every mixed graph (G) over tilde. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in Luo et al. (2018); Wong et al. (2016) may be deduced consequently. (C) 2019 Elsevier B.V. All rights reserved.