On the relation between the H-rank of a mixed graph and the matching number of its underlying graph

摘要

Amixed graph G is obtained by orienting some edges of G, where G is the underlying graph of G. Let H( G) denote the Hermitian adjacency matrix of G and m(G) be the matching number of G. The H-rank of G, written as rk( G), is the rank of H( G). Denoted by d(G) the dimension of cycle spaces of G, that is d(G) = | EG| -| VG| +.(G), where | VG|, | EG|,.(G), respectively, denotes the size, order and the number of connected components of G. In this paper, we concentrate on the relation between the H-rank of G and the matching number of G. Firstly, it is proved that -2d(G) = rk( G) -2m(G) = d(G) for every connected mixed graph G. Secondly, the mixed graphs that attain the upper and lower bounds are characterized, respectively. By these obtained results in the current paper, as a unified approach, all those main results obtained in [ Linear Algebra Appl. 465 (2015) 363-375; J. Inequal. Appl. 20 (2016) 71; Eur. J. Combin. 54 (2016) 76-86; Discrete Appl. Math. 215 (2016) 171-176] can be deduced consequently.