A mixed graph G is obtained from a simple undirected graph G,
the underlying graph of G, by orienting some edges of G. Let
c(G)=β£E(G)β£ββ£V(G)β£+Ο(G) be the cyclomatic number of G with Ο(G)
the number of connected components of G, m(G) be the matching number of
G, and Ξ·(G) be the nullity of G. Chen et al.
(2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that
β£V(G)β£β2m(G)β2c(G)β€Ξ·(G)β€β£V(G)β£β2m(G)+2c(G),
respectively, and they characterized the mixed graphs with nullity attaining
the upper bound and the lower bound. In this paper, we prove that there is no
mixed graph with nullity Ξ·(G)=β£V(G)β£β2m(G)+2c(G)β1. Moreover,
for fixed c(G), there are infinitely many connected mixed graphs with nullity
β£V(G)β£β2m(G)+2c(G)βs(0β€sβ€3c(G),sξ =1) is proved