No signed graph with the nullity $\eta (G,\sigma )=\mathrm{\mid }V(G)\mathrm{\mid }-2m(G)+2c(G)-1$

Let $G^{\sigma }=(G,\sigma )$ be a signed graph and $A(G,\sigma )$ be its adjacency matrix. Denote by $m(G)$ the matching number of $G$. Let $\eta (G,\sigma )$ be the nullity of $(G,\sigma )$. He et al. [Bounds for the matching number and cyclomatic number of a signed graph in terms of rank, Linear Algebra Appl. 572 (2019), 273–291] proved that $$\mathrm{\mid }V(G)\mathrm{\mid }-2m(G)-c(G)\le \eta (G,\sigma )\le \mathrm{\mid }V(G)\mathrm{\mid }-2m(G)+2c(G),$$ where $c(G)$ is the dimension of cycle space of $G$. Signed graphs reaching the lower bound or the upper bound are respectively characterized by the same paper. In this paper, we will prove that no signed graphs with nullity $\mathrm{\mid }V(G)\mathrm{\mid }-2m(G)+2c(G)-1$. We also prove that there are infinite signed graphs with nullity $\mathrm{\mid }V(G)\mathrm{\mid }-2m(G)+2c(G)-s,(0\le s\le 3c(G),s\mathrm{\ne }1)$ for a given $c(G)$.