No signed graph with the nullity η(G,σ)=|V(G)| − 2m(G)+2c(G)−1

Let Gσ=(G,σ) be a signed graph and A(G,σ) be its adjacency matrix. Denote by m(G) the matching number of G. Let η(G,σ) be the nullity of (G,σ). He et al. (2019) [6] proved that |V(G)|−2m(G)−c(G)≤η(G,σ)≤|V(G)|−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 there are no signed graphs with nullity |V(G)|−2m(G)+2c(G)−1. We also prove that there are infinitely many signed graphs with nullity |V(G)|−2m(G)+2c(G)−s,(0≤s≤3c(G),s≠1) for a given c(G). © 2021 Elsevier Inc.