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

Abstract

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 $$|V(G)|-2m(G)-c(G)\le \eta (G,\sigma )\le |V(G)|-2m(G)+2c(G),$$ 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 no signed graphs with nullity $|V(G)|-2m(G)+2c(G)-1$. We also prove that there are infinite signed graphs with nullity $|V(G)|-2m(G)+2c(G)-s,(0\le s\le 3c(G),s\ne 1)$ for a given $c(G)$.

15pages, 2 figures