• Corpus ID: 219177437

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

@article{Lu2020NoSG,
  title={No signed graph with the nullity \$\eta(G,\sigma)=|V(G)|-2m(G)+2c(G)-1\$},
  author={Yong Lu and Jingwen Wu},
  journal={arXiv: Combinatorics},
  year={2020}
}

• Yong Lu, Jingwen Wu
• Published 29 May 2020
• Mathematics
• arXiv: Combinatorics

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 $$|V(G)|-2m(G)-c(G)\leq\eta(G,\sigma)\leq |V(G)|-2m(G)+2c(G),$$ 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 $|V(G)|-2m(G)+2c(G)-s,~(0\leq s\leq3c(G), s\neq1)$ for a given $c(G)$.