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

TL;DR

This paper proves that no signed graph can have nullity exactly equal to the upper bound minus one, and constructs infinite examples for certain nullity values within a specific range.

Contribution

It establishes the non-existence of signed graphs with nullity at the upper bound minus one and provides infinite examples for other nullity values.

Findings

No signed graph with nullity |V(G)| - 2m(G) + 2c(G) - 1 exists.

Infinite signed graphs exist with nullity |V(G)| - 2m(G) + 2c(G) - s for 0 ≤ s ≤ 3c(G), s ≠ 1.

Bounds for nullity are tight except for the case s=1, which is impossible.

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 $$|V(G)|-2m(G)-c(G)\leq\eta(G,\sigma)\leq |V(G)|-2m(G)+2c(G),$$ 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 $c(G)$.