Further results on the nullity of signed graphs

TL;DR

This paper explores the spectral properties of signed graphs, providing formulas for nullity related to graph structure, and characterizes nullity sets for specific bicyclic signed graphs, advancing understanding in spectral graph theory.

Contribution

It introduces new formulas for the nullity of signed graphs with cut-points and characterizes nullity sets for unbalanced bicyclic signed graphs.

Findings

Derived the coefficient theorem for the characteristic polynomial of signed graphs.

Established formulas for nullity of signed graphs with cut-points.

Determined the nullity set of bicyclic signed graphs.

Abstract

The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a signed graph, give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graph $\Gamma(\infty(p,q,l))$, obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.