The Nullity of Bicyclic Signed Graphs

TL;DR

This paper characterizes certain nullity values of signed graphs, introduces a nullity-preserving transformation, and applies these results to classify unbalanced bicyclic signed graphs with specific nullity levels.

Contribution

It provides a characterization of signed graphs with nullity n-2 or n-3 and introduces a graph transformation that preserves nullity, with applications to bicyclic signed graphs.

Findings

Characterized signed graphs with nullity n-2 and n-3.

Introduced a nullity-preserving graph transformation.

Classified unbalanced bicyclic signed graphs with nullity n-3 or n-4.

Abstract

Let \Gamma be a signed graph and let A(\Gamma) be the adjacency matrix of \Gamma. The nullity of \Gamma is the multiplicity of eigenvalue zero in the spectrum of A(\Gamma). In this paper we characterize the signed graphs of order n with nullity n-2 or n-3, and introduce a graph transformation which preserves the nullity. As an application we determine the unbalanced bicyclic signed graphs of order n with nullity n-3 or n-4, and signed bicyclic signed graphs (including simple bicyclic graphs) of order n with nullity n-5.