# Signed Graphs with Maximum Nullity at Most Two

## Author ORCID Identifier

https://orcid.org/0000-0003-1790-2804

5-2-2022

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mathematics and Statistics

Hein van der Holst

Marina Arav

Guantao Chen

Frank J. Hall

Zhongshan Li

## Abstract

A signed graph is an ordered pair (G,Σ), where G = (V, E) is a graph and Σ ⊆ E. The edges in Σ are called odd, and the edges in E \ Σ are called even. The family of matrices S(G,Σ) is defined such that if [ai,j] = AS(G,Σ), then ai,j < 0 if there is at least one edge between i and j and if all edges between i and j are even; ai,j > 0 if there is at least one edge between i and j and if all edges between i and j are odd; ai,jR if there is at least one even edge and at least one odd edge between i and j; and ai,j = 0 if there are no edges between i and j. The maximum nullity of a signed graph M(G,Σ) is the largest corank(A) for AS(G, Σ). The matrix AS(G, Σ) has the Strong Arnold Property with respect to (G,Σ) if X = 0 is the only matrix such that AX = 0, and xi,j = 0 if i is adjacent to j or i = j. The stable maximum nullity of a signed graph ξ(G,Σ) is the largest corank(A) for AS(G,Σ) where A has the Strong Arnold Property. Here, we present a combinatorial characterization of signed graphs with maximum nullity at most two, extending a result of Johnson, Loewy, and Smith. We also find the forbidden minors for signed graphs with stable maximum nullity at most two, extending a result of Hogben and van der Holst. We generalize the notion of zero forcing to signed graphs. We find the zero forcing number of signed graphs with maximum nullity at most two, extending a result of Row.

## DOI

https://doi.org/10.57709/28777057