On a question of Haemers regarding vectors in the nullspace of Seidel matrices

TL;DR

This paper constructs infinite graph families showing that singular Seidel matrices can have nullspace vectors with entries larger than any given bound, answering Haemers' question negatively.

Contribution

It provides counterexamples to Haemers' question and characterizes nullspace vectors of Seidel matrices, establishing necessary conditions for singularity.

Findings

Existence of graphs with singular Seidel matrices and nullspace vectors with arbitrarily large entries.

Necessary conditions for the singularity of Seidel matrices.

Properties of graphs that relate to the nullspace vectors of their Seidel matrices.

Abstract

In 2011, Haemers asked the following question: If $S$ is the Seidel matrix of a graph of order $n$ and $S$ is singular, does there exist an eigenvector of $S$ corresponding to $0$ which has only $\pm 1$ elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number $N$, there exists a graph whose Seidel matrix $S$ is singular such that for any integer vector in the nullspace of $S$, the absolute value of any entry in this vector is more than $N$. We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel matrices. Finally, we obtain some properties of the graphs which affirm the above question.