Signed $(0,2)$-graphs with few eigenvalues and a symmetric spectrum

Abstract

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two distinct eigenvalues $\pm \lambda$. Next, we consider to what extent induced subgraphs of signed graph with two distinct eigenvalues $\pm \lambda$ are determined by their spectra. Lastly, we classify signed $(0,2)$-graphs that have a symmetric spectrum with three distinct eigenvalues and give a partial classification for those with four distinct eigenvalues.