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 rectagraphs (triangle‐free signed ( 0 , 2 ) $(0,2)$‐graphs) with vertex degree at most 6 that have precisely two distinct eigenvalues ± λ $\pm \lambda $. Next, we consider to what extent induced subgraphs of signed graph with two distinct eigenvalues ± λ $\pm \lambda $ are determined by their spectra. Lastly, we classify signed rectagraphs that have a symmetric spectrum with three distinct eigenvalues and give a partial classification for signed ( 0 , 2 ) $(0,2)$‐graphs with four distinct eigenvalues.