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

TL;DR

This paper classifies signed (0,2)-graphs with few eigenvalues and symmetric spectra, focusing on those with limited degrees and eigenvalues, and explores spectral determination of subgraphs.

Contribution

It provides a complete classification of signed (0,2)-graphs with degrees up to 6 and two eigenvalues, and investigates spectral determination of their subgraphs.

Findings

Classified all signed (0,2)-graphs with degree ≤6 and two eigenvalues

Characterized signed (0,2)-graphs with symmetric spectra and three or four eigenvalues

Explored spectral determination of induced subgraphs

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.