On signed graphs with at most two eigenvalues unequal to $\pm 1$

TL;DR

This paper investigates signed graphs whose adjacency matrices have all but at most two eigenvalues equal to ±1, exploring various classes and providing examples that cannot be derived from unsigned graphs.

Contribution

It is the first to analyze signed graphs with adjacency matrices having all but two eigenvalues equal to ±1, including disconnected, bipartite, and complete cases.

Findings

Characterization of signed graphs with at most two eigenvalues not equal to ±1.

Examples of signed graphs not obtainable from unsigned graphs by switching.

Initial steps towards classifying such graphs based on their eigenvalues.

Abstract

We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.