The signed graphs with two eigenvalues unequal to $\pm 1$

TL;DR

This paper characterizes signed graphs whose adjacency matrices have all but at most two eigenvalues equal to ±1, completing the classification for the remaining cases not covered in prior work.

Contribution

It provides a complete classification of signed graphs with all but two eigenvalues equal to ±1, extending previous results to the remaining cases.

Findings

Complete classification of such signed graphs.

Identification of remaining cases not covered before.

Extension of previous spectral graph theory results.

Abstract

We complete the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$. The unsigned graphs and the disconnected, the bipartite and the complete signed graphs with this property have already been determined in two earlier papers. Here we deal with the remaining cases.