The graphs with all but two eigenvalues equal to $-2$ or $0$

TL;DR

This paper classifies all graphs whose adjacency matrices have at most two eigenvalues different from -2 or 0, and identifies which are uniquely determined by their spectra.

Contribution

It provides a complete characterization of such graphs and analyzes their spectral uniqueness, advancing understanding in spectral graph theory.

Findings

Identified all graphs with at most two eigenvalues not equal to -2 or 0.

Determined which of these graphs are uniquely identified by their adjacency spectra.

Contributed to the classification problem in spectral graph theory.

Abstract

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $-2$, or $0$, and determine which of these graphs are determined by their adjacency spectrum.