Graphs with three eigenvalues and second largest eigenvalue at most 1

TL;DR

This paper classifies connected graphs that have exactly three distinct eigenvalues and a second largest eigenvalue not exceeding 1, providing a complete characterization of such graphs.

Contribution

It offers a complete classification of connected graphs with three eigenvalues and a second largest eigenvalue at most 1, filling a gap in spectral graph theory.

Findings

Complete classification of such graphs.

Characterization based on eigenvalue constraints.

Insights into spectral properties of these graphs.

Abstract

We classify the connected graphs with precisely three distinct eigenvalues and second largest eigenvalue at most 1.