Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult

TL;DR

This paper extends the classification of graphs based on their smallest eigenvalue, covering infinitely many new cases where the eigenvalue lies between -λ* and -2, broadening the scope of Cameron et al.'s 1976 theorem.

Contribution

It provides a complete classification of connected graphs with smallest eigenvalues in the interval (-λ*, -2), generalizing the classical classification for eigenvalues at least -2.

Findings

Classified all connected graphs with smallest eigenvalue in (-λ*, -2)

First to classify infinitely many graphs with eigenvalues in this range for any λ > 2

Extended the eigenvalue classification beyond the -2 bound

Abstract

In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least $-2$ by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in $(-\lambda^*, -2)$, where $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, and $\rho$ is the unique real root of $x^3 = x + 1$. Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in $(-\lambda, -2)$ for any constant $\lambda > 2$.