On graphs with smallest eigenvalue at least -3 and their lattices

TL;DR

This paper proves that large, connected graphs with a smallest eigenvalue at least -3 are 2-integrable, extending Hoffman's 1977 result for graphs with eigenvalue at least -2.

Contribution

It generalizes Hoffman's theorem to a broader class of graphs with smallest eigenvalue at least -3.

Findings

Connected graphs with smallest eigenvalue ≥ -3 and large minimal degree are 2-integrable.

Extends Hoffman's 1977 result for eigenvalue ≥ -2.

Provides new insights into the spectral properties of graphs.

Abstract

In this paper, we show that a connected graph with smallest eigenvalue at least -3 and large enough minimal degree is 2-integrable. This result generalizes a 1977 result of Hoffman for connected graphs with smallest eigenvalue at least -2.