A generalization of a theorem of Hoffman

TL;DR

This paper extends Hoffman's theorem to graphs with smaller smallest eigenvalues using Hoffman graphs, revealing structural properties of such graphs and applying results to specific well-known graph classes.

Contribution

It generalizes Hoffman's characterization to a broader class of graphs with smaller eigenvalues using Hoffman graphs, providing new structural insights.

Findings

Graphs with smallest eigenvalue at least λ satisfy certain local conditions.

The results classify graphs cospectral with Hamming, Johnson, and grid extension graphs.

The approach links spectral properties to combinatorial structure.

Abstract

In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least $-2$. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph, introduced by Woo and Neumaier in 1995. Our result says that for every $\lambda \leq -2$, if a graph with smallest eigenvalue at least $\lambda$ satisfies some local conditions, then it is highly structured. We apply our result to graphs which are cospectral with the Hamming graph $H(3,q)$, the Johnson graph $J(v, 3)$ and the $2$-clique extension of grids, respectively.