Hoffman's ratio bound

TL;DR

Hoffman's ratio bound provides an eigenvalue-based upper limit on the independence number of regular graphs, but its history and generalizations are not well documented, leading to confusion among users.

Contribution

This paper clarifies the history of Hoffman's ratio bound and discusses some of its generalizations, making the concept more accessible and better understood.

Findings

Hoffman's ratio bound is a useful eigenvalue-based upper bound.

The paper clarifies the historical development of the bound.

It discusses generalizations of Hoffman's bound.

Abstract

Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With this note I hope to clarify the history of this bound and some of its generalizations.