The History of Hoffman’s (Ratio) Bound

Hoffman’s bound (or: ratio bound) on the size of a coclique (or: independent set, stable set) in a graph is one of the most important bounds in spectral graph theory. At the same time it is often misattributed. Primary reason for is that Hoffman never published it, but people want to cite something for it. A few weeks ago, Willem Haemers published a nice article which presents the history of Hoffman’s ratio bound (here is the journal version).

As I have probably misattributed the bound myself in the past and even one of my favorite books, “Distance-Regular Graphs” by Brouwer, Cohen and Neumaier does so too (but they correct it online), I wanted to make this quick post.

The sad occasion of Willem’s article is that Alan J. Hoffman past away on the 18th January 2021 at the age of 96. May he rest in peace.

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Don’t be a Square (but count them)

One of the structures investigated in finite geometry are related to quadratic forms over finite fields (see below for definitions). Knowledge on the geometry of singular points of quadratic forms is very common and covered in many textbooks on finite geometry, but one cannot say the same for the geometry on non-singular points. This short post tries to amend this a little.

(This is no surprise as the geometry with singular points is much nicer than the geometry associated with various types of non-singular points. Also, everything in the following is well-known for decades. It is simply a bit more obscure than other facts about finite quadrics. Lastly, my title is a terrible pun on slang from the mid-20th century (and Pulp Fiction).)