I’ve often said that trying to explain your work to others is the best way to check you understand it, and whilst preparing for this week’s talk at Cambridge (which was apparently well received!) I started to have some doubts about an algorithm I’d been using. In the end I didn’t go into enough technical detail during the talk for the issue to come up, but those concerns nagged at me during the commutes and lunchbreaks: eventually, I confirmed that there’s quite a big flaw in my proof as it stands.
For cyclotomic matrices over the rational integers, it turns out that all maximal examples satisfy the equation M2=4I. However, this is a side effect of the classification, rather than an independent result which can be used to arrive at that classification. Which is a problem, as (despite attempts to guard against it) my approach implicity assumes this condition…
Thus what I currently have is a classification not necessarily of all the maximal cyclotomic graphs, but of all the graphs whose matrix representation squares to 4I; no less substantial a piece of work, but a less significant result.
It’s still my belief that this is the full classification, but to prove that, I’ll need to demonstrate the equivalence of these two conditions (the reverse direction I already have, but that’s no help); or at least the weaker result that any cyclotomic graph with a vertex of weighted degree less than four admits a cyclotomic supermatrix, which is the hidden assumption in my existing algorithm. Suggestions welcome!