# How can I prove that a sequence of squares of graph norms is never cyclotomic?

The norm of a graph is the largest eigenvalue of the adjacency matrix. I'll write ||G|| for the norm of G.

Now, fix some graph G with a chosen vertex *, and consider the family of graphs G_k obtained by adding a chain of k edges to *.

For many such examples, the sequence {||G_k||^2}_k appears to be never cyclotomic; I'd like some ideas as to how I might try to prove such statements for particular graphs G.

I know how to show individual algebraic integers aren't cyclotomic -- modulo any prime not dividing the discriminant, the minimal polynomial of a cyclotomic integer must factor into factors with uniform degree. This approach seems very hard to make work for a family of numbers, although I'm aware of the work of Asaeda-Yasuda in which they did this for the graph

        o-o-o
/
*-o-o-o
\
o-o-o


(with the exception of k=4, where the norm-square is in fact cyclotomic). If anyone has ideas about how one should attack such a question, or examples of similar problems, please let me know!

Finally -- the application here is to subfactors; Etingof-Nikshych-Ostrik proved that the index of a subfactor must be a cyclotomic integer, and the index is just the norm square of the principal graph. When we look for possible new examples of subfactors, we tend to get results constraining the principal graph to lie in such a sequence {G_k}.