The adjacency matrices of graphs form a special subset of the set of all
integer symmetric matrices. The description of which graphs have all their
eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most
2) has been known for several decades. In 2007 we extended this classification
to arbitrary integer symmetric matrices.
In this paper we turn our attention to symmetrizable matrices. We classify
the connected nonsymmetric but symmetrizable matrices which have entries in
Z that are maximal with respect to having all their eigenvalues in [-2,2].
This includes a spectral characterisation of the affine and finite Dynkin
diagrams that are not simply laced (much as the graph result gives a spectral
characterisation of the simply laced ones).Comment: 20 pages, 11 figure
