Semi-affine Coxeter-Dynkin graphs and G ⊆ SU2(C). Dedicated to

Abstract. The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types. Semi-Affine Graphs It is profitable to treat the so-called Coxeter-Dynkin diagrams as graphs. A classification of finite graphs with an adjacency matrix having 2 as the largest eigenvalue is made in a paper of John Smith [JHS]. It is in a combinatorial context and no reference is made to Coxeter-Dynkin diagrams there. This maximal eigenvalue property is a defining property of the affine diagrams. What is introduced in this note is a more weakly constrained graph in which we interpret the rational functions which arise in terms of a q-deformation of my correspondence [M1], [K]. Since these semi-affine graphs do not have symmetrizable matrices, this appears to imply a connection with singularities rather than Lie algebras. Here we shall deal only with those of type A, D, andE. Undirected edges are treated as a pair of edges directed in opposing directions as in [FM], [M1], [M2]. By so doing, we can introduce the semi-affine graph which may be defined in terms of a graph of finite type with an additional edge (two for A-type) directed toward the affine node; equivalentl